Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers

CLASSICAL RECURSION THEORY The Theory of Functions and Sets of Natural Numbers STUDIES I N LOGIC AND THE F O U N D A ...

Author: Piergiorgio Odifreddi

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CLASSICAL RECURSION THEORY The Theory of Functions and Sets of Natural Numbers

STUDIES I N LOGIC AND THE F O U N D A T I O N S OF MATHEMATICS V O L U M E 125

Editors.

ELSEVIER AMSTERDAM

LALJSANNE N E W Y O R K

OXFORD SHANNON

SINGAPORE TOKYO

CLASSICAL RECURSION THEORY The Theory of Functions and Sets of Natural Numbers Piergiorgio ODIFREDDI Uiii\v?rsirj, of' Tirriti Tir rin, Itcil!

ELSEV IHR AMSTbKDAM

LAUSANNE

N E W YORK

OXF*ORD SHANNON

SINGAPORE

TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 21 I , I000 AE Amsterdam. The Netherlands

0 1989, 1992 Elsevier Science B.V. All rights reserved This work

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Fint edition I989 / Second impression 1992

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To Lidia W h a t thou lovest well remains, t h e rest is dross W h a t t h o u lov’st well shall not be reft from t h e e W h a t thou lov’st well is t h y t r u e heritage. (Pound, Cantos, LXXXI)

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Foreword Odifreddi has written a delightful yet scholarly treatise on recursion theory. Where else can one read about mezoic sets? His book constitutes his answer t o the central question of recursion theory: what is recursion theory? His answer, I am pleased t o note, is idiosyncratic. He makes numerous references t o set theory, for example Baire’s category theorem, the analytical hierarchy, the constructible hierarchy, and the axiom of determinateness. To my mind a n understanding of recursion theory, even a t the level of Turing degrees and recursively enumerable sets, is incomplete until the connection t o higher levels is made via set theory. If recursion theory is about computations, then the familiar finite case allows only a shallow view of the matter. Infinitely long computations, as in Kleene’s account of finite type objects, or as in Takeuti’s version of recursive functions of ordinals, permit a deeper insight into the nature of computation. This is borne out by t h e work of Slaman and others on fragments of arithmetic and polynomial reducibility, in which ideas from high up are applied low down. The author’s use of ‘classical’ in his title is partially meant, in this volume, t o date the material he covers. He concentrates on the early days of recursion theory. Perhaps those were the glory days. Perhaps only the early results will survive. The author makes the set theoretic connection but does not pursue it fully here. Let us hope he writes his next volumes on ‘modern’ recursion theory. His sparkling first volume proves him worthy of the task.

G.E. Sacks Harvard University and M.1.T November 1987

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Preface T h e origins of this book go back to fourteen years ago when, having done my studies in a country t hat , as Kreisel later remarked t o me, was ‘logically underdeveloped’, I thought I could learn Recursion Theory by writing it. There were at t h e time a few textbooks, prominent among t hem Kleene [1952] a n d Rogers [1967], but I was unsatisfied with them because papers I was interested in, on progressions of formal systems, seemingly required many results t h a t they did not cover. Thus I sat down, read a lot, and wrote a first version in Italian. Fortunately, I did not publish it. Meanwhile, I had gotten in touch with some recursion theorists, a n d decided I would go to t h e United States t o study some more. T h e Italian Center of Researches (C.N.R.) provided support, and in 1978 I landed in UrbanaChampaign, where t h e world opened up t o me. I found there a very sensitive a n d kind teacher, Carl Jockusch, who taught me in one pleasant year more t h a n I could have taught myself in a lifetime. And I found a friend in Dick Epstein, from whom I learned how t o write mathematics. Then, having read some of their papers, I went t o t he U.C.L.A. people for a year, and I’m afraid I tried their patience with my many questions. There I learned what I know of Set Theory an d Generalized Recursion Theory, through t h e teaching and help of Tony Martin, Yannis Moschovakis and John Steel. Back in Italy, I rewrote t h e whole book, this time in English. In t h e meantime, I had grown aware of t he fact t h a t mathematics was not t h e universal science t hat I had once thought it was: not only personal, but also social a n d historical influences shape the work of t h e researchers. More specifically, I had learned t h a t t h e Soviets were doing, in Recursion Theory, work t h a t t h e Westerners did not know much about, and they themselves were largely unaware of what people did in t h e West. I found this a n odd situation, a n d decided I would go t o t h e Soviet Union t o bridge t h e gap, at least in my knowledge. T h e Italian and Soviet State Departments provided support, and I stayed in Novosibirsk for one and a half years, in 1982-83, again learning a lot, in both mathematical and human terms. In particular, great help was provided by Marat Arslanov, Sergei Denisov, Yuri Ershov, and Victor Selivanov. Despite

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Preface

some difficulties there, which cost me a marriage among other things, I came back with more experience, and the book was now ready. A final and unexpected touch was added by Anil Nerode and Richard Shore, who invited me t o Cornell for a year in 1985, a n d in the following summers. With them I started a (for me) very fruitful collaboration, partly financed by a joint N.S.F.-C.N.R. grant. In particular, Richard Shore has relentlessly proved theorems t h a t covered blank spots in t h e book. In cornell I also met Juris Hartmanis, who changed my perspective in Complexity Theory. In addition to all t h e people mentioned above, I was greatly helped by those who have read, and commented upon, substantial parts of t h e manuscript, or have taught me different things, including Klaus Ambos-Spies, Felice Cardone, Alexander Degtev, Leo Harrington, Georg Kreisel, Georgi Kobzev, Manny Lerman, Jim Lipton, Gabriele Lolli, Flavio Previale, Mark Simpson, a n d Bob Soare. Many other people have provided various kinds of help a n d corrections, in particular those who attended classes and seminars on various parts of t h e book in Torino and Siena (Italy), Urbana, U.C.L.A. and Cornell (United States), Novosibirsk and Kazan (Soviet Union). I t would take t o o much space t o mention them all, but t o everybody go my sincerest thanks. Since Gutenberg, books have usually been written to be printed. In my case this was made possible by Solomon Feferman a n d Richard Shore, who introduced t h e book t o different editors. Michael Morley convinced me t h a t I could type it myself in UT)-$, a t a time when I did not even know how t o t u r n a computer on, and he and Anil Nerode helped afterwards with t h e machines, in many ways. While I was preparing t h e typescript, t h e amazing Bill Gasarch and Richard Shore provided overall corrections in real time. Finally, support for typesetting was provided by the C.N.R. Thanks to all of them, too. Different and very special thanks go t o Lidia. She was there before it all, saw the book taking shape, and heard about it more than anybody else. She followed me on my pilgrimages, and I could perceive how great a toll this was taking on her only when it was too late. She is not here anymore, to see t h e end of it, and this is most sad. T h e immense amount of time stolen from her and devoted t o this work is partly responsible for her absence. No doubt it was a stupid trade, b u t now, after fourteen years, here is t h e book: devoted t o her as a partial, late compensation for what she deserved, a n d I was unable t o give. Torino - Urbana - Los Angeles Novosibirsk - Ithaca 1974-1988

Preface to the Second Edit ion After its first appearance in 1989, Classical Recursion Theory has seen a second printing and a paperback edition in 1992, all sold out. With the appearance of Volume I1 in 1999, the time has come to publish a second edition of Volume I, revised and corrected of all the misprints and mistakes that have been noticed through the years. I seize this opportunity t o thank all the readers who have kindly pointed out imperfections in the text, thus helping me t o make it better. Recursion Theory has seen many exciting developments in the last decade, but most of them concern advanced areas which are treated in Volume 11. Since Volume I has been left largely untouched by these developments, I have not attempted any major rewriting of the original text. Instead, I have inserted in t h e two volumes a number of forward and backward pointers (to pages, results and chapters) t o make the whole book more interconnected, following the new Gospel of the Net. Torino, June 1999.

xi


Contents Foreword by G.E. Sacks

vii

Preface

ix

Preface to the Second Edition

xi

Introduction W h a t is ‘Classical’ . . . . . . . . . . . . . . . . . . . . W h a t is in t h e Book . . . . . . . . . . . . . . . . . . . Applications of Recursion Theory . . . . . . . . . . . . How to Use t h e Book . . . . . . . . . . . . . . . . . . Notations a n d Conventions . . . . . . . . . . . . . . . I

1 . . . . . . . .

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RECURSIVENESS AND COMPUTABILITY 1.1 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions by inductions . . . . . . . . . . . . . . . . . . . . . . Proofs by induction . . . . . . . . . . . . . . . . . . . . . . . . . Recursiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical roots of Recursion Theory * . . . . . . . . . . . . . . Formal Arithmetic * . . . . . . . . . . . . . . . . . . . . . . . . Some primitive recursive functions and predicates . . . . . . . . Codings of t h e plane . . . . . . . . . . . . . . . . . . . . . . . . Elimination of primitive recursion . . . . . . . . . . . . . . . . . 1.2 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . T h e formalism of equations . . . . . . . . . . . . . . . . . . . . Definability by systems of equations . . . . . . . . . . . . . . . Derivability from systems of equations . . . . . . . . . . . . . . A logical programming language * . . . . . . . . . . . . . . . . I .3 Arithmetical Formal Systems . . . . . . . . . . . . . . . . . Notions of representability . . . . . . . . . . . . . . . . . . . . . xiii

1 2 5 11 13

17 18 20 20 21 22 23 24 26 28 31 31 33 36 38 39 39

Contents

xiv

1.4

1.5

1.6

1.7

1.8

Formal systems representing Invariant definability * . . . Definability of functions * . Turing Machines . . . . .

t h e recursive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Variations of t he Turing machine model . . . . . Physical Turing machines * . . . . . . . . . . . . . Finite automata * . . . . . . . . . . . . . . . . . . Turing machine computability . . . . . . . . . . . Machinedependent programming languages * . . Flowcharts . . . . . . . . . . . . . . . . . . . . . . Unstructured programming languages * . . . . . Unlimited register, random access machines * . . Flowchart computability . . . . . . . . . . . . . . . Structured programming languages * . . . . . . . P r o g r a m for primitive Petri nets * . . . . . . Functions as Rules A-calculus . . . . . . .

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recursion * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other formulations of t h e A-calculus * . . . . . . A-definability . . . . . . . . . . . . . . . . . . . . . Functional programming languages * . . . . . . . Arithmetization . . . . . . . . . . . . . . . . . . Historical remarks * . . . . . . . . . . . . . . . . . Numerical tools for arithmetization . . . . . . . . T h e Normal Form Theorem . . . . . . . . . . . . .

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Equivalence of t h e various approaches t o recursiveness . . . . . T h e basic result of t he foundations of Recursion Theory . . . . Church’s Thesis * . . . . . . . . . . . . . . . . . . . . . . . . Introduction t o Church’s Thesis . . . . . . . . . . . . . . . . . . Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . Computers and physics . . . . . . . . . . . . . . . . . . . . . . . Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic physics . . . . . . . . . . . . . . . . . . . . . . . . computers and thought . . . . . . . . . . . . . . . . . . . . . . T h e brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructivism . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 44 45 46 49 51 52 53 59 61 63 64 65 68 70 74 75 76 82 83 86 87 87 88 90 97 100 101 102 105 106 107 109 113 115 118 122

Contents I1 BASIC RECURSION THEORY 11.1 Partial Recursive Functions . . . . . . . . . . . The notion of partial function . . . . . . . . . . . . Partial recursive functions . . . . . . . . . . . . . . Universal Turing machines and computers * . . . .

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....... . . . . . . . . . . . . . .

Recursively enumerable sets . . . . . . . . . . . . . . . . . . . . R.e. sets as foundation of Recursion Theory * . . . . . . . . . . A programming language based on r.e. sets * . . . . . . . . . .

11.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . The essence of diagonalization . . . . . . . . . . . . . . . . . . . Recursive undecidability results . . . . . . . . . . . . . . . . . . Limitations of mechanisms * . . . . . . . . . . . . . . . . . . . . Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . Limitations of formalism * . . . . . . . . . . . . . . . . . . . . . Self-reference * . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-reproduction and cellular automata * . . . . . . . . . . . . 11.3 Partial Recursive Functionals . . . . . . . . . . . . . . . . . Oracle computations and Turing degrees . . . . . . . . . . . . . The notion of functional . . . . . . . . . . . . . . . . . . . . . . Partial recursive functionals . . . . . . . . . . . . . . . . . . . . First Recursion Theorem . . . . . . . . . . . . . . . . . . . . . . Recursive programs * . . . . . . . . . . . . . . . . . . . . . . . Topological digression . . . . . . . . . . . . . . . . . . . . . . . Iteration and fixed-points * . . . . . . . . . . . . . . . . . . . . Models of A-calculus (part I) * . . . . . . . . . . . . . . . . . . Different notions of recursive functionals * . . . . . . . . . . . . Higher Types Recursion Theory * . . . . . . . . . . . . . . . . . Computability on abstract structures * . . . . . . . . . . . . . . 11.4 Effective Operations . . . . . . . . . . . . . . . . . . . . . . Effective operations on partial recursive functions . . . . . . . . Effective operations on total recursive functions . . . . . . . . . Effective operations in general * . . . . . . . . . . . . . . . . . . Recursive analysis * . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Indices and Enumerations * . . . . . . . . . . . . . . . . . . Acceptable systems of indices . . . . . . . . . . . . . . . . . . . Axiomatic Recursion Theory * . . . . . . . . . . . . . . . . . . Models of A-calculus (part 11) * . . . . . . . . . . . . . . . . . . Indices for recursive and finite sets . . . . . . . . . . . . . . . . Enumerations of classes of r.e. sets . . . . . . . . . . . . . . . . The Theory of Enumerations * . . . . . . . . . . . . . . . . . . 11.6 Retraceable and Regressive Sets * . . . . . . . . . . . . . . Retraceable versus recursive . . . . . . . . . . . . . . . . . . . .

125 126 127 127 132 134 143 144 145 145 146 149 152 158 165 170 174 175 177 178 181 185 186 192 194 196 199 202 205 205 208 210 213 214 215 221 223 225 228 236 238 239

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Contents Regressive versus r.e. . . . . . . . . . . . . . . . . . . . . . . . . Existence theorems and nondeficiency sets . . . . . . . . . . . . Regressive versus retraceable . . . . . . . . . . . . . . . . . . .

IIIPOST’S PROBLEM AND STRONG REDUCIBILITIES 111.1 Post’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . Origins of Post’s Problem * . . . . . . . . . . . . . . . . . . . . Turing reducibility on r.e. sets . . . . . . . . . . . . . . . . . . 111.2 Simple Sets and Many-One Degrees . . . . . . . . . . . Many-one degrees . . . . . . . . . . . . . . . . . . . . . . . . . . Simple sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effectively simple sets * . . . . . . . . . . . . . . . . . . . . . . 111.3 Hypersimple Sets and lkuth-Table Degrees . . . . . . . Truth-table degrees . . . . . . . . . . . . . . . . . . . . . . . . . Hypersimple sets . . . . . . . . . . . . . . . . . . . . . . . . . . The permitting method * . . . . . . . . . . . . . . . . . . . . . 111.4 Hyperhypersimple Sets and Q-Degrees . . . . . . . . . Q-reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperhypersimple sets . . . . . . . . . . . . . . . . . . . . . . .

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Maximal sets * . . . . . . . . . . . . . . . . . . . . . . . . . . .

111.5 A Solution to Post’s Problem . . . . . . . . . . . . . . . . . Semirecursive sets . . . . . . . . . . . . . . . . . . . . . . . . . 7-hyperhypersimple sets . . . . . . . . . . . . . . . . . . . . . . 111.6 Creative Sets and Completeness . . . . . . . . . . . . . . . Effectively nonrecursive sets . . . . . . . . . . . . . . . . . . . . Creative sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasicreative sets * . . . . . . . . . . . . . . . . . . . . . . . . Subcreative sets * . . . . . . . . . . . . . . . . . . . . . . . . . Effectively inseparable pairs of r.e. sets . . . . . . . . . . . . . . 111.7 Recursive Isomorphism Types . . . . . . . . . . . . . . . . Mezoic sets and l-degrees . . . . . . . . . . . . . . . . . . . . . Recursive isomorphism types . . . . . . . . . . . . . . . . . . . Recursive equivalence types and isols * . . . . . . . . . . . . . . 111.8 Variations of Truth-Table Reducibility * . . . . . . . . . . Bounded truth-table degrees . . . . . . . . . . . . . . . . . . . . Weak truth-table degrees . . . . . . . . . . . . . . . . . . . . . Other notions of reducibility * . . . . . . . . . . . . . . . . . . 111.9 The World of Complete Sets * . . . . . . . . . . . . . . . . Relationships among completeness notions . . . . . . . . . . . . Structural properties and completeness . . . . . . . . . . . . . . III.10Formal Systems and R.E. Sets * . . . . . . . . . . . . . . Formal systems and r.e. sets * . . . . . . . . . . . . . . . . . . .

242 245 248

251 252 253 254 256 257 259 263 267 269 272 277 280 281 282 288 294 294 299 304 304 306 311 314 316 319 320 324 328 330 331 337 339 341 341 348 349 350

Contents

xvii

Undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . Essential undecidability . . . . . . . . . . . . . . . . . . . . . . Independent axiomatizability . . . . . . . . . . . . . . . . . . .

352 353 357

IV HIERARCHIES AND WEAK REDUCIBILITIES IV.l The Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . T h e definition of t r u t h * . . . . . . . . . . . . . . . . . . . . . . T r u t h in First-Order Arithmetic . . . . . . . . . . . . . . . . . T h e Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Arithmetical Hierarchy . . . . . . . . . . . . . A; sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativizations * . . . . . . . . . . . . . . . . . . . . . . . . . . IV.2 The Analytical Hierarchy . . . . . . . . . . . . . . . . . . . Truth in Second-Order Arithmetic . . . . . . . . . . . . . . . . T h e Analytical Hierarchy . . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Analytical Hierarchy . . . . . . . . . . . . . . II: sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 363 363 363 365

A: sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 392 394

Descriptive Set Theory * . . . . . . . . . . . . . . . . . . . . . . Relativizations * . . . . . . . . . . . . . . . . . . . . . . . . . .

Post's Theorem in t h e Analytical Hierarchy * . . . IV.3 The Set-Theoretical Hierarchy . . . . . . . . . Truth in Set Theory . . . . . . . . . . . . . . . . . . Standard structures . . . . . . . . . . . . . . . . . . T h e Set-Theoretical Hierarchy . . . . . . . . . . . . AFKp functions . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Set-Theoretical Hierarchy . . . . W F a n d t h e Arithmetical Hierarchy . . . . . . . .

367 373 374 375 376 377 380 381

. . . . . . .

395 . . . . . . . 397 . . . . . . 397 . . . . . . 401 . . . . . . . 405 . . . . . . 406 . . . . . . . 411 . . . . . . . 414

Absoluteness a n d t h e Analytical Hierarchy . . . . . . . . . . . . 418 Admissible sets * . . . . . . . . . . . . . . . . . . . . . . . . . . 421 IV.4 The Constructible Hierarchy . . . . . . . . . . . . . . . . T h e Constructible Hierarchy . . . . . . . . . . . . . . . . . . . . T h e levels of t h e Constructible Hierarchy . . . . . . . . . . . T h e structure of L . . . . . . . . . . . . . . . . . . . . . . . . . Constructible sets of natural numbers . . . . . . . . . . . . . Cksets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1C a n d t h e Analytical Hierarchy . . . . . . . . . . . . . . . . Recursion Theory on t h e ordinals * . . . . . . . . . . . . . . . Relativizations * . . . . . . . . . . . . . . . . . . . . . . . . . .

. 422 422 . 424 425 . 432 437

. 441 . 443 444

xviii

Contents

V TURING DEGREES V . l The Language of Degree Theory . . . . . . . . . . . . . . . T h e join operator . . . . . . . . . . . . . . . . . . . . . . . . . . T h e jump operator . . . . . . . . . . . . . . . . . . . . . . . . . First properties of degrees . . . . . . . . . . . . . . . . . . . . . T h e Axiom of Determinacy * . . . . . . . . . . . . . . . . . . . V.2 The Finite Extension Method . . . . . . . . . . . . . . . . Incomparable degrees . . . . . . . . . . . . . . . . . . . . . . . Embeddability results . . . . . . . . . . . . . . . . . . . . . . . T h e splitting method . . . . . . . . . . . . . . . . . . . . . . . . Forcing t h e jump . . . . . . . . . . . . . . . . . . . . . . . . . . V.3 Babe Category * . . . . . . . . . . . . . . . . . . . . . . . . . Topologies on total functions . . . . . . . . . . . . . . . . . . . Comeager sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bake Category a n d Degree Theory . . . . . . . . . . . . . . . . Meager sets of degrees . . . . . . . . . . . . . . . . . . . . . . . Measure Theory and Degree Theory * . . . . . . . . . . . . . . V.4 The Coinfinite Extension Method . . . . . . . . . . . . . . Exact pairs and ideals . . . . . . . . . . . . . . . . . . . . . . . Greatest lower bounds and least upper bounds . . . . . . . . . Extensions of embeddings . . . . . . . . . . . . . . . . . . . . . V.5 The n e e Method . . . . . . . . . . . . . . . . . . . . . . . . Hyperimmune-free degrees . . . . . . . . . . . . . . . . . . . . . Minimal degrees . . . . . . . . . . . . . . . . . . . . . . . . . . Minimal upper bounds * . . . . . . . . . . . . . . . . . . . . . . Konig's Lemma and @ classes * . . . . . . . . . . . . . . . . . Complete extensions of Peano Arithmetic * . . . . . . . . . . .

V.6 Initial Segments * . . . . . . . . . . . . . . . . . . . . . . . . Uniform trees . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447 448 448 450 451 453 456 457 459 463 467 471 472 473 477 481 484 484 485 488 490 493 495 498 502 505 510 516 516

Minimal degrees by recursive coinfnite extensions . . . . . . . . 520 T h e threeelement chain . . . . . . . . . . . . . . . . . . . . . . 523 T h e initial segments of the degrees * . . . . . . . . . . . . . . . 528 530 V.7 Global Properties . . . . . . . . . . . . . . . . . . . . . . . . Definability from parameters . . . . . . . . . . . . . . . . . . . 530 T h e complexity of t h e theory of degrees . . . . . . . . . . . . . 536 Absolute definability . . . . . . . . . . . . . . . . . . . . . . . . 540 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 V.8 Degree Theory with Jump * . . . . . . . . . . . . . . . . . 550

Contents VI MANY-ONE AND OTHER DEGREES VI.l Distributivity . . . . . . . . . . . . . . . . . . . . . . . . . . .

XiX

555

555

Distributive upperseniilattices . . . . . . . . . . . . . . . . . . . 556 Ideals of distributive uppersemilat. tices . . . . . . . . . . . . . . 558

V1.2 Countable Initial Segments . . . . . . . . . . . . . . . . . . Finite initial segments . . . . . . . . . . . . . . . . . . . . . . . Countable initial segments . . . . . . . . . . . . . . . . . . . . . VI.3 Uncountable Initial Segments . . . . . . . . . . . . . . . . Strong minimal covers . . . . . . . . . . . . . . . . . . . . . . . Uncountable linear orderings . . . . . . . . . . . . . . . . . . . Uncountable initial segments . . . . . . . . . . . . . . . . . . . VI.4 Global Properties . . . . . . . . . . . . . . . . . . . . . . . . Characterization of t h e structure of many-one degrees . . . . . Definability, homogeneity, a n d automorphisms . . . . . . . . . . T h e complexity of t h e theory of many-one degrees . . . . . . . VI.5 Comparison of Degree Theories * . . . . . . . . . . . . . . 1-degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truth-table degrees a n d weak truth-table degrees . . . . . . . . Elementary inequivalences . . . . . . . . . . . . . . . . . . . . . VI.6 Structure Inside Degrees * . . . . . . . . . . . . . . . . . . Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inside many-one degrees . . . . . . . . . . . . . . . . . . . . . . Inside truth-table degrees . . . . . . . . . . . . . . . . . . . . . Inside Turing degrees . . . . . . . . . . . . . . . . . . . . . . . .

561 562 566 569 569 570 571 574 575 575 577 582 582 584 590 591 591 594 598 600

Bibliography

603

Notation Index

643

Subject Index

649


Introduction Classical Recursion Theory is the study of real numbers or, equivalently, functions over the natural numbers. As such it has a long history, a n d a number of notions a n d results t h a t were originally proved in different fields a n d for different purposes are incorporated, unified a n d extended in a systematic study. We are thinking here, for example, of t h e different equivalent definitions of real number, of Cantor’s theorem t h a t the real numbers a r e uncountable, of Gadel’s class of constructible real numbers, and so 011. All of these are now part of Recursion Theory and of our study, but t h e theory also provides new tools of its own, the origins of which can be traced back to Dedekind [1888]: he introduced t h e study of functions definable over t h e set w of t h e natural numbers by recurrence using the well-ordered structure of w , whence t h e name Recursion Theory. T h e power of recursion as a tool for defining functions was analyzed in detail by Skolem [1923], Peter [1934], and Hilbert a n d Bernays [1934], but its limitations were also pointed out. Gradually the collective work of Post [1922], Church [1933], Godel [1934], Kleene [1936], and Turing [1936], led to t h e identification of t h e most general form of the recursion principle and to what we now call recursive functions. In a bold philosophical abstraction Church [1936] proposed t o identify the notion of ‘effectively computable function’ of natural numbers with t h a t of recursive function, thus providing a feeling of absoluteness to t h e notion. With Post [I9441 Recursion Theory became a n independent branch of mathematics, studied for its own sake.

What is ‘Classical’ In more recent decades Recursion Theory has been generalized in various ways t o different domains: ordinals bigger than w ,functionals of higher order, abstract sets. All these subjects belong t o what we call Generalized Recursion Theory. We use the word ‘classical’t o emphasize the fact t h a t we confine our treatment t o t h e original setting, and we will deal with notions of Generalized 1

2

Introduction

Rccursion Theory only when t h e theory provides results for t h e case we a r e interested in. If we see classical mathematics as t h e s t u d y of concrete structures, like t h e set of natural numbers in Number Theory, or t h e set of functions over t h e real or complex numbers in Analysis (as opposed to modern mathematics, where t h e emphasis is on abstract structures, like algebraic or topological ones), t h e n Classical Recursion Theory is p a r t of classical mathematics, a n d s i t s between Number Theory a n d Analysis. T h i s provides another reason for t h e word ‘classical’ in our subject. Mathematics is usually formalized in well-established systems of Set Theory such as Z F C ( t h e Zermelo-Fraenkel system, together with t h e Axiom of Choice). Our final use of t h e word ‘classical’ emphasizes t h e fact t h a t we will be working mostly in Z F C . I t is not surprising, d u e to t h e well-known independence results of Gijdel [1938] a n d Cohen [1963], t h a t only a p a r t of t h e s t u d y of real numbers can b e carried o u t in Z F C a n d we will point o u t t h e limits of our approach, together with possible extensions of Z F C suitable for Recursion Theory, at t h e e n d of t h e book.

What is in the Book T h e basic methods of analysis of t h e real numbers t h a t we a r e going to use a r e two:

Hierarchies. A hierarchy is a stratification of a class of reals built from below, starting from a subclass t h a t is taken as primitive (either because well understood, or because already previously analyzed), a n d obtained by iteration of a n operation of class construction. Degrees. Degrees a r e equivalence classes of reals under given equivalence relations, t h a t identify reals with similar properties. O n c e a class of reals has been studied a n d understood, degrees a r e usually defined b y identifying reals t h a t look t h e s a m e from t h a t class point of view. Degrees were used for t h e purpose of a classification of r e d s already in Euclid’s Book X (w.r.t. a geometrical equivalence relation, between rational a n d algebraic dependence). See Knorr [1983] for a survey. As might be imagined t h e two methods a r e complementary: first a class is analyzed in terms of intrinsic properties, for example by appropriately stratifying it in hierarchies, a n d t h e n t h e whole structure of real numbers is studied modulo t h a t analysis with t h e appropriate notion of degrees induced by t h e given class. T h e two methods also have a different flavor: t h e first is essentially definitional, t h e second essentially computational.

Introduction

3

To give t h e reader a n idea of what (s)he will find in the book we outline its bare skeleton, referring t o the introductions of the various chapters for more detailed outlines. T h e starting point of our study is t h e class of recursive functions i n t r e duced in Chapter I. T h e idea of its definition is simple: we t r y to isolate t h e functions over w t h a t are 'computable' in ways appealing both to t h e mathematician a n d t o t h e computer scientist. Having many different approaches available, and various different intuitions of t h e notion of computability, we t r y them all, a n d discover t h a t they all produce, once appropriately formalized, t h e same class of functions (and sets, through characteristic functions). Chapter I1 considers two fundamental generalizations of t h e notion of recursiveness. Partial recursive functions are the natural formalization of algorithms: these, in t h e common use of t h e term, do not necessarily define total functions but only provide for specifications t h a t allow t h e computation of values if particular conditions are satisfied. Partial recursive functionals take care of a different aspect of computations, namely t h e interactive procedure according t o which a machine can be piloted, in its behavior, by a human agent. This can be formalized by the use of oracles t h a t help t h e computation when requested by the machine. A set is recursive if membership in it is effectively computable. T h e next level of complexity is reached when a set is effectively generated. I n this case membership still can be effectively determined by waiting long enough in t h e generation of t h e set until the given element appears, but nonmembership r e quires waiting forever, a n d thus does not have effective content. Such sets are called recursively enumerable, and are the subject of Chapter 111. But t h e emphasis of t h e study here is on the relative difficulty of computation. In other words, we identify sets which are equally difficult t o compute. Then we attack t h e problem of whether t h e only relevant distinction among recursively enumerable sets, from a computational point of view, is between recursive a n d nonrecursive. T h e answer is t h a t the world of recursively enumerable sets is a variegated one, in which different nonrecursive effectively generated sets may have different computational difficulty. Chapter IV introduces t h e first hierarchies, by building on t h e fact t h a t t h e recursively enumerable sets are exactly those definable in t h e language of FirstOrder Arithmetic with exactly one existential quantifier (coding t h e fact t h a t a n element is in a given recursively enumerable set if a n d only if there is a stage of t h e enumeration in which it appears). A natural hierarchy is thus obtained by looking at t h e arithmetical sets as those sets which are definable in FirstOrder Arithmetic, counting the number of alternations of quantifiers. Other hierarchies in t h e same vein are possible: counting alternations of function quantifiers in Second-Order Arithmetic which stratifies the analytical sets; or measuring t h e complexity of t h e definition of a set of natural numbers in

4

Introduction

t h e language of Set Theory in terms of previously defined sets which defines t h e constructible sets of integers. Hierarchies are, by their nature, only partial tools of analysis. T h e notion of degree is instead a global one, classifying all sets modulo some equivalence relation. Chapters V and VI study t he structure of t h e continuum with respect t o two notions of relative computability, Turing degrees a n d m-degrees, a n d obtain two structural results. T h e first equates t h e complexities of t h e decision problem for t h e theories of Turing and m-degrees with t h a t of SecondOrder Arithmetic, t he second gives a complete algebraic characterization of t h e continuum in terms of t h e structure of m-degrees. T h e Baire Category method, in both its original version and generalized forms, is t h e basic method of proof. This completes Volume I, which introduces t h e fundamental notions a n d methods. Volumes I1 a n d I11 are a deeper and more sophisticated study of t h e same topics, in which t he structures already introduced are revisited a n d analyzed more carefully and thoroughly. Volume I1 deals with sets of t h e arithmetical hierarchy, Volume 111 with t he rest. Chapters VII and VIII resume t he analysis of t h e fundamental objects in Recursion Theory, t he recursive sets and functions, a n d provide a microscopic picture of them. We start in Chapter VII with a n abstract st udy of t h e complexity of computation of recursive functions. T hen in Chapter VIII we will attempt t o build from below t h e world of recursive sets a n d functions t h a t was previously introduced in just one go. A number of subclasses of interest from a computational point of view are introduced a n d discussed, among them: th e polynomial time (or space) computable functions which provide an upper bound for t h e class of feasibly computable functions (as opposed t o th e abstractly computable ones); t h e elementary functions, which are t h e smallest known class of functions closed under time (deterministic or not) and space computations; t h e primitive recursive functions, which are those computable by t h e ‘for’ instruction of programming languages like PASCAL, i.e. with a preassigned number of iterations (as opposed to t h e recursive functions, computable by t h e ‘while’ instruction, which permits a n unlimited number of iterations). Chapters IX and X return to t h e treatment of recursively enumerable sets. In Chapter 111 a good deal of information on their structure had been gathered, but here a systematic study of t h e structures of both t h e lattice of recursively enumerable sets and of t h e partial ordering of recursively enumerable degrees is undertaken. Special tools for their treatment are introduced, most prominent among them being t he priority method, a constructive variation of t h e Baire Category method. Chapter XI deals with limit sets, also known as A! sets, which are limits of recursive functions. They are a natural formalization of t h e notion of sets for which membership can be determined by effective trials a n d errors, unlike

Introduction

5

recursive sets (for which membership can be effectively determined), and recursively enumerable sets (for which membership can be determined with at most one mistake, by first guessing that a n element is not in the set, and then changing opinion if it shows up during the generation of the set). T h e following chapters produce an analysis of the sets introduced, and only touched upon, in Chapter IV, in particular arithmetical, hyperarithmetical, A;, and constructible sets, and various other classes. In all these chapters the study proceeds by first analyzing the classes themselves, and then looking at t h e notions of degree associated with them (respectively: arithmetical degrees, hyperdegrees, A;-degrees, constructibility degrees, as well as degrees with respect to appropriate admissible ordinals). T h e final chapter deals with nonclassical set-theoretical worlds in order t o point out t h e limitations of the classical approach, t o exactly establish its limits, and t o reach beyond it by adding appropriate axioms (prominent among them the Axiom of Projective Determinacy). Starred subsections deal with topics related t o t h e ones at hand thought sometimes quite far away from the immediate concern. They provide those connections of Recursion Theory t o the rest of mathematics and computer science which make our subject part of a more articulate and vast scientific experience. Limitations of our knowledge and expertise in these fields make our treatment of t h e connections rather limited, but we feel they add important motivation and direct the reader to more detailed references. Particular themes on which continuous commentary is made throughout t h e book are relationships with computers, logic, and t h e theory of f o r m a l systems, in particular the results known as Gadel’s Theorems. As our development becomes more technical, connections t o fields outside logic in general, a n d other branches of Recursion Theory in particular, become less important. As will be clear by now, we have opted for breadth rather than depth, a n d have provided rudiments of many branches of Classical Recursion Theory, rather than complete and detailed expositions of a small number of topics. In this respect our book is in t h e tradition of Kleene [1952] and Rogers [1967], and differs from recent texts like Hinman [1978], Epstein [1979],Moschovakis [1980], Lerman [1983], and Soare [1987], which can be used as useful complements and advanced textbooks in their specialized areas.

Applications of Recursion Theory No sound mathematical theory is self-contained or detached from t h e rest of mathematics or science. It takes inspiration from, and provides matter of reflection to other branches of knowledge. Recursion Theory is no exception and, despite this being a book on the pure theory, we will touch on applications

6

Introduction

a n d connections whenever possible. Here we give a n idea of t h e applications t h a t our subject can have in other branches of science some of which will be taken u p again in more detail in t h e book.

Philosophy If one of t h e main goals of Philosophy of Science is t h e conceptual analysis of epistemological notions, t h e n t h e foundations of Recursion T h e o r y provide sonie astounding successes for it. O n e of t h e original concerns of Recursion Theory had been t h e analysis of t h e notion of effective computability a n d of t h e related concept of algorithm. T h e isolation of t h e technical notion of recursiveness as a formal proposal intended to capture t h e essence of computability on natural numbers (see Chapter I) is a first success of t h e philosophical side of t h e theory, b u t by no means t h e only one. After all, computability o n natural numbers is j u s t one p a r t of t h e whole story. A great deal of work has been spent on axiomatizing t h e a b s t r a c t notion of computability (see p. 22l), a n d on analyzing t h e role of t h e special properties of natural numbers in computations. Decent notions of elementary computability have been proposed for abstract domains (see p. 202), a n d deeper properties have been shown to extend to a variety of domains more general t h a n w (such as c?dmissible ordinals, see p. 443). T h i s has required a n analysis of t h e role of finiteness in computations, a n d a n isolation of its essential properties. T h e familiarity of t h e notion involved, which is usually used unconsciously, magnifies t h e success obtained. T h e concern of Recursion Theory with predicativity predates even i t s concern with computability (see p. 22), a n d it is reflected in i t s widespread use of hierarchies as a mean of building classes of functions from below. O n e of these hierarchies ( t h e hyperarithmetical, see p. 391) has t u r n e d o u t to be particularly interesting a n d to provide for a n upper bound to t h e notion of A predicatively defined set of natural number (Kreisel [1960]). Related work h a s sulisequently been able to isolate a precise analogue of this notion (Feferman [1964]), t h u s doubling t h e success obtained with computability,

Computer Science T h e area of Recursion Theory t h a t deals with recursiveness is p a r t of Theoretical Computer Science. Turing’s analysis of computability i n t e r m s of machines provided t h e conceptual basis for t h e construction of phystcal computers in t h e late Forties: in t h e United States through Von Neumann, who knew Turing’s work, a n d in t h e United Kingdom through Turing himself ( s e e p . 132). Different approaches to recursiveness generate different types of programming languages, a n d we discuss (Chapter I a n d Section 11.1) how t h e computational core of

Introduction

7

PASCAL, LISP, PROLOG, and SNOBOL can easily be obtained from t h e appropriate versions of recursiveness. Finally, a good deal of Recursion Theory is devoted t o t h e analysis of t he complexity of algorithms and t o a classification of recursive functions according t o t h e tools needed t o compute them. This is rapidly becoming a field of its own, called Complexity Theory, with methods a n d results strongly influenced by other parts of Recursion Theory (see Chapters VII a n d VIII).

Number Theory T h e very origins of Recursion Theory place it close t o Number Theory: t h e m G tivation of Dedekind [1888] was t he analysis of t h e concept of natural number (see p. 22), while Skolem [1923] wanted t o present a formulation of Arithmetic t h a t avoided t h e difficulties of t h e common solutions t o t h e paradoxes. But perhaps t h e most striking application of Recursion Theory t o Number Theory is t h e solution of Hilbert’s Tenth Problem (see p. 135) which asked for a d e cision procedure t o determine the existence of solutions of given diophantine equations. Matiyasevitch [1970] proved a representation theorem, showing t h a t t h e sets of (non-negative) solutions of diophantine equations are exactly t h e re cursively enumerable sets. A negative solution t o Hilbert’s Tenth Problem then follows from t h e existence of a recursively enumerable, nonrecursive set.

Algebra Until t h e second half of t h e last century, including t he work of Lagrange, Gauss, Abel, and Galois, algebra had been developed in a strictly constructive way. T h e dichotomy between constructive and nonconstructive methods arose with t h e notion of prime ideal, which both Kronecker and Dedekind discovered from th e usual constructive approach, but which Dedekind published in t h e now coninion set-theoretical framework. After t hat , nonconstructive methods which may produce less informative b u t more easily graspable arguments have become standard (see Metakides and Nerode [1982] for more historical background). Recursion Theory makes t h e analysis of t h e constructive content of classical results possible, as t h e following typical case illustrates. Steinitz Theorem shows t h a t a field has a n algebraic closure which is unique u p to isomorphism. I ts original proof does not constructivize: this is an accident for t h e existence part, but necessary for t he uniqueness. T h e former follows from Rabin [1960] who, using a different existence proof, showed t hat a recursively presented field (i.e. a field with recursive set of elements and field operations, including equality) always has a recursively presented algebraic closure. T h e latter comes from Metakides and Nerode [1979], who showed t hat uniqueness

8

Introduction

(up to recursive isomorphism) of the recursively presented algebraic closure is equivalent to t h e existence of a splitting algorithm (to determine whether a polynomial is irreducible or not), a result t h a t uses t h e priority method introduced in Chapter X. T h e analysis of t h e effective content of classical algebra has been thoroughly pursued: see Ershov [1980], Crossley [1981], Nerode a n d Remmel [1985] for references. T h e usefulness of Recursion Theory in the analysis of constructivity in algebra is plausible. But there are unexpected uses too, such as in Higman (19611 who shows t h a t t h e finitely generated groups embeddable in a finitely presented group are exactly t h e recursively presented ones (i.e. those for which t h e set of words equal t o 1 is recursively enumerable), thus linking a purely algebraic notion with t h e notion of recursiveness. Higman’s representation theorem easily implies t h e undecidability of t h e word problem (to determine whether two words are equal) for finitely presented groups, proposed by Dehn in 1911 and solved by Novikov (19541 a n d Boone [1959]. T h e undecidability of the easier word problem for semigroups, proposed by Thue [1914] and solved by Post [1944] and Markov [1947], is historically important, being t h e first undecidability result of a problem from classical mathematics. These results started a whole area of research, devoted to t h e determination of which properties of algebraic structures a r e (un)decidable. See Tarski, Mostowski a n d Robinson [1953], Ershov, Lavrov, Taimanov a n d Taislin [1965] and Ershov [1980] for detailed treatments a n d references.

Analysis Bore1 [1912] introduced t h e notion of computable real number, using t h e intuitive notion of computability. T h e very paper in which Turing introduced his influential approach to computability was motivated by t h e search for a formal definition of computable reals, and was thus t h e beginning of recursive analysis. Turing isolated a class of recursive reds t h a t is independent of t h e proposed constructivization (in the sense t h a t all classically equivalent definitions of real number remain equivalent when appropriately constructivized), contains all commonly used reals, and is algebraically closed. Subsequent work extending the notion of recursive functional (Section 11.4) defined t h e notion of a recursive function of a real variable as a function defined on all reals, not only on the recursive ones. This provided the needed tools t o analyze t h e effective content of analysis: a result is constructive if whenever it has recursive data it provides us with recursive solutions. As a typical example, Weierstrass proof of t h e existence of a maximum for a continuous real function on a closed interval is constructive; b u t a n argument at which t h e maximum is attained cannot be constructively found

Introduction

9

(Lacombe [1957],Specker [1959]). Another example is provided by the ordinary differential equation y' = f ( z , y ) : the original proof of Picard that i f f satisfies a Lipschitz condition t h e solution exists and is unique is constructive, but Aberth [1971] and Pour El and Richards [1979] showed that even t h e existence alone is not constructive if f is only uniformly continuous. See p. 213 for more on the subject.

As for algebra, one can look for undecidability results as well, some of which have been obtained by Richardson [1968], Adler [1969] and Wang [1974]. As an example, the latter proves that there is no recursive procedure t o decide whether a real elementary function has a zero.

Set Theory Recursion Theory and Set Theory have a large overlap in the study of sets of integers of high complexity: the material dealt with in Volume I11 could hardly be classified as solely belonging to one of them; it is rather a new field sprung from their marriage. But Recursion Theory does have successful applications t o pure Set Theory in areas were the latter seems to be classically impotent. The best-developed applications have been two theories about cardinals: recursive equivalence types, and admissible ordinals. The former deals with sets that, in a constructive sense, are infinite but Dedekind-finite, i.e. can be one-one mapped neither t o a proper initial segment of w, nor t o a proper subset of themselves. Classically such sets do not exist in the presence of the Axiom of Choice, but their recursive versions have generated a rich theory that provides new insights into the notion of finiteness (see p. 328). Another branch of Set Theory which is classically unmanageable is the theory of large cardinals: even the inaccessible ones, the smallest proposed type, cannot be proved to exist i n classical Set Theory. The lack of examples different from w forces one to resort to trivial cases, such as considering 1 as weakly but not strongly inaccessible because ' 0 = 1 (Godel [1964]). Recursion Thcory provides a well-developed analogue of the theory of large cardinals, in which the role of the first regular cardinal is taken by the first ordinal which is not the order type o f a recursive well ordering of w (see p. 385). The notion of adtnissible ordinal (11. 443) takes care of the analogue of regular cardinal i n general as an ordinal closed under recursive operations on ordinals, and analogues of a great variety of large cardinals can already be seeii to exist among the countable ordinals. The existence of analogues of Ramsey cardinals can be disproved which might prompt some reflection on the role of very large cardinals in Set Theory (see Volume I11 for details).

10

Introduction

Descriptive Set Theory Cantor's Set Theory, a n d in particular t h e unlimited use of t h e power set, provoked various reactions at t h e turn of t h e century, one of which produced Descriptive Set Theory as a study of larger a n d larger classes of sets of reals which were explicitly defined (see p. 392). This approach, in which hierarchies a r e one of t h e main tools, is obviously a forerunner a n d a n analogue of various recursion theoretical hierarchies (see Chapter IV), t h e main difference being one level of complexity: sets of reals a r e considered in t h e first case, sets of integers in t h e second. B u t Addison [1954], [1959] discovered t h a t not only a r e there analogies: t h e full classical theory can be obtained by relativization of t h e recursive hierarchy theory by substituting continuous functions a n d open sets for recursive functions a n d recursively enumerable sets (see p . 392). T h i s implies t h a t all classical theorems have recursive versions of which they a r e consequences (but not conversely). This allows a unified approach, with recursion theoretical methods applicable to t h e classical case, a n d t h e theory has been resurrected from t h e state of lethargy in which it had fallen in t h e Forties.

Constructive Mat hemat ics T h e use of constructivism in classical mathematical theories is conservative: nonconstructive methods a r e accepted, a n d t h e issue is only whether given proofs a r e constructive as they stand, or can be replaced by constructive ones, a negative answer being interesting a n d acceptable. B u t constructivism can be taken more seriously as a philosophy of mathematics t h a t would simply banish nonconstructive notions a n d proofs from practice. One possible approach to constructive mathematics consists of using t h e notion of recursiveness as a substitute for t h e notion of constructivity. This can be taken literally, as in Markov's school (see p. 214), which considers only those mathematical objects a n d operations on them t h a t can be effectively described by recursive p r o c o dures as existing. But it can also be taken as a tool of analysis to compare different approaches. For example, in Kolmogorov [1932] intuitionism is seen as a logic of problems: a V P means to solve one of cr a n d p, cr --+ p to reduce t h e problem of solving /3 to t h a t of solving a , 3za(z) to solve a(.) for some z, a n d so on. Kleene [ 19451 then introduced t h e notion of recursive realizability for Intuitionistic Number Theory: numbers realize formulas if they code, inductively, recursive procedures t h a t prove t h e formula according to t h e constructive meaning of t h e logical operations. Realizability has been extended t o Intuitionistic Set Theory by Kreisel a n d 'Ikoelstra [1970] and, even if not accepted as t h e only possible way of interpreting intuitionistic provability, it has become a common tool of analysis since it provides for constructive models of theories. See Troelstra

Introduction

11

[1973] and Beeson [1985] for detailed treatments of t h e subject.

Logic After t h e first fifty years in which Recursion Theory was mainly motivated by mathematical problems about Arithmetic, t h e logicians took over. Their main interest was still in Arithmetic, but their point of view was metamathematical. In their hands t h e theory obtained its most astonishing and revolutionary results which a r e also the best known applications of t h e subject and one of t h e main impulses to its growth. By a balanced use of two of t h e most fundamental methods of proof of Recursion Theory, arithmetization and diagonalization, a complete characterization of the expressiveness of formal systems was obtained, t h e result being that (as in the case of diophantine equations) exactly t h e recursively enumerable sets are (weakly) representable in any consistent formal system having a minimal arithmetical strength. T h e existence of a recursively enumerable, nonrecursive set then implies t h e undecidability and incompleteness of any such system (see Section 11.2), thus showing t h e inadequacy of t h e concept of formal system. These are t h e highlights of t h e extensional analysis of formal systems provided by recursion theoretical methods, b u t by no means t h e only ones (see p. 349). A result of Myhill [1955] (111.7.13) points out t h e limits of this analysis and shows that, from a n extensional point of view, all formal systems of common use look alike in the sense of being all recursively isomorphic.

How to Use the Book This book has been written with two opposite, and somewhat irreconcilable, goals: t o provide for both an adequate textbook, and a reference manual. Supposedly, t h e audiences in t h e two cases are different, consisting mainly of students in t h e former, and researchers in the latter. This has resulted in different styles of exposition, reflecting different primary goals: self-containment and detailed explanations for textbooks, and completeness of treatment for manuals. We have tried to solve the dilemma by giving a detailed treatment of t h e main topics in t h e text, and sketches of t h e remaining arguments in t h e exercises a n d in t h e starred parts. T h e exercises usually cover material directly connected to t h e subject just treated and provide hints of proofs in the majority of cases, in various degrees of detail. In a few cases, for completeness of treatment a n d easiness of reference, some of t h e exercises use notions or methods of proof introduced later in t h e book. T h e starred chapters and sections treat topics t h a t can be omitted on a

12

Introduction

first reading. T h e starred subsections deal with side material, usually giving broad overviews of subjects t h a t a r e more or less related to t h e main flow of thought, b u t which we believe provide interesting connections of Recursion Theory with other branches of Logic or Mathematics. T h e style is mostly suggestive: we t r y to convey t h e spirit of t h e subject by quoting t h e main results a n d , sometimes, t h e general ideas of their proofs. Detailed references a r e usually given, both for t h e original sources a n d for appropriate u p d a t e d treatments. T h e general prerequisite for this book is a working knowledge of first year undergraduate mathematics. When dealing with applications, knowledge of t h e subject will be assumed b u t , since t h e treatment is kept separate from t h e main text, there will be no loss in skipping t h e relative parts. T h e chapters have been kept self-contained as far as possible. We have d o n e our best to keep t h e style informal a n d devoid of technicalities, a n d we have resorted to technical details only when we have not been able to avoid t h e m , no doubt because of our inadequacy. Instead of t h e usual complicated diagrams of dependencies, we give suggestions on how t h e first two volumes of t h e book can be used as a textbook for classes in which Recursion Theory is t h e main ingredient.

Elementary Recursion Theory Chapters I a n d I1 provide a number of alternative approaches to recursiveness a n d t h e baqic development of t h e theory. Scctions 2 to 6 of C h a p t e r I a r e independent a n d can be chosen according to t h e audience in t h e class. More precisely, mathematicians can concentrate on Sections 2 a n d 3 a n d cover also t h e Incompleteness a n d Undecidability Results, treated in Section 11.2. On t h e other hand, computer scientists will find inore interest in Sections 4 to 6 of Chapter I a n d Section 11.1, where t h e foundations of a number of prograniniing languages a r e laid, a n d can also cover self-reproducing machines, touched upon in Section 11.2, a n d t h e tools needed to build models of A-calculus a n d combinatory logic, covered i n Sections 11.3 a n d 11.5. Section 1.0 t r e a t s Church’s Thesis in a less simpleminded way t h a n usual (i.e. facing t h e problems, instead of sweeping them under t h e rug), a n d i t is perhaps more appropriate for philosophers.

Recursively Enumerable Sets T h e elementary theory of r.e. sets a n d degrees is contained in C h a p t e r 111 which requires only sonic background in elementary Recursion Theory. T h e chapt,er goes u p to t h e solution t o Post’s problem (Sections 1 to 5 ) a n d t h e basic classes of r.e. sets. It can be used either as a final section of a course

Introduction

13

on elementary Recursion Theory (not dealing with alternative definitions of recursiveness), or as t h e initial segment of a n advanced course on r.e. sets. In t h e latter case, it should be followed by Chapter IX, dealing with t h e lattice of r.e. sets, an d a choice of material from Chapter X, in which priority arguments are introduced. Some of t h e material here, e.g. t he theory of r.e. m-degrees, is not standard, b u t is useful in various respects: intrinsically, this structure is much better behaved than t h e schizoid one of r.e. T-degrees, and it reflects t h e global structure of degrees, which t he latter does not; moreover, arguments on T-degrees (such as t h e coding method) ar e better understood in their simpler versions for m-degrees.

Degree Theory Elementary degree theory is treated in Chapter V which, with some background in elementary Recursion Theory, can be read autonomously. We develop t h e theory u p to a point where it is possible t o prove a number of global results. This forms t h e nucleus of a course, a n d it can be followed by a number of advanced topics including a choice of results from Chapters XI a n d XII, on degrees of A! a n d arithmetical sets. Chapter VI,on m-degrees, is often unjustly neglected, b u t it does provide for t he only existing example of global characterization of a structure of degrees. It can be read independently of Chapter

V.

Complexity Theory Chapters VII an d VIII deal with abstract complexity theory and complexity classes, an d do not require any background, except for a working knowledge of recursiveness and Turing machines (like Sections 1 and 4 of Chapter I). T h e treatment is fairly complete but , going beyond t h e usual unbalanced confinement t o polynomial time and space computable functions, it also covers unjustly neglected classes of recursive functions, such as elementary, primitive recursive, a n d €0-recursive ones which are of interest t o t he computer scientist.

Not at ions and Conventions w = {0,1,. . . } is t h e set of natural numbers, with t he usual operations of plus (+) a n d times ( X or .), a n d t h e order relation 5. P(w)is t h e power set of w , i.e. t h e set of all subsets of w. ww and p are, respectively, t h e sets of total a n d partial functions from w t o itself. We reserve certain lower or upper case letters t o denote special objects: 0

a, b , c, . . . ,~ , yz ,, . . . for natural numbers

14

Introduction 0

f,g, h, . . . for total functions of any number of variables

0

a,@,r,. . . ,(p, $,

0

x,. . . for partial functions of any number of variables

F, G, H , . . . for functionals, i.e. functions with some variables ranging over numbers, an d some over functions

0

A , B , C , .. . , X , Y ,2 , ., . for sets of natural numbers

0

P, Q, R , . . . for predicates of any number of variables

. . for strings, i.e. partial functions with finite domain a n d values in {0,1).

0 CT,~,.

Regarding sets: 0 2

E A means t hat z is an element of

A

0

(A1 is t h e cardinality of A , i.e. t h e number of its elements

0

A

0

0

0

0

0

0

0

CB

and A

c B ar e t he relations of inclusion a n d strict inclusion

is t h e complement of A , and t he prefix ‘cc-’ in front of a property of a set means th a t the complement has this property (i.e. a set is c e i m m u n e if its complement is immune)

A U B is th e union of A and B , i.e. t h e set of elements belonging t o at least one of A and B

A

@ B is the disjoint union of A a n d B , i.e. t h e set of elements of t h e form 2 2 if 2 E A , and 2 2 + 1 if 5 E B

A n B is th e intersection of A and B , i.e. t h e set of elements belonging t o both A an d B A x B is th e Cartesian product of A and B , i.e. t h e set of pairs whose first and second components are, respectively, in A a n d B A . B is the recursive product of A and B , i.e. t h e set of codes pairs ( 2 , ~E)A x B (see p. 26 for codings)

(2,~)

(2, y)

of

is t h e characteristic function of A , with value 1 if t h e given argument is in t h e set, arid 0 otherwise.

CA

Regarding predicates:

Introduction 0

15

-

- P , P A Q, P V Q, P --+ Q , P Q, VxP, 3xP are t h e usual logical operations of negation, conjunction, disjunction, implication, equivalence, universal an d existential quantification. T h e symbols -+ and H will be used in a formal way, t o build new properties from given ones. T h e symbols =+ and will be used informally, as abbreviations for ‘if . . .then’, and ‘if and only if’.

*

We use bounded quantifiers as abbreviations:

(32 I Y)P(4 for (Vx 5 y ) P ( x ) for 0

(3x)[x I Y A q x ) ] (Vx)[x 5 y + P(x)].

cp is t h e characteristic function of P , with value 1 if P holds for t h e given argument and 0 otherwise.

Regarding binary relations on a set A , R is: 0

reflexive if x R z for every x E A

0

antireflexive if ~ ( x R z )for , every x E A

0

symmetric if xRy

0

transitive if xRy A yRz =+ xRz for every x , y, z E A

3

yRa: for every x , y E A

0

a (weak) partial ordering if it is reflexive and transitive (weak partial orderings are indicated by 2,5 , or C)

0

a (strict) partial ordering if it is antireflexive and transitive (strong partial orderings ar e indicated by

must have been derived. Note t h a t by definition there c a n be no equation with t,he left-hand side like t h e right-hand side of t h e above equation. Also, R2 only allows for replacement of ternis of t h e form: functional letter followed by nunierak. T h e n equations like -

-

f;+'(zl,.. . ,xn,z ) = Ti

fg+2(z1,.. . ,zn,z,z) =6

must, have been derived. B u t then again t h e induction hypothesis applies. 0

I. Recursiveness and Computability

38

In Section 7 we will prove t h a t t h e syntactical notion of Herbrand-Gadel computability a n d t h e semantical notion of finite definability a r e globally equivalent, by showing them equivalent to recursiveness. Also, by t h e proof just given, every recursive function is Herbrand-Godel computable f r o m a system of equations finitely defining it. But the two notions are not locally equivalent, in t h e sense t h a t given a system & t h e following may happen: 1. g can be finitely defined by & without being Herbrand-Godel computable f r o m it, as t h e system

f(z) = 0

f(z)= h ( z ) g ( z ) = h(z).

a n d t h e function g ( z ) = 0 show. This simply results from t h e fact t h a t t h e rules R1 and R2 are of a very specific form, a n d do not even allow for full logical substitution of equal entities.

2. g can be Herbrand-Godel computable f r o m & without being finitely defined by it, as t h e system f(0) = 0

f(S(z))= S(f(S(z)))

g ( 4 = f(0)

and t h e function g ( z ) = 0 show. Here Herbrand-Gadel computability follows because only f(0) is used among t h e values o f f , but finite definability fails because there is no total function f satisfying every equation (since f(z) = f(z) 1 for z > 0).

+

Kreisel and Tait [I9611 isolate a notion of derivability from systems of equations, which is locally equivalent t o finite definability. Basically, t h e rules correspond t o t h e logical axioms for equality a n d successor. See Statman [1977] for a proof-theoretical analysis. Herbrand-Gadel computability has t h e advantage of using simple rules, and t h e disadvantage of not being complete, in t h e sense of not allowing t h e derivation of everything which is logically derivable.

A logical programming language * Note t h a t a n equation can be put into a normal form of t h e kind R1

A*-.AR,,

+

Q

with R,, Q atomic equations of t h e form fy(z1,.. . ,z), = y a n d xi, y variables or constants (the interpretation of variables being t h a t they are all universally quantified). For example, f(4= g ( h ( 4 )

1.3 Arithmetical Formal Systems

39

can be written as

h ( z )= y A g(y) = z

+

f(z)= Z.

T h en t h e previous results show t hat t he values of recursive functions can be logically deduced from axioms of t he described kind. T h e programming language PROLOG (Programming in Logic, Colmerauer, Kanoui, Pasero and Roussel [1972], Kowalski and Van Emden [1976]) is based on logical deductions from clauses of t he form above, with R,, Q atomic relations holding of terms. These are called Horn clauses, and are especially interesting because proof procedures for them are particularly manageable. They can be thought of as conditions breaking up a goal Q into a series of subgoals R,. T h e results of this section show t hat PROLOG, although concocted to handle deductive more than computational problems, has nevertheless the power of computing all the recursive functions.

1.3

Arithmetical Formal Systems

T h e general trend of this century’s mathematics has been to work in formal systems which are supposed t o capture, more or less accurately, some aspects of t h e objects in which we are interested. From a formalistic point of view we can thus consider a function as computable when we have a consistent formal system representing it, i.e. allowing us t o prove for t h e appropriate numbers (and for nothing else) t hat they ar e t h e function’s values for given arguments. Certainly th e approach of Herbrand-Godel computability falls in this trend, b u t t h e formal system involved there, concocted for different goals, is somewhat unnatural from a purely arithmetical point of view. T h e same will be t rue of t he approach of A-definability, see Section 6 . Since we are considering arithmetical functions, it is natural t o investigate which functions are representable in t he usual logical systems for arithmetic, for example in Peano Arithmetic (see p. 24). We attack th e problem in a general way, by isolating minimal conditions (which will tu r n o ut t o be very weak) sufficient t o represent every recursive function. I n this section, ‘formal system’ will always mean ‘formal system

extending first-order logic with equality, and having constants terms ii, called numerals, for each n ’. For more details on formal systems, see p. 350.

Notions of representability Definition 1.3.1 (Tarski [1931], Godel [1931], [1934], Tarski, Mostowski and Robinson [1953]) Given a formal system F and a function f , we say that:


40

1. f is weakly representable ,in F if, f o r some f o n n u l a 'p of t h e language

of

F, f ( 5 1 ,... ,%)

= ?J

FF

@

-

2. f i s representable in .F 2% f o r some f o r m u l a

=+

f(:c,,. . . ,Zn) =y

f(XI,. ..,Xll)

# y =+

FF

FF

-

' ~ ( TI .I. . , %,Y) 'p,

Cp(S1,. . . ,:n,u)

-

y'p(z*,. . . , x n , p )

if f o r some fonriida 'p, f is repre3. f is strongly representable in sentable hy cp, and inoreoiler t h e following uniqueiiess c o n d i t i o n holds:

'I'he relationships arnoiig t,he various notions are: if f is strongly representable then it is rel)re;eiitabk, and i f f k represenl.&k in a consistent formal syst,eni then it is wcakly rc:present,al,le (becausc if .F is c.orisisf,ent, and k~ "p the11

y>-cp).

Exercises 1.3.2 a) ?'he two cmiditioiw /(ZI , . - . , ~ n ) ~ ?/: *

~p(-Cl,...,+n,S)

FF

-

F F (VY)(vz)['P(il,...r~n,~)ACp(~lr ...,Zn,Z) nrr cquivnlerit to the unique condition

F.F (VY)l'P(i,,. . . , Z , , Y ) b) If .F zs such that I

# y =+

FI

-

y

~

f(.l,,

..

+

Y

--I

.Tn)l.

l(Z = 5 )

t,Iwii strung repi-eseiitability off iii .F is iquiiralent to the unique condition

Proposition 1.3.3 (Tarski, Mostowski and Robinson [1953])I f .F is a coiisisteiit. foriiial system with a predicate < satisfying thc axiom s c h e m a t a 1.

2.

T(Z

< 0) -

x[~p(~l,. . . ,z n , y )

++

Y = f(z1,.. zn)1.

We have I-F (p(Z1,.. . ,Z,,!(XI,. . . , 5 , ) ) from t h e first part of t h e proof. Suppose now FF 'p(z1,. . . ,zn,y).By axiom 3 t h e only possibilities are y

< f(z1,.. . ,xn) v Y = f ( 2 1 , . . . , G I ) v

. . ,Xn) < Y.

f(x1,.

T h e first one is ruled out, since from l - ~cp(T1,. . . ,Tn,f(zl,. . . ,zn))we have t3 +J(Z~,. . . ,Zn,y),while from I-F (p(T1,. . . ,Z,,y) (assumed by hypothesis) we have I-F @(?El,.. . ,Z,,y), and F is consistent. Similarly we can rule out f(z1,.. . ,zn)< 9. Then y = f ( ~ 1 ,... ,zn). 0 T h e notion of representability makes sense for predicates as well:

Definition 1.3.4 Given a formal system F and a relation R, we say that: 1. R is weakly representable iJ for some

R ( ~ i , . - - , z n@ ) FF 2. R is representable if, for some

(p,

(p,

(~(%,...,2n)


42

Note that i f tlic charac:terist,ic function CIz of I< is (weakly) rcprcsetited by z),t,lic:n 11’ is (weakly) reprcsetit,tible l)y cp(z’:l ,. . . rxL’ll, 1). Also, if .T is such that, 2,’# y =$ t :,- 7(z = 5 ) ‘p(z1,.. . , x,,

atid R is represetitotl by ‘ p ( z l , . . . , : i z T L )then , A h =

‘112

is (strongly) represetitable by

-

I ) v (7cp(z,, . . . ,&)

A 3 =Ti)

(tliis follows from 1.3.2.1)). Note t.liat the axionis a r e tiecxldetl evcti for simplr rel)txsetiiabiIity of (:R.I)ctc.aitsc: w l i w c ~ ~ ( z l ., . : I ’ ~ ~#) 2 and 2 # O, 1 wc nctxl to ktiow 3 # t,o be able to i t i f e r t l i w t , thc formula i i i t , r : n t l e t l t.o r e p r c x n t C R

o,r

is tiot prova1)Ie.

Formal systems representing the recursive functions N o t c that if j(.t.,. . . . , n.,,) = pyR(.rl , . . . ,.T,,, !/) then

Proposition 1.3.5 (Tarski, Mostowski and Robinson [1953]) 1s F jOl.lllol a‘lJ.5tClIL S U ( h 11101

Proof. Suppose

j(.q = y ( l t l ( T ) ,. . . ,h,n(.?))

LI

(I

1.3 Arithmetical Formal Systems

43

if kF cp(??,y) let yl, . ~.,ym be such that

0

x (Y1, . . . Ym,Y). By strong representability it must be yi = hi(Z)and thus l-F $1

(2,y1) A . . . A $m (2,Y m

A

y = g(h1(Z),. . . , h,(Z))

= f(?).

1

0

We are now ready t o conclude our search for axioms which allow representability of every recursive function.

Theorem 1.3.6 (Godel [1936], Mostowski [1947], Tarski, Mostowski and Robinson [1953]) I n any formal system 3 with a predicate < and functions and . satisfying the following axiom schemata, any recursive function is (strongly) representable (and thus, i f the system is consistent, also weakly representable):

+

B1 B2 B3 B4 B5 B6

y(T=jj), f o r x f y V A < x

x t i i i i i a version callccl thc XI-calculus, i n which fii iict ions with fictitious argii 11ICIit,s are excluded. This rcst r ict,io11 has 1 )ccii int rotliiced t o avoid soltie patliologies, like t,he existence o f noriiializable, i i o t , st roiigly noriiializable t>ernis,as S C Y : ~ above. The: N arid j j rules define a system called the calculus of p-conversion. Varioiis riiotlific;it,ioiis o f it are possible, e.g. by adding aii extensionality law tcritis,

for ,\-t.cmns:

77-rule. \Vc can itlcntify evcry

t ~ r i i iwitli

a fnwt,ion:

1)rovidctl z is not frec i n 121. No1.c that., I)y t,lie rules of term forniaLioii, for a n y term A1 t,lie expr"?isiori Xz. A1:c is also a teriii, a i i t l thus it r q ) r e scnts a fiinctioii: (,lie r n l e wsiires that, this is exactly thc fiiriction t,h;it, is rc:presentetl by the term A1 itself.

1.6 Functions as Rules

83

T h e reason we call this a n extensionality law is that it implies that if M and N behave extensionally in t h e same way, i.e. Ax. M x and Ax. N x are equal, then so are M and N . An equivalent but different approach to the X-calculus (called the theory of combinators, Schonfinkel [1924], Curry [1930]) consists in postulating some, actually very few, of t h e combinators as primitive, and t o deduce all t h e others from these. This produces a kind of synthetical (bottom-up) analysis of t h e global concept of X-definability. In particular, it can be shown t h a t only two combinators are needed:

S = Xzyz. z z ( y z )

K = Xxy. z.

S can be seen as a kind of interpreted application, where x and y are first interpreted in t h e environment z, and then applied one t o the other. Since t h e identity I can be defined as S K K : I X= S K K X= K x ( K 2 )=x , the X operator can be defined by induction on the terms obtained from t h e variables by application, as follows:

Xx.x = I Ax. c = K c if c is K , S or a variable y A X . M N = S(Xx.M ) ( X z .N ) .

#x

We can then prove t h e P-reduction rule, by induction. Standard references on t h e subject are Church [1941], Curry and Feys [1958], Curry, Hindley and Seldin [1972], Barendreght [1981], Hindley and Seldin [1986]. Historical accounts on t h e origins of X-calculus and its interaction with Recursion Theory are in Kleene I19811 and Rosser [1984].

X-definability We are interested in numerical functions, but it would seem that until now we have just set up a logical basis, and that we still need to add t o t h e language of X-calculus numerical terms and some basic numerical functions. But then we would face the problem of not knowing exactly what t o add, and we would have to turn back to different approaches, with the X-calculus relegated to a mere role of convenient notation. Instead, and this is a most interesting aspect of this approach, it turns out that there is no need of additional notions. T h e natural numbers appear obliquely in this general setting, when we consider the number of iterations of a function application. We can thus define X-terms T i that, applied t o a function f and an argument x , give t h e result


81

f(")(z)o f R iterations of the function t l i r iiit,rgers. Since we want

f

o n z, and take them as representing

wc ('a11just let:

Definition 1.6.3 (Peano [1891], Wittgenstein [1921], Church [1933]) vw iiiiiriera/ ii is the ,\-tr:mri ~f.. f(")(~). We write f and z to help the intuitive underst,aiiding, but note that we just, have one type of variable (with no distinctiori I)et,weeti functions and argunierits), arid t h u s we slioultl just write -

71

= Azy. z(747,

Not,c that the teriiis TE and ii are different if m # n, by the theorem o f ('1inrc:h aiid Itosser quoted on 1). 78, I)ecaiise they are in normal form arid tlist,iiict. Iiidiictively we have, by tlcfinit,ion of iteration, -

0 = Afx.2:

m

= Af:c.f(7if2:)

I)ccaiise f ( " ) ( ~ ) = z ant1 f(lt+')(:c) = f(f("'(2;)). Since these terms rcprcseiit. i n sonie way the constarit, function 0 and the sii( s o r operation S,we gel. froni t,liis t,he idea of reprc;cnt,ing iiunierical f i i r i c t i o i i s :

Definition 1.6.4 (Church [1933], Kleene [1935]) A I L?t,-ary f u n c t i o n f is A-definable if there i s a A - t e r m 1' such that

.

f ( ~ , l , .. ,a,)

=b

holds if m d only if lG, . . . ?in

5.

Theorem 1.6.5 (Church [1933], Rosser [1935], Kleene [1935], [1936b]) I < i w r y i r ~ c ~ ~ ~ i :fuiictioii siiie i s X-deJinahle. Proof. We proceed by iiidiictioii o n the definition of recursive function. '1'0 simplify tlic technical clctnils, wc rely o n t,he alt.eriiatjivec,liaract,erizationof thc class o f primitive recursive functions given in 1.5.10. 1. initid functioris 0 , S arid Tp are, respectively, A-defined by: AX.

ii

Azfz. f ( z f z ) A z 1 ' . ' X 7 & .2 ,

1.6 Functions as Rules

85

2. coding and decoding functions for pairs This is obtained, in analogy with the representation of natural numbers, as: (n,m)= Afgx. f'"'(g'"'(x)). Thus t h e pairing function is

AY zfgx. Y f ( W ) . Decoding is then immediate: if Z is the representation of T i , i.e. of the identity function, then

Thus t h e decoding functions are Xy f x . y f I x and Aygx. ylgx. 3. composition Suppose h, gi are A-defined by H , Gi. Then

is A-defined by

4. iteration This is a n immediate consequence of the representation of natural numbers: if T is a term representing the function t, then the iteration

is represented by

Axy. yTx. 5 . p-recursion This is obtained as in the proof of 1.2.3. Recall that if


86 where

h(S’ y, =

{ :(?,

y

+

if g(5,y) = 0 1) otherwise

By t h e first part of the proof case definition, successor, a n d composition are all A-representable, because primitive recursive. By induction hypothesis, so is g. Thus there is a term M representing Ah-y. F , where

F(Z,Y,h) =

{ :(?,

y

+ 1)

if g(5,y) = 0 otherwise.

Then t h e function h defined above, being a fixed-point of F , is represented by y M . And f is finally represented by A?. ( y M ) & . To conclude t h e proof we note that: 0

T h e completeness property, stating t h a t t h e values can be deduced from the appropriate A-terms, is quite evident from t h e informal discussion just given.

0

T h e consistency property, stating t h a t no other value can be deduced, follows from t h e theorem of Church and Rosser quoted on p. 78, which ensures t h a t if the process of P-reduction of a t e r m produces a term in normal form (like T i ) , then this term is uniquely determined (up to renaming of t h e bound variables). 0

Exercises 1.6.6 a) Alternative coding and decoding functions are, respectively, Xxyu.uxy, Xz.zT and Xz.zF, where T = Xxy.x and F = Axy.y. Being distinct Xterms, T and F can be taken as representation of the truth values ‘true’ and ‘false’. b) 6 = Xz.z(Xu. F)T represents a function that, when its argument is a numeral, is T i f z is and F otherwise. Then Axyz. ( 6 z ) x y represents definition by cases, i.e. a function that returns the first or the second argument, according t o whether the -third is 0 or not. (Hint: a numeral applied to two terms produces the second if it is 0, and applies the first at least once otherwise.) c) The predecessor function can be directly represented, using only repwsentations of successor, coding, and decoding functions. (Kleene [ 19351) (Hint: the predecessor of n is the second component of the n-th iteration of the function on pairs defined as t ( ( z y, ) ) = (z 1, z), started on (0,O).) By 11.2.15 this provides an alternative proof of 1.6.5, not using 1.5.10.

a,

+

Functional programming languages * T h e programming languages discussed in Sections 4 a n d 5 a r e imperative in t h e sense t h a t they specify a sequence of instructions a n d a n order of execution to be followed t o produce a n output f(5).T h e functional approach, suggested by A-calculus, defines f directly, and computes t h e required values by

1.7 Arithmetization

87

P-reductions. T h e name ‘functional’ reflects the emphasis put on t h e functions themselves as intensional objects, as opposed t o extensional emphasis on t h e values. Relying on the characterization of recursive functions given in 11.2.15, McCarthy [I9601 notes t h a t a function is recursive if and only i f 0

there is a definition of it from identities, successor, and predecessor by means of A-abstraction, composition and the conditional operator

if t(5)= 0 then g(Z) else h(Z). 0

t h e function is computable from this definition by a call by value procedure, i.e. a computation which evaluates first the innermost (and leftmost, if there is more than one) occurrence of the letter defining t h e function.

Clearly t h e conditional operator replaces the definition by cases, and t h e fixedpoint operator is eliminated, in favor of a computational approach t h a t just produces it. This formulation of recursive functions can be extended from numerical functions t o functions on words of a given alphabet and, as such, it has furnished t h e computational basis for t h e programming language LISP (List Processing, McCarthy [1960]; see Sammett [1969] a n d Wexelblat [I9811 for history a n d references). Since there is no reason t h a t functional programming languages should be forced to run on machines designed for imperative ones, work has been done t o design hardware directly inspired by t h e functional approach, based on A-calculus ( t h e SECD machine, Landin [1963], implementing P-reductions) or on t h e theory of combinators (the SK machine, Turner [1979],implementing reductions of graphs t h a t represent t h e definition of a function, in terms of t h e combinators S and K , see p. 83).

I.?

Arithmetization

Arithmetization simply means translation into the language of arithmetic. We will give one detailed example of t h e method, and show how to code the machinery of computation of recursive functions in a primitive recursive way. T h e details are quite cumbersome, and in t h e rest of our work we will content ourselves t o sketch similar arguments, leaving the details t o the reader.

Historical remarks

*

T h e first attempt t o find number-like connections between propositions of various sorts probably goes back t,o Lullus’ Ars Mugna, b u t it was Leibniz [1666]


88

who dreamt of arithmetization as a general method to replace reasoning in natural language by arithmetical propositions, with t h e goal of substituting arguments with computations. Leibniz went further than just this oneiric activity, and devised (see [1903]) a precise coding method, by first assigning numbers t o primitive notions, and then showing how t o associate composite numbers (e.g. by multiplication) t o composite notions. However, his tentative work was left unpublished until 1903, and did not influence modern developments. Hilbert [1904] again envisaged arithmetization in his idea of formalizing consistency proofs into Arithmetic, but it was Godel [1931] who first used it explicitly and formally t o translate t h e concepts relative to formal systems into a n arithmetical language. Tarski [1936] independently arrived at t h e method in his investigations of t h e concept of truth. Contrary to Leibniz's dreams, the effect of arithmetization was, ironically, not t o shield the language against its oddities, but rather to spring a leak in Arithmetic, through which the linguistic paradoxes poured only to reveal t h e inadequacies of formalism (see 11.2.17).

Numerical tools for arithmetization Primitive recursive functions and predicates are more than enough to carry out arithmetizations. We will use prime numbers a n d factorizations (recall, see p. 26, t h a t the sequence { p s } , E w of prime numbers is primitive recursive), because this coding is particularly simple. It should however be noted t h a t we could use functions from much smaller classes: this will be done in Chapter VIII, when t h e necessity for more efficient codings will arise. To code the sequence (50,.. . ,zm), t h e simplest way would be t o use t h e number pgo - .. p 2 . But then we could not uniquely decode a number, since we would not know whether a prime in the decomposition has exponent 0 accidentally or meaningfully (in t h e sense that it is coding t h e number 0). Thus, either we rule out 0 as a meaningful exponent, a n d let (20,.. . ,z),

= p;0+1

*

. . pEn+',

or we tell in advance how many numbers we are coding, and let

..., zn)= p ; * p ; 1 . . . p?.

(XI,

Since we have t o make a choice for the following, we decide t o use t h e second proposal. T h e decoding system is given by t h e following functions a n d predicates, all primitive recursive (see p. 26):

89

1.7 Arithmetization

We call h ( z ) t h e length of 2,and ( x ) the ~ n-th component of z. If S e q ( z ) holds then we say t h a t z is a sequence number. In this case, z = ((XI1 9 .

*.

7

(.)In(,)>.

We will also need a concatenation operation (21,.

*, such t h a t

. . , z n )* (?/I,.. . ,Ym)= h,.. . ,zn,Yl,.* . ,Y,>.

if S e q ( z ) A S e q ( y ) holds, a n d 0 otherwise. Finally, we will need t h e notion of initial subsequence, defined as

As a first use of sequence numbers we get t h e following useful result. Proposition 1.7.1 Course-of-values recursion (Skolem [1923],Peter [1934])The class of primitive recursive functions is closed under recursions in which the definition of f (2,y f 1) may involve not only the last value f (2,y), but any number of (and possibly all) the values { f (2,z ) } ,-< ~ already obtained. Formally, let f be the history function o f f , defined as: f(.',Y)

= (f(.',O),...,f(.'>!4).

Then, i f f is defined as

and g , h are primitive recursive, so is f . Proof. It is enough to show that

f is primitive recursive, since then also f

is:


90

But this is imrriediat,e. since

+

1) only uses t h e last previous valuc f (.',y), Thus f(?,y recursive because so a r e coding arid coricatenation. 0

arid

f is primitive

Exercise 1.7.2 Simultaneous primitive recursion. The clnss of primit,ive w cursive functions is c l o d under simultaneous recursion on mom than one function. (Ililbert arid Rernays [ 19341) (Hint: reduce simultaneous primitive recursion of, say, f l (i) arid f2(z+),to primitive recursion of the single function (fl(?), f'~(.t)))), 1Jy coding.)

The Normal Form Theorem We are now in position t o give our first a n d only complete example of a r i t h n i o tization, by reducing t h e reciirsivc runc:t,ions to a normal form. Any approach to recursiveiicss would produce similar results, a n d we will skct,c:li t,he versions relative t o t h e approadies o f Sections 2--6 later in this sect,ioii, but, here we give a self-cont,ained treatment based or1 recursiveness alone.

Theorem 1.7.3 Normal Form Theorem (Kleene [1936]) Th,ere ,is a primitive recursive f u n c t i o i i U a n d (for each n I) primitiiie recursive predicates such that for every recursive f u n c t i o n f o f 71 variables there is n miirrber e (called index o f f ) for which the following hold:

>

zl,

1.

VXl

...Vxn3y?;l(f:,xl, . . . ,xn,g)

2. f ( X h , .

. ,%) '

= ~ ( P Y T 1 ( c l x l.>. ,.G L l Y ) ) .

Proof. The idea of t h e proof is to associate Iiuinbers to fuiictioiis a n d m n i putations i n such a way t h a t t h e predicate ~ l ( e l x.~. ,lx T . L l yt)r a n s l a t e t h e assertion: ?j is t h e tiuriiber of a computation of t h e value of t h e function with . this, pyTn(e,xl, . . . , xTL,y) associated nurriber e , on input,s z1,. . . ,z,,Ilaving will give t h e riurriber of one such coriipiitat~ion,arid t h e fiiiiction U will extract t h e value of t h e o u t p u t from it. T h i s is more easily said than done, and to achieve it we need to carry out a riuml~erof steps.

1.7 Arit hmetization

91

1. associate numbers t o recursive functions

This is done according to the inductive procedure that generates the recursive functions. T h e details are obviously irrelevant, and we just give one possible assignment: 0

(0) t o 0

0

(1) to

0

(2,n,i)t o Tin, for 15 i 5 n

0

0

0

s

( 3 , b l ,... ,b,,a) to f(Z)= g(hl(Z),..., h,(Z)), where b l , ... ,b, and a are numbers respectively associated t o hl , . . . ,h,, and g (4, a, b) to f(Z, y) defined by primitive recursion from g and h, where a and b are respectively associated t o g and h ( 5 , a ) t o f(2) = py(g(Z,y) = 0), if VZ3y(g(P,y) = 0) and a is associated to g.

Any number associated t o a recursive function is called a n index of this function. Since there are many ways t o define the same function, there will be many indices for each recursive function (see 11.1.6). Also, many numbers are not indices of recursive functions, either because they are not sequence numbers of t h e right form, or because some of their relevant components are not indices of recursive functions. We will see (p. 146) that in general there is no effective way t o tell whether a number is indeed the index of a recursive function, because basically there is no way t o tell whether VZ3y(g(Z,y) = 0).

2. put computations in a canonical form A natural way t o organize a computation for t h e values of a given recursive function, is by way of computation trees. Each node of such a tree will tell how a value needed in t h e computation can be inductively obtained. Of course, t h e only possibilities are those given by t h e permissible schemata of definition I. 1.7, namely: 0

nodes without predecessors:

f (4= 0 iff=U if f = S f(z)= z 1 f ( z 1 , . . . ,zn) = zi if f = Tr

+

0

composition If f(Z)= g(hl(Z),. . . ,h,(Z)) predecessors:

then the node f(2) = z has r n

+1


92

f(2)= z

hl(Z)= z1 0

...

h,(Z) = z , g ( z * , . . . , z,,,) = z

primitive recursion I f f(Z,y) is dcfiried by primitive recursion from g a n d h t h c r c a r e two cases, respfxtively with one o r two predecessors:

p-recursion If f(2) = py(g(S,?y) = 0 ) t h e n there is r i o fixed p a t t e r n to t h e predecessors of f(Z)= y. T h e general situation is:

f(2)= z

where l o , . . . , t,-

I

a r e all different from 0.

3. associate numbers to cornputatjoris This is done by induction on t h e c.onstriic.l,ion of t h e computn1,iori tree. First of all, we assign numbers to nodes: since t h e y a r e expressions of t h e kind f(xl,. . . , z T L=) z,we give t h e m numbers

where e is a given index of f . T h u s a tiode is represented by t,hree numbers, correspotiding respectively to t h e firtiction, t h e inputs i i r i t l t h e olll,put.

We then assign niiinl)ers t o trees: each tree T consists of a vertex v with associated numl>er u,a n d of a certain number (finite, arid possibly equal to zero) of orderrd predecessors, each o n e being a siibtree T,. By

93

1.7 Arit hmetization V

Figure 1.12: A tree T with vertex v and subtrees Ti's induction, we assign to this tree t h e number

5

is t h e number assigned t o the subtree Ti. In particular, if t h e where vertex with number 21 does not have predecessors, then it has number (v) as a tree.

4. translate in a primitive recursive predicate 7 ( y ) the property that y is a number coding a computation tree To increase readability we use commas instead of nested parentheses, and write e.g. ( a ) i , j , k in place of ( ( ( a ) ; ) j ) k . To keep track of what we are doing, check Figure 13 and recall t h a t h

Y = (V,Tl,...,Trn), a n d hence:

(Yh (y)l,l (Y)1,2 (Y)1,3

(Y)i+l (Y);+~,]

= (e, (21 ,. . . ,%), 4 = various types, depending on e

=

(21 1 . .

. ,GJ

z

= = Ti = number of the vertex of T;. A

First we let:

This expresses t h e most trivial properties of y. We then have four cases, corresponding to t h e possible situations spelled o u t in Part 2 above.

I. Recursiveness and Computability ((3, b , , . . . , ( I , , , , a),(x,,. . ,:c,J, f

2)

a) composition

( 0 ,( X I , .

I

. . ,&),

2)

I,) primitive recursion

c ) p-recursion

1.7 Arithmetization

For composition we let:

For primitive recursion, recall that there are two possible cases:

For p-recursion we let:

95

I. Recursiveness and Computability 'I'liese cwiditions t a k e c a r e o f all possible cases, a n d thus we may define

iiitliictively:

'I'lien T is priniitive recursive because it is definccl by using only pririiit,ive rwiirsive clauses arid values (of its cliaracteristic function) for prcvious iwgumerits because, by definit,ioii of cwling, (y), < y . T h a t is, 7 is defined by coursc of value rtxmrsioii, wliicli is a priiiiitive recursive o p e r a t i o n by

1.7.1.

z,

5 . drfine arid U We a r e now rciidy to concliiclc: o u r work. Namely, for each n

2

1 we let:

ittid

U(Y) = ( Y ) l , 3 r 1

I lime are obviously pritnitivc: r m u r s i v e

Lct. iiow f 11c a recursive n-ary function with index e . Since Is total, for every . I : ~, . . . , x l I there is ii c u i n p i i t i i t i o i i t r e e for j ( x l , .. . , x1,)relative to t h e c o m p u t a t i o n procculure c:otlctl by c . This is forinally expressed by:

Moreover, from any coriipitatiori ( , r e (in par1,iciiIar from t.he one wit.li t h e sinallrst code iiiiniber) w c c a n extract thc value of t l i c funct,ioii by lookiiig at t h e t h i r d coniponeiit of its vertex. ThLs is foriiially expressed by:

Exercise 1.7.4 There is a rrcirrsivc Jwictioia munieratirq the uriary primitiiic rrarrsive fuirctions, i.e. a recursive function j ( e ,T ) such that: for every c the function Ax. f ( e , s) is prirriitive recursive, arid every primitive recursive function is cqual to As. J ( c ,s) for snmc e . (Petrr [ 19351) (Hint: let

J(e,x)

~

{

1A(,u~y?; ( e , 5 , y))

if e is a primitive recursive index o th erw is?,

wherc k i n g a primitive rcc:iirsive iridcx mcnns to define n recursive fiincticm from a r i d prirriitive recursion alone, without using the the iriitial functions by corrip~~+itioii

/i-operator.)

97


Equivalence of the various approaches to recursiveness By using t h e method of arithmetization we can get a cascade of results relative to t h e various approaches introduced in Sections 2-6. We will not go beyond sketches because we believe t h a t , once t h e method is understood, t h e translation of these into formal proofs should not present theoretical difficulties. I t is, however, a very useful exercise to t r y to fill in t h e cumbersome details of some of these sketches. We begin by dealing with t h e notions of Section 2.

Proposition 1.7.5 (Kreisel and Tait [196l]) Every finitely definable function is recursive. Proof. Suppose f is finitely definable by & w.r.t. t h e n fl(Z) = Z is a logical consequence of 4

f1.

We know t h a t if f(2)= z

for some Zl, . . . , Zp. By t h e completeness of t h e predicate calculus, a n d t h e fact t h a t whenever we have a model we also have a n w-model (i.e. one with domain t h e integers, a n d with 0 a n d S interpreted as zero a n d successor), t h i s is equivalent of saying t h a t if f(2)= z then f1 ( T ) = Z is derivable in a n y complete formalization of t h e predicate calculus with equality, from t h e premises 4

a n d t h e axioms for t h e successor operation:

(with t h e f l , . . . ,fm held fixed in t h e derivation). By arithmetization we can define a primitive recursive predicate 7,(e,Z,y) (where n is t h e number of components of t h e vector )’. meaning: 4

y codes a derivation of a n equation of t h e form fi(Z)= Z,in t h e predicate calculus with equality, from t h e axioms for successor a n d a finite conjunction of substitution instances of t h e system of equations coded by e. Let U be a primitive recursive function such t h a t whenever y codes a derivation, t h e n U ( y ) gives t h e value of t h e numeral on t h e right-hand side of t h e layt equation coded by y.

98


Then we have that f is recursive, because

fF)= u(PY%(e,%Y)).

0

Proposition 1.7.6 (Kleene [1936]) Every Herbrand-Godel computable function is recursive. Proof. By arithmetization we can define 7,(e,z1,. sive, meaning:

. . ,xCn,y)primitive recur-

y codes a derivation, by means of the rules R1 a n d R2 and from the system of equations coded by e , of a n equation of t h e form f,'"(Zl,. . . ,z), = Z, where f? is t h e leftmost letter in t h e last equation of t h e system coded by e. Let

U be a primitive recursive function such that if y codes a derivation then U(y) is t h e value of t h e numeral in the right-hand side of the last equation coded by y.

If f is Herbrand-Gdel computable from the system of equations coded by e then f is recursive, because f ( x 1 , .. . ,x,) = U(pLy%(e,zl,. . . ,x , , ~ ) ) .

0

We turn now t o t h e representability approach of Section 3.

Proposition 1.7.7 (Godel [1936], Church[1936]) Every function weakly representable in a consistent formal system (with recursive sets of axioms and of recursive rules) is recursive. Proof. By arithmetization we can define 7,(e,x1,. . . ,z,,y) primitive recursive, meaning: y codes a derivation of a sentence of t h e form $ ( T I , . .. ,Z, Z), where $ is the formula coded by e, from the axioms of t h e given system and by means of its rules. Let U be a primitive recursive function such t h a t if y codes a derivation then U(y) is t h e value of t h e numeral which instantiates the last variable of the last formula of the derivation coded by y.

If f is weakly representable by the formula coded by e then f is recursive, because 0 f ( z 1 , .. . ,x,) = U ( P Y z ( e , x l , .. . ,Z,,Y)).

99


Corollary 1.7.8 For any consistent formal system extending R,the following arc equivalent: 1. f is weakly representable

2. f is representable 3. f is strongly representable.

Proof. Indeed, recursive =+ strongly representable =+ representable =+ weakly representable recursive

(by (by (by (by

1.3.6) definition) consistency) t h e proposition).

0

W e now t u r n to t h e computational approaches of Sections 4 a n d 5 .

Proposition 1.7.9 (Turing [1936], [1937]) Every Turing machine computable function is recursive. Proof. By arithmetization we can define 7 , ( e , z l , . sive, meaning:

. . ,z,,y)

primitive recur-

y codes a computation carried o u t by t h e Turing machine coded by

el on inputs

51,.

. . , 2,.

Let U be a primitive recursive function such t h a t if y codes a computation then U ( y ) is t h e value of t h e number written on t h e t a p e to t h e left of t h e head, in t h e last configuration of t h e computation coded by y.

I f f is computed by t h e Turing machine coded by e t h e n f is recursive, because .f(x11..

. ,G)

= U ( W ; T , ( ~ , X..I ~ x n , ~ ) ) . 0

Proposition 1.7.10 (Wang [1957], Peter [1959]) Every flowchart computable function is recursive. Proof. To simplify t h e details we can make t h e following conventions: any program has only variables named X O ,X I , . . .; t h e inputs a r e indicated by X I , .. . ,X,, a n d t h e o u t p u t by X O . By arithmetization we can define a primitive recursive predicate 7 , ( e , 5 1 , . . . ,x,, y), meaning: y codes a computation of t h e program coded by e when, at t h e beginning, t h e input variables a r e set equal to 5 1 , . . . ,z, a n d all t h e remaining variables a r e set equal to 0.


lo0

Let, U be a priiiiitivc rccursive function siich t h a t

if y codrs a c.onil~iit.iitioii,t , I i c i r U(,y) Ls the value of tlic. varinble, in t,lie last step of the compiitatioii c.od~xlby y .

I f f is coniputecl

I)y t h e program c d e d by f(.l,.

I‘

oiit,piit

then J is recursive, becaiisc:

0 . . ,:En) = U(IL!/75I(P,.Tl,~. . ,GI ,!I)).

Wc filially t u r n to the X-defiiial)ility approach o f Sedioii G .

Proposition 1.7.11 (Church [1936], Kleene [1936b]) Ehcry A-dcJfiiiahlr: /iinc!ioii % s recursive.

Proof. I3y arit,Iiirrctizatioii we sive, rricmiiiiig:

caii

define 7 ; , ( c , z l , . . . , x T L , y primitivc ) recur-

y cotles a rculiictiori t,o a i i i i i i w r a l , via cr or /jr i i l t s , o f t h e t,erin codctl hy c , applictl to Z1, . . . ,x,.

I d , U I)(! a primitive rccursive

fiiric.t.ioir

such that

if y (.odes a rcxluctioii theii U ( y ) is the valiie of the iiii~ncralobtairietl in its last stc:p.

I f f is A-tlefinable by t h e tcrrir cotltxl 1)y e

t,heii

f is recursivc, brcausc

The basic result of the foiindations of Recursion Theory I’lie rc:siilt,s provctl in this first part, o f the Iwok iiirply t,hat all tlic a p p r o d i e s t,o offertive c.i)nipiital)ility irit,rotliic.etl SO far are equivalent, and thus sliow that /.he iiotioii of rf:ciirsiiii:ii~ss is absolute and very stable. ‘I‘his is a stxiking fact,, and t o s t r c w i1.s irriportance we isolate it in a thcorern o f its o w n , which c a p t u r e s thc esseii(~:o f t,liis chapter: anew tlirb, perch6 tii veggi pura

la vrritk rhe Ih giu si corifonde, fatta I e t t u r a (Dantc, Paradzso, XXIX)

q i i i v o ( w i ( 1 o i i i si

Theorem 1.7.12 Basic result. 7’hc folloiiriiig arc: cqiiiimlmt: ‘ I shall say more, so that you may s w clear ~y the truth that, t l i w p lwlow, has lwei1 cr)rifiised by tcachirrg that rriay I w arnbigiicius.

‘

1.8 Church’s Thesis

*

101

1. f is recursive 2. f is finitely definable

3. f is Herbrand-Godel computable

4. f

is representable in a consistent formal system extending R

5. f is %ring computable

6. f is flowchart (or ‘while’) computable

7. f is A-definable.

Proof. T h e direction showing t h a t all these notions a r e not more extensive t h a n recursiveness is given by t h e results j u s t proved in t h i s section. T h e opposite direction, showing t h a t all these notions a r e at least as extensive as recursiveness, is given by Theorems 1.2.3, 1.2.5, 1.3.6, 1.4.3, 1.5.4a n d 1.6.5.

0

I t should be noted t h a t the equivalence proofs among different notions of computability are effective and eficient. Effectiveness means t h a t for a n y pair of notions t h e r e is a recursive function t h a t , given t h e code of a recursive function relative to one notion, produces a code of t h e s a m e recursive function relative to t h e other notion. T h i s is t h e basis of t h e definition of acceptable system of indices, see 11.5.2. A precise statement of efficiency requires concepts introduced in Chapter VII, a n d will be given there. T h e intuitive idea is t h a t t h e code of a function not only defines t h e function, b u t also shows a method t o compute it, a n d t h e translation roughly preserves t h e computational efficiency of t h e methods.

1.8

Church’s Thesis *

In t h i s section we discuss t h e assertion t h a t every effectively computable function is recursive, by considering physical a n d biological computers. For t h e former we rely on physical theory, a n d t r y first to determine how far t h e particular model of determinism provided by recursiveness accounts for t h e general model of determinism, a n d then to establish t h e extent of determinism itself. For t h e latter, d u e to lack of theory, we pursue t h e synthetical a n d analytical approaches, b y analyzing t h e brain structure a n d formulations of constructive reasoning. D u e to t h e generality of t h e discussion, we will freely quote results which either will be proved later in t h e book, or will not be proved at all, being outside t h e scope of our work. However, appropriate references will be given, whenever needed.

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Introduction to Church’s Thesis T h e work done so far shows t h a t t h e class of recursive functions is certainly a very basic one, since it arises in fields as varied as mathematics, logic, computer science a n d linguistics, with quite independent approaches ( a n d each one interesting in its own right), t h a t t u r n o u t to be equivalent a posteriori (by t h e Basic Result 1.7.12). T h e generality of t h e method of arithrnetization, t h a t allows for these equivalence results, also leads us to believe t h a t o t h e r possible approaches to t h e notion of computability a r e likely to produce notions not more extensive t h a n recursiveness, if not outright equivalent. T h e fact t h a t many variations in t h e details of t h e various approaches do not produce changes in t h e defined class (see e.g. t h e discussion on p. 49), shows t h a t t h e notion of recursiveness is very stable. By Theorem 1.7.12,t h e class of recursive functions is not sensitive to changes in t h e formal systems considered to represent its functions: t h a t is, t h e s a m e functions a r e representable in any consistent formal system having a least minimal power, independently of t h e system strength. A n d even more is true: Kreisel [1972] shows t h a t riot only in formal systems, b u t even in vast classes of recursive transfinite progressions of formal systems, only recursive functions a r e representable. T h u s t h e notion is absolute in a certainly astonishing way, with few (if any) analogues among other logical notions. To q u o t e Godel [1946]: With this concept o n e has for t h e first t i m e succeeded in giving a n absolute definition of a n interesting epistemological notion, i.e. o n e not depending on t h e formalism chosen. In all other cases treated previously, such as demonstrability or definability, o n e h a s been able to define t h e m only relative to a given language, a n d for each individual language i t is clear t h a t t h e one t h u s obtained is not t h e one looked for. For t h e concept of computability however, although it is merely a special kind of demonstrability or definability, t h e situation is different. Giidel referred t o t h e situatioii as ‘a kind of miracle’. These facts point out t h e exceptional importance of t h e class of recursive functions, a n d have led (see t h e ncxt subsection for historical notes) to propose t h e following as a working hypothesis:

Church’s Thesis (Church [1936], Turing [1936]) Every eflectively computable function is recursive. T h e Thesis, if true, would have a great relevance as a piece of applied philosophy, since it imposes a precise, mathematical upper b o u n d to t h e vague, intuitive b u t basic notion of algorithm t h a t underlies t h e concept of effective


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computability, an d t h a t has permeated technique and mathematical experience for thousands of years. Post [1944] emphasizes t hat if general recursive functions is t h e formal equivalent of effective calculability, its formalization may play a role in t he history of combinatorial mathematics second only t o t hat of t he formulation of t h e concept of natural number. In applications, t h e Thesis has a n essential use in metamathematics. Limiting t h e extension of t h e concept of algorithm allows for proofs of absolute u n s o lv a b ility : if we prove t h a t a function is not recursive then, by t h e Thesis, it is not computable by any effective means. Thus, t o see t hat a problem is effectively unsolvable, is enough t o faithfully translate it into a function, a n d prove t h a t this is not recursive. Like t h e classical unsolvability proofs, these proofs are of unsolvability by means of given instruments. What is new is t h a t in t h e present case these instruments, in effect, seem t o be t h e only instruments at man’s disposal. (Post [1944]) T h u s undecidability proofs rest on two conceptually different bases: a mathematical proof of recursive unsolvability (independent of the Thesis), and a n

appeal to the Thesis, to deduce from it absolute unsolvability. There is another avoidable use of the Thesis, in Recursion Theory. Giving a n algorithm for a function amounts, by t he Thesis, t o showing t h a t this function is recursive. Although theoretically not important, and in principle always avoidable (if t h e Thesis is true), this use is often quite convenient, since it avoids t h e need for producing a precise recursive definition of a function (which might be cumbersome in details). Strictly speaking, however, this use does not even require a Thesis: it is just a n expression of a general preference, widespread in mathematics, for informal (more intelligible) arguments, whenever their formalization appears t o be straightforward, and not particularly informative. We will d o this (and have already done it) throughout t h e book. T h e meaning of t he above formulation of Church’s Thesis is ambiguous in at least two respects. First of all, t h e statement can be taken as saying t h a t each effectively computable function is extensionally equivalent t o a recursive one or, more strongly, th at every effective rule is intensionally equivalent to, say, some program for a n idealized computer. Following Kreisel [1971]we will distinguish t h e two meanings, and refer t o them, respectively, as Thesis and Superthesis. Second, an d this is t he crux of t h e matter, there are various possible meanings for t h e word ‘effective’, partly depending on one’s philosophy of mathematics. ECxtremal attitudes ar e possible. One (Church [1936]) is t o take recursiveness as a precise definition of t he otherwise vague notion of effective computability: this makes t h e Thesis empty. An opposite one (Kalmar [1959]) is

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t o consider effective computability as an open concept, t h a t can only be successively approximated: this only allows for partial verifications of t h e Thesis, relative to given approximations (although it would allow for a disproval of it). T h e popular attitude is, however, to consider t h e facts t h a t various attempts t o characterize t he notion of effectiveness have all led to t h e sam e class of functions, and th at no counterexample to t he Thesis has ever been found, as conclusive arguments in favor of it, a n d to regard t h e matter as (positively) settled. Kreisel [1972] has stressed t h e fact t h a t equivalence of at t em pt s is not particularly significant (there might be systematic errors), a n d t h a t only t h e intrinsic values of each model can be relevant (one good reason is better t han many bad ones). Church’s Thesis can be analyzed from various points of view, some of which dealt with in th e special issue of t h e Notre Dame Journal of Formal Logic (vol. 28, no. 4, 1987) on t h e subject. T h e task we set for ourselves in this section is t o analyze some meanings of t h e word ‘effective’, a n d t o discuss t h e (lack of) evidence of t h e Thesis for these meanings. Precisely, we consider physical and biological computers. A physical computer, as described here, is a discrete physical system together with a theory for its behavior (according to which t h e values are under experimental control). We restrict our attention to discrete systems because we are considering discrete functions (from natural numbers to natural numbers), although continuous systems can be treated via approximations (see below). T h e fact th at we have a theory (physical laws) to work with, is what makes t h e Thesis in this case less pretentious, therefore less simpleminded, than in t h e original intended meaning (considered afterwards): it allows us t o compare a n abstract model of computability with descriptions of classes of physical devices. Obviously we d o not question here t h e validity of t h e world description in terms of (present day) physical laws: t h e relevance of our discussion will be proportional t o t h e degree of confidence we have in it. Since Turing machines are locally deterministic devices, t o ask whether any physical computer computes only recursive functions actually splits into two questions: it means first t o determine how far a particular model of determinism accounts for all of it, then t o establish the extent of determinism itself. Clearly t h e former is less problematic, and it therefore produces a more satisfactory analysis. For th e biological computer, we d o not have yet a theory, and discussions of human computability are mostly rambling talk. We pursue bot h t h e synthetical (bottom-up) and the analytical (top-down) approaches, by analyzing t h e brain structure and theories of constructive reasoning, but we reach a dead end soon in both cases. Before we plunge into our work, we would like t o warn about what effective computability does not mean: it is not practical (feasible) computability. T h e relationship between these two notions is t he distinction between Recursion


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Theory a n d Computer Science, i.e. between ideal computers a n d real ones. T h e issue in Computer Science is not Church’s Thesis (whether t h e class of recursive functions is broad enough), b u t its dual (which recursive functions a r e practically computable): from this t h e a t t e m p t to define restricted versions of recursiveness, like polynomial-time computability (considered in C h a p t e r VIII). Actually, from a strict point of view, practical computability is not even interested in asymptotical behavior, a n d it will never use infinitely many values. I n t h i s respect even t h e a t t e m p t to restrict t h e class of recursive functions to other abstract models may be irrelevant.

Historical remarks Post was working, at t h e beginning of t h e Twenties, toward a general formulation of t h e undecidability results h e h a d obtained. He defined t h e notion of canonical systems (p. 143) as a n abstraction of t h e notion of formal s y s t e m , a n d proposed (on t h e basis of reductions h e had of known formal systems to canonical ones) t h e identification of t h e notions of a set of strings effectively generable on o n e side, a n d generable by canonical systems on t h e other. T h i s is equivalent to saying, in modern terminology, t h a t t h e effectively generable sets a r e t h e recursively enumerable ones, a n d it is t h u s indirectly equivalent to Church’s Thesis (for partial functions). However, Post had a Platonist philosophical view, a n d saw his proposal as something t h a t h a d to be proved somehow, by a kind of psychological analysis of t h e mental processes involved in combinatorial niathematical processes. I n particular, h e believed t h a t t h e analysis he had at t h e moment was ‘fundamentally weak’, a n d t h u s t h a t t h e proposal was not completely convincing. All this work (Post [1922]) was left unpublished, a n d so did not influence later developments. At t h e beginning of t h e thirties Church formulated t h e A-calculus, in a foundational a t t e m p t to develop a system of logic from t h e primitive notion of function (Section 6 ) . I t gradually turned o u t (in 1932-33) t h a t there was a natural way to represent integers in A-notation, a n d t h a t a great number of functions were A-definable (ultimately t h a t all t h e recursive ones were, Theorem 1.6.5). I n 1934 Church proposed his Thesis (Church [1936]),as a mathematical definition of t h e informal concept of computability. Meanwhile Godel, dissatisfied with Church’s approach, believed (somewhat following Hilbert [ 19261) t h a t t h e computable functions could all be defined by some general kind of recursion. This again turns o u t to be equivalent to Church’s Thesis, through t h e Fixed-Point Theorem. Godel [1934] even vent u r e d to forniulate t h e notion of Herbrand-Godel computability (Section 2) as


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a test, b u t was not at all convinced t h a t this concept really comprises all possible recursion. His proposal was similar to Post’s: to analyze t h e notion of computability, aiming at the isolation of its essential features. T h i s was done (in a way satisfactory to Godel) b y Turing [1936], who unaware of t h e work referred t o above - proposed his model of a n a b s t r a c t computer (Section 4), a n d a n equivalent version of Church’s Thesis. Simultaneously a n d independently, Post [1936] attained a very similar analysis a n d stated t h a t a fundamental discovery in t h e limitations of t h e mathematicizing power of Homo Sapiens has been made. G d e l thought otherwise (see Davis [1965],p. 73): T h e results mentioned . . . do not establish any b o u n d s for t h e power of human reason, b u t rather for t h e potentiality of p u r e formalism in mathematics.

For more information on t h e history of Church’s Thesis see Kleene [1981], [198la], [1987], Davis [1982], Shanker [1987], Webb [1980] a n d t h e original papers in Davis (19651.

Computers and physics T h e notion of a deterministic reality t h a t evolves according to mathematically explicit laws is typical of classical mechanics. Galilei [1623], [1638] introduces t h e modern scientific methodology of experimenting in order to verify t h e results of theoretical reasoning, a n d stresses t h e importance of mathematics (‘the language t h e book of nature is written in’). Newton [1687]achieves a n informal axiomatization of mechanics, for t h e first t i m e unifying large t r a c t s of experience into a coherent, picture. W i t h him t h e mechanization of t h e world picture (see Dijksterhuis [1961] for a n historical account) is accomplished: a system with k degrees of freedom needs only 2k parameters (positions a n d moments) to completely specify every value of physical quantity for t h e system at a given t i m e a n d t h e evolution in t i m e of t h e system s t a t e . If s o m e of t h e parameters a r e unknown, by averaging over t h e m in s o m e way it is still possible to obtain statistical prevision (like in thermodynamics, where the hidden 2k parameters needed to describe a system of k molecules produce a statistical description in terms of pressure a n d temperature alone). Plank’s discovery in 1900 of energy packets ignited a new physics (quantum mechanics, see Feynman, Leighton a n d Sands [1963] for background), with philosophical foundations as distant from those of classical mechanics as they can be. T h e concept itself of reality is at stake: m a t t e r h a s a double


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appearance] as waves and as particles (Einstein, De Broglie), and its physical quantities cannot in general be simultaneously measured with absolute precision (Heisenberg). A system with Ic degrees of freedom is now described by a wave function \ k ( q l , . .. ,q k ) , which still evolves deterministically in time (Schrodinger), but allows only statistical prevision on the values of physical quantities for the system at a given time (Born). This last fact can be variously interpreted as necessary (change at subatomic level is casual, and can only be accounted for probabilistically), accidental (as in t h e case of thermodynamics, hidden variables might deterministically account for change, a n d give quantum-mechanics states as average), or contingent (the wave function represents not only possibilities, but realities in simultaneously coexisting worlds: a measurement, forcing a possibility into a n actuality, corresponds t o choosing a path in t h e tree of all possible universes, see DeWitt a n d Graham [1973]).

Classical mechanics T h e first aspect t h a t we examine of Church’s Thesis can be phrased as follows: t h e notion of recursiveness (a technical isolation of a restricted class of mechanical processes) captures the essence of mechanism. We can formulate, more precisely:

Thesis M (for ‘mechanical’) (Kreisel [1965])The behavior of any discrete physical system evolving according to local mechanical laws is recursive. This clearly implies our real interest: t h a t any function computable by such a device is recursive as well (each output being obtained by a finite iteration of

a recursive procedure applied t o t h e input). T h e Thesis formulation in terms of behavior of physical systems, in addition t o being more general, has t h e advantage of being directly suitable for analysis (since we d o not need t o know details on how the device computes a function). T h e Turing-Post analysis of Section 4 is certainly not sufficient to prove Thesis M since, being explicitly patterned on human behavior, it sees computations as well-ordered sequences of atomic steps, and thus (at least) it does not account for parallel computations. Arguments in favor of Thesis M fall into three distinct categories] which we analyze separately. a) A general theory of discrete, deterministic devices T h e analysis (Church [1957], Kolmogorov and Uspenskii (19581, Gandy [1980]) starts from the assumptions of atomism and relativity. T h e former reduces t h e structure of matter t o a finite set of basic particles of bounded dimensions] a n d thus justifies t h e theoretical possibility of dismantling a machine down t o


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a set of basic constituents. T h e latter imposes a n upper bound ( t h e speed of light) on t h e propagation speed of causal changes, a n d t h u s justifies t h e theoretical possibility of reducing t h e causal effect produced in a n instant t on

a bounded region of space V , to actions produced by t h e region whose points a r e within distance c . t from some point of V . Of course, t h e assumptions do not take into account systems which a r e continuous, o r which allow unbounded action-at-a-distance (like Newtonian gravitational systems). Gandy’s analysis shows t h a t the behavior is recursive, for a n y device with a fixed bound on the complexity of its possible configurations (in t h e sense t h a t both t h e levels of conceptual build-up from constituents, a n d t h e number of constituents in any structured part of a n y configuration, a r e bounded), and jixed finite, deterministic sets of instructions for local and global action ( t h e former telling how to determine t h e effect of action on structured parts, t h e latter how to assemble t h e local effects). Moreover, t h e analysis is optimal in t h e sense t h a t , when made precise, any relaxing of conditions becomes compatible with any behavior, a n d it thus provides a sufficient a n d necessary description of recursive behavior.

b) Numerical approximations of the local differentiable equations of classical mechanics T h e work in classical mechanics, from Newton to Hamilton, has led to a description of t h e evolution of mechanical systems by local differentiable equations. More precisely, a conservative Hamiltonian system is defined, in local coordinates, by Hamilton’s equations:

where q = ( 4 1 , . . . , q k ) a n d p = ( p l , . . . , p k ) a r e t h e vectors of, respectively, positions a n d momenta of t h e system, k being t h e number of degrees of freedom of t h e system. T h e n t h e evolution in continuous t i m e of t h e system state s (completely describing t h e relevant variables of t h e system) can be expressed by a vector differential equation of t h e form S = f(s). By assuming sufficient smoothness conditions on t h e derivative involved, a n d stepping from continuous to discrete time (in which t h e evolution of t h e system is sampled at regular, sufficiently small, discrete time intervals) we can linearly approximate t h e r a t e of change given by t h e previous equation, as

s(t

+ A t ) N s ( t ) + f ( s ( t ) ). A t .

By taking A t as t h e unit interval of sampling, we get

s(t

+ 1) = s(t) + f ( s ( t ) )


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a n d t h i s gives, for mechanical (not necessarily discrete) systems, a recursively described system evolution (in t h e form of a simultaneous recursive definition of all t h e relevant variables explicit in s) (Kreisel [1965]). Note t h a t t h e other, equivalent, way t o describe t h e evolution of systems in classical mechanics, namely by global variational principles (like Maupertuis’ principle of least action), does not, seem to b e useful for a similar analysis, because of i t s teleological approach (see Von Neumann [1954]).

c) Discrete models for classical mechanics I n classical mechanics discrete data (coming froni experiments) a r e used to build continuous models, from which discrete data have to be deduced by numerical approximation methods. T h e s t e p from discrete to discrete through a continuous model seems logically unbalanced: as Feynman [1982] p u t s it,

It is really true, somehow, t h a t t h e physical world is representable in a discretized way, a n d . . . we a r e going to have to change t h e laws of physics.

Discrete models, studying the dynamical behavior of systems entirely in terms of (high speed) arithmetic, have been obtained for classical mechanics, including special relativity a n d conservative Ilamiltonian theory (Greenspan [1973], [1980], [1982], L a Budde [1980]). Their dynamical equations a r e difference (opposed to differential) equations, whose solutions a r e discrete functions. T h i s approach still yields various conservation a n d synimetry laws of continuous mechanics, a n d it also has direct applications to non-linear physical behavior. Related t o this, cellular automata have been investigated as a basis

for the representation of partial differential equation models in a direct computer simulation, again avoiding indirect numerical approximation (Vichniac [198,4],Toffoli [1984]). See also (3rd-Smith a n d Stephenson [1975] for a general t r e a t m e n t of computer simulation of continuous systems.

To s u m u p t h e discussion above, it is plausible that the behavior of a discrete physical system, evolving according to the local and causal laws of classical mechanics, can be simulated by a computer, and it is thus, in particular, recursive.

Probabilistic physics We t r y now to formulate Church’s Thesis for abstract machines, in t h e most general way. We will have to account for analog computers, t h a t is any physical system computing some function, by representing numerical data ‘by analogy’ (based on any physical, a n d possibly continuous, quantity, like intensity of a n electrical current, or rotation angles of a watch hand). More

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precisely, a n analogue computat,ion is a combination of physical processes, behaving (mathematically) in t h e same way as some other process, which is t h e real object of study, b u t which for some reason is more manageable or b e t t e r observable t h a n it (e.g., because of difference in scale). In t h e e x t r e m e case, a n y physical process is a n analog calculation of i t s o w n b e h a ~ i o r . ~We t h u s formulate:

Thesis P (for ‘probabilistic’) (Kreisel [1965]) A n y possible behavior of a discrete physical s y s t e m (according t o present d a y physical theory) i s recursive. Possible behavior means a sequence of states with non-zero probability: we cannot simply talk of behavior according to present d a y physical theory, because this (as opposed t o classical mechanics) is formulated also in ternls of probability. We collect our observations under categories parallel to those used for classical mechanics.

a) A general theory of analog machines Nothing similar in spirit to t h e theory of Gandy [I9801 for discrete deterministic devices has yet been developed. A first step has been undertaken by Shannon (19411, who generalized t h e iiotion of finite a u t o m a t o n into t h a t of general purpose analog computer. This is a device consisting of electronic circuits, a n d a series of black-boxes (hooked u p with lots of instantaneous feedback), of four elementary kinds: constant (producing any desired constant voltage), adder a n d multiplier (producing sum a n d product of t h e inputs), a n d integrator u(s)dv(s) C , where C is t h e (producing, given inputs u a n d v , t h e o u t p u t ‘initial setting’ of t h e integrator). Once t h e connections a n d t h e initial settings a r e made, t h e device is permitted to r u n in real time, a n d a n y voltage t h a t can be read in t h e circuit (as a function of time) is a n o u t p u t . A characterization of t h e behavior of these devices has been obtained (Shan11011 [1941], Pour El [1974], Lipschitz a n d Rubel [1987]): a f u n c t i o n f(x) is t h e output of a general purpose analog computer i f a n d o n l y if it is differentially algebraic, i.e. t h e solution of a n algebraic differential equation

s,’

+

where P is a polynomial (over t h e complex field) in all its variables. Such functions provide a n extremely rich class, including almost all t h e special functions 51n this case, Church’s Thesis amounts to saying that the universe is, or at least can be simulated by, a computer. This is reminiscent of similar tentatives to assimilate nature to the most sophisticated amilable machine, like the mechanical clock in the 17th Century, and the heat engine in the 19th Century, and it might soon appear to be as simplistic.


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in common use (algebraic, trigonometric, Bessel functions), and with strong closure properties (see Rubel [1982], Rubel and Singer [1985]), including the existence of analogues of the universal function (Rubel [1981]). Also, despite t h e fact t h a t some transcendental functions (notably, Euler’s r and Riemann’s 6) a r e not exact solutions of algebraic differential equations, any continuous function can be approximated, with arbitrary preassigned accuracy, by differential algebraic functions. This approach thus describes a wide.variety of physical phenomena, but it is still only a first step toward a general theory of analog computers. Some extensions have been recently proposed, e.g. allowing black-boxes for convolution (which would add memory t o t h e device, since convolutions involve t h e whole past history of their inputs and introduce time delays). Digital simulations with arbitrary precision (which are our real concern, since we talk about functions of integral, not real or complex, variables) should be possible, by replacing t h e black boxes by digital approximations t o them (e.g., integration can be performed by some appropriate numerical integration, say via Simpson’s rule), b u t details have not yet been worked out.

b) Analysis of the formulation of probabilistic physical laws T h e dynamics of classical physical systems with probabilistic behavior may be described by Markov chains, which consist of a finite set of states { q l , . . . ,q n } , together with a n x n stochastic matrix P = ( p i j ) 1 5 , , j j n , whose interpretation is: if t h e system is in a given state qi at a certain instant of time, t h e probability t h a t it be in state qj at t h e next instant (in a discrete time scale) is p i j . A system described by a Markov chain satisfies Thesis P, and actually something more general holds: any sequence of states with non-zero probability, in a stochastic

process with infinitely m a n y discrete states and recursive matrix of transition probabilities, i s recursive (Kreisel [1970a]). T h e r e a o n is simply t h a t such a sequence is a n isolated branch of a finitely branching recursive tree (the tree of possible sequences of states), since there are only finitely many possible sequences of a given non-zero probability. T h e remark just made teaches a more general lesson. Suppose we consider a structurally stable system, i.e. such t h a t slight changes of t h e parameters in t h e equations describing t h e system behavior produce only slight changes in t h e behavior itself. T h e stability of the solutions tends t o require t h a t they be isolated in t h e relevant spaces, and if these spaces are recursively described, t h e solutions a r e recursive as well. Thus it is likely that i f Thesis P fails,

counterexamples have to be looked f o r in unstable systems. It can be noted t h a t it is known t h a t some differential equations in recursive analysis (of t h e kind arising in the description of physical phenomena) have recursive data b u t no recursive solutions (see e.g. Pour El and Richards [1979],

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[1981], [1983]). These results are, however, not directly relevant to Thesis P, since their data a r e mathematically concocted, a n d do not apparently arise from t h e description of physical phenomena (see Kreisel (19821 for a review).

c ) Deterministic models of quantum mechanics We have already noted above t h a t quantum mechanics is not deterministic, as it stands. Hidden-variables theories were postulated to leave o p e n t h e possibility of a deterministic description of subatomic phenomena: their existence would prove quantum mechanics observably inadequate, b u t at t h e s a m e t i m e q u a n t u m theory - although incomplete - could be complemented to obtain a full description of individual systems. Impossibility proofs of t h e existence of hidden-variables theories h a d been proposed, from Von Neumann [1932] on, b u t with unsatisfactory features analyzed in Bell [1966]. A breakthrough was Bell [1964]: h e proved that realism and hidden variables are n o t only philosophically, but also theoretically incompatible with quantum theory. He devised (in t h e style of Einstein, Podolski a n d Rosen [1935]) a simple experiment, a n d computed probabilistic lower b o u n d s to t h e outcome predictions, assuming t h a t well-defined states really exist, prior to their observation. This bound is greater t h a n t h e o n e obtained by q u a n t u m theory considerations. Recently (1981-82) t h e experiment h a s been actually carried o u t , a n d seemingly conclusive evidence provided t h a t the quantistic predictions are correct (see Mermin [1985] for a n elementary description, a n d references). T h i s moves t h e incompatibility of realism with q u a n t u m theory from philosophical a n d theoretical ground t o t h e experimental one, a n d seems to settle t h e matter. At first sight it might seem impossible to simulate Bell’s experiment deterministically, since t h e theoretical outcome predictions would clash with experimental evidence, b u t we should not forget t h a t these predictions are obtained by using a particular kind of inductive inference based, in particular, on classical probability theory. Now t h e s a m e inference theory is used in q u a n t u m mechanics, a n d this flatly produces its incompatibility with determinism. But, as Feynman [1982] points o u t , there could be a problem with probability theory itself, at quantum level: we assume t h a t we can always do a n d repeat a n y experiment t h a t we want, without taking into account t h e constrictions (stressed by q u a n t u m thcory!) imposed by t h e fact t h a t we a r e all p a r t of t h e s a m e universe, a n d t h a t t h e universe does not remain t h e same. O n t h e o t h e r h a n d , we do not know of any version of Bell’s experiment t h a t avoids probabilistic computations. Bwides probability, logic is t h e other tool used in t h e inductive inference of Bell’s theorem, a n d classical logic itself seems to be inadequate to describe phenomena at quantuin level. See e.g. Birkoff a n d Von Neumann [1936], where


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i t is argued t h a t t h e experimental propositions concerning a system in classical mechanics form a Boolean algebra, while (due to t h e fact t h a t only compatible observations commute, a n d incompatible observations cannot be independently performed) t h e y a r e a complemented b u t nondistributive lattice in quantum mechanics.

To sum u p t h e discussion, we have only scarce evidence in favor of Thesis P and, despite the fact that n o outright refutation exists, there is plenty of room t o doubt its validity.

Computers and thought T h e reduction of soul to (atomistic) physics has a n old pre-Socratic tradition, centering around Leucippus a n d Democritus. T h e Socratic revolution, a n d t h e standing success of Plat0 a n d Aristotle, has led to a tradition of organismic physics t h a t has left little space for pure mechanism until more recent times. A notable exception was Lucretius, who devoted two books of his De Rerum Natura to a n atomistic account of t h e mind a n d its functions. Descartes [1637] laid t h e foundation of t h e modern mechanistic world view, by trying to devitalize t h e human organism as much as was logically possible, in particular by including into physics (as he envisaged it) a great deal of what later c a m e to be called psychology, but not t h e mind itself. He saw self-consciousness a n d language sophistication (in particular, t h e ability to see t h e meaning of signs a n d events) as t h e privilege a n d exclusive ability of a n immaterial, unextended mind. Hobbes [1655] provided t h e dissenting note: by relying on t h e apparently effective manipulation, in reasoning, of names as symbols for thoughts, a n d on Pascal’s construction of t h e first calculating machine in 1645, he defended a global mechanism, a n d did not hesitate to obliterate t h e difference between mind a n d matter. In his extreme dedication to mechanism, Hobbes was a rather lonely figure in his day, b u t with t h e advent and t h e success of machines, it was inevitable t h a t mechanism would attract more advocates. L a Mettrie [1748] provided a most notorious a t t e m p t , aimed at wholesale reductionism. Once t h e terms of t h e debate had been set up, endless arguments developed, a n d new life to t h e dispute has been provided by modern advances in t h e areas t h a t supported Hobbes a n d Descartes, respectively. On t h e one hand, much current mathematical practice has been formalized (from Boole t o Bourbaki), a n d thus indeed mechanized (and Godel Completeness Theorem [1930] shows t h a t , for what concerns first-order logic, t h e formalization is complete). M o r e over, t h e quality of computers (from Babbage to t h e Fifth Generation) has improved enormously, a n d machines a r e now capable of quite sophisticated behavior themselves. O n t h e other hand, t h e undecidability a n d incompleteness

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results (Section 11.2) expose the limitations of t h e formalization a n d mechanization programs. Moreover, these results have sometimes been used t o infer the superiority of men over machines, basically with t h e following arguments:

Undecidability. Church’s Theorem 11.2.18 shows t h a t first-order logical reasoning is not mechanically decidable. It might thus appear t h a t man, t h e bearer of this reasoning, is capable of nonmechanical behavior, and thus not a machine. It is easy t o see t h e weak point of this argument: a mechanism is such because of local mechanical behavior, and the Completeness Theorem for Predicate Calculus (Godel [1930]) does indeed show t h a t classical logical reasoning can be formalized, and thus simulated by 1ocally mechanical steps. But a mechanism does not need t o have a global, mechanically predictable behavior, as discussed on p. 151. Undefinability. Tarski’s Theorem (see p. 166) shows that, for a classical formal system, truth is not representable in it. Again, it would seem t h a t man has a notion of truth, and thus that thought has nonmechanical elements. The difficulty of this argument is two-fold: on t h e one hand, it is only global truth that it is not representable, while for each fixed bound of logical complexity there is a representable notion of truth (at t h e next level of complexity); on t h e other hand, not only does man not appear to have a global notion of truth: he even seems unable to handle (by direct intuition) more than four or five alternations of quantifiers, and hence local truth itself, beyond very small levels of complexity. Incompleteness. Godel’s Theorem 11.2.17 tells us that any consistent and sufficiently strong formal system is incomplete, in the sense t h a t it does not prove some numerical sentence which we know is true. It would seem that man, being able to produce, for any machine (formal system), a task that can be solved by him but not by t h e machine, is not himself a machine. The first objection t o this is t h a t the proof of t h e result is effective, i.e. there is a machine that, given the number of a formal system, produces the undecidable sentence (this effectiveness is actually one of t h e crucial features of Godel’s result, since - by showing t h e incompleteness of every sufficiently strong formal system - it points t o inadequacies in t h e concept of formal system itself). Post [1922] has noted that such a n effectiveness does not show up by chance: given an argument intended to prove that man can fool any machine, if this argument can be made sufficiently precise, then it becomes itself mechanizable, a n d it backfires. Another important feature of Godel’s proof is that t h e undecidable sentence is shown to be true only under the hypothesis of t h e consistency of


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the system. Certainly the problem of deciding t h e consistency of given formal systems is not recursively solvable, but there is no hint that man has a decision procedure for such a problem (and hence for t h e truth of the undecidable sentence relative to a given consistent system). If it were so, this would be a direct refutation of mechanism. Finally, although Godel’s Theorem does not by itself refute mechanism, it does so when combined with t h e assumptions that there are no numbertheoretical questions undecidable for the human mind, and that the mind is somehow consistent. But this does not solve the problem, it just moves it t o a different level. Also, although the second assumption seems quite reasonable, t h e first one is certainly more problematic and controversial, even if understood (as it should be) as saying t h a t the human mind has no limitation regarding the problems it poses itself in the limit (given enough time and resources), and taking the words ‘deciding a question’ as meaning ‘settling the problem’, possibly by showing it t o be unsolvable.

See the papers in Anderson [1964], and Hofstadter and Dennett [1981] for some discussions on the relevance of these topics for t h e mind-brain debate, and Popper and Eccles [1977] for a modern philosophical and neurological introduction to the latter.

The brain Our first global approach to Church’s Thesis for human thought is to look at t h e brain, the physical basis of intelligence. We begin by discussing:

Superthesis B (for ‘brain’) (Descartes (16371)The brain is a machine.

It should be stressed that the statement is not t o be taken as reducing t h e brain complexity t o the roughness of present-day machines: a proof of Superthesis B would probably revolutionize the contemporary idea of machine, and precisely in this lies its interest. In particular, Turing (see p. 164) has stressed the significance of the limitation results (Section 11.2) in showing how a purely deterministic model of machine cannot fully account for intelligence. It is, however, instructive t o compare the brain and the most sophisticated available machine, the computer6 (for information on t h e brain see Von Neu‘History teaches us that this should not be taken too literally: Descartes 116641 saw the brain as a complex hydraulic system, permitting the periodic flow of vital spirits from the central reservoir into the muscles; Pearson [1892]described it as a telephone exchange, consisting of fixed wires and mobile switches (a model that proved useful for an understanding of spinal reflex response); Ashby [1952]provided a cybernetic model as a collection of selfcontrolling systems. The computer model might look as simplistic in the not too distant future.

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mann [1958], Arbib [1964], [1972], Eccles [1973], Young [1978], Kandel and Schwartz [1981]). Computers are electromagnetic devices with fixed wiring between more or less linearly connected elements, operating mostly sequentially, a n d at high speed. Brains are dynamical electrochemical organs with extensively branched connections, operating with massive parallel action at a slow speed and low energetic cost, continuously capable of generating new elements, a n d perhaps making new connections. T h e architectural differences a r e great: see e.g. Haddon a n d Lamola (19851 for a survey of t h e technical foreseeable advances regarding chip dimensions. T h e holistic logic employed by t h e brain is simply o u t of reach: we d o not know how it concentrates on essential information and experiences it as structured. We are thus forced t o suspend judgement on the validity of Superthesis B , until enough might be known on these problems. If ever, since it is certainly conceivable (La Mettrie [1748], Von Neumann [1951]) t h a t , due to t h e extreme complexity involved, a linguistic (mechanical) description of t h e cerebral functions might be simply unfeasible or uninforming: t h a t is, t h e system itself could be its own most intelligible description (in t he terminology of p. 151, the brain

could be a random object). On th e positive side there are some results worth mentioning which show t h e mechanical behavior of some simplified neuron nets. As a first approximation t o t h e great complexity of natural neuron systems, McCulloch and Pi t t s [1943] introduce regularity assumptions for artificial neurons: they are infallible all-or-nothing devices with fixed synaptic threshold, firing synchronously at discrete intervals (when t he algebraic sum of t h e adjacent neurons effects reaches th e threshold). T h e behavior of an isolated system of artificial neurons is completely characterized by t h e input conditions, a n d t h e system t hen works as a n abstract machine (actually, this is simply a n equivalent description, and t h e original one, of finite automata, see p. 52). Since t h e control box of a Turing machine can be regarded as a finite automaton, it can thus be seen as a n

abstract brain of the Turing machine. W e thus have two complementary analyses: Turing’s analytical (top-down) approach describes the functioning of the computing device, without further analyzing the way it is actually built, while McCulloch and Pitts’s synthetical (bottom-up) analysis shows how t o obtain the same functioning b y organizing, in a possibly very complex way, simple parts o f described structure. Much work has been done toward a relaxation of t he restrictive assumptions on artificial neuron nets. Von Neumann [1956], a n d Winograd a n d Cowan [1963] consider systems in which t he unreliability of the components does not affect t h e reliability of t h e whole net, by transmitting t h e same information in a highly redundant way, along multiple parallel lines or, respectively, blocks of components. Hebb [1949] and Eccles [1953] permit variable synaptic thresholds: their systems have feedback information, by which they can


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somehow realize whether t h e o u t p u t s a r e conforming to t h e expectations. By trials a n d errors it is possible to gather sufficient information a n d determine thresholds t h a t give t h e expected o u t p u t , t h a t is t h e systems have t h e ability to learn. T h i s represents short-term memory by t h e variable states of t h e system, a n d long-term memory by t h e level of t h e synaptic thresholds. Hopfield [1982], [1984] has considered very general neural networks, with backward coupling (where neurons can a c t indirectly, through other neurons, on themselves), asynchronous firing, graded continuous response (in t h e form of a sigmoid input-output relation, as opposed to a 0,l-valued step function) a n d integrative t i m e delays (due to capacitance). He has shown t h a t (if t h e connect,ions between neurons a r e symmetric) these general networks a r e still computational devices, since a n y set of inputs leads to stable states. T h i s provides a model of content-addressable memory, where a stable state represents memorized information, t h a t can be retrieved by setting t h e right input t h a t would lead to it. Various other models have been proposed, see e.g. Arbib (19731, Bienestock, Fogelman SouliC a n d Wekbuch [1985], Selverston [1985], McClelland a n d Rumelhart [ 19861, a n d recent issues of Bzological Cybernetics. T h e models j u s t discussed a r e all digital (based on neuron nets), a n d have been obviously inspired by t h e structure of digital computers (finite a u t o m a t a , in particular). B u t it is known t h a t p a r t s of t h e central nervous system function analogically, e.g. many neurons never fire, a n d are engaged in different activities (see Rakic [1975], Shepherd [1979], Roberts a n d Bush [1981], Crick a n d Asanuma [1986]). To be closer t o reality, t h e digital model of t h e brain should t h u s be supplemented, a n d substituted by a hybrid one, partly digital a n d partly analog. A natural approach seems to be t h e use of t h e general purpose analogue computer of p. 110 (Rubel (19851): neurons or neuron-circuits t h a t perform t h e functions of t h e black-boxes have been already identified, a n d t h u s at least t h e basic components of t h e general purpose analog computer a r e present in t h e central nervous system. It is only fair to note t h a t (some of) t h e results quoted might be more relevant, to t h e problem of whether machines can think (in t h e operative sense, introduced by Turing [1950], of being able to simulate aspects t h a t we believe to be characteristic of thought) t h a n to our discussion of Superthesis B. I t is not clear whether t h e models proposed above really describe t h e brain’s own s e lutions to problems of unreliability, learning, memorization a n d organization of information. IIowever, they a r e certainly relevant to t h e following ‘Promethean irreverence’:

Thesis A1 (for ‘Artificial Intelligence’) (Wiener [1948], Turing [1950])T h e mental functzons can be simulated by machines.

All work in Artificial Intelligence (pattern recognition, language reproduction, problem solving, theorem proving, game playing, learning a n d under-


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standing) produces inductive evidence for Thesis AI. Note t h a t Thesis A1 is not simply t h e extensional version of Superthesis B. T h e step from t h e latter to t h e former is not automatic: it requires psyche logical materialism, in t h e form (La Mettrie [1748]) of human t h o u g h t being completely determined by t h e brain, with no intervention of a n extraphysical mind. But, stepping down to t h e simulation level introduced by Thesis AI, we a r e not interested - in our discussion of Church’s Thesis - in i t s full version, since our present concern is just mathematical thought.

Constructivism We t h u s isolate our real interest in:

Thesis C (for ‘constructive’) (Kleene [1943], Beth [1947]) Any constructive function is recursive. In view of t h e discussion in t h e last p a r t of t h e previous subsection, even establishing Superthesis B would not automatically establish Thesis C. Conversely, failure o f Thesis C would not disprove Superthesis B. Also, a failure of Thesis C docs rio t, disprove rriaterialism, unless ( t h e extensional version of) Superthesis B holds simultaneously. Actually, establishing b o t h Superthesis B a n d Thesis A1 is probably as far as we can go toward a possible justification of ( t h e above form of) materialism. It is olwiously impossible to disprove t h e existence of mind without using Ockham razor, i.e. beyond showing its unusefulness in explaining thought activities. O n t h e other hand, as Godel has suggested (Wang [1974], p. 326), i t might be possible to disprove mechanism by showing t h a t there is not sufficient s t r u c t u r e (at nerve level) to perform all tasks actually performed by m a n . We t u r n now to a discussion of Thesis C. A first step h a s already been carried o u t by Turing a n d Post, with a n analysis of routine computations (Section 4). T h i s provides, at t h e s a m e time, more a n d less of w h a t we need: it gives a n intensional argument, b u t it concerns only a portion of t h e intended meaning of ‘constructive’. We can however say t h a t in t h e limited ront,ext to which it applies, t h i s analysis is conclusive W e isolate what is proved in t h e following:

Superthesis R (for ‘routine’) (Turing [1936], Post [1936]) Any computation perfonnea‘ b y a n abstract human being working i n a routine way, is isomorphic to a computation performed b y a Tiiring machine. But t h i s is still a far cry from Thesis C. We cannot rely on t h e analysis of mcchanical reasoning given by Turing machines: constructive and mechanical


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are apparently independent concepts. It seems t h a t we understand (by grasping abstract objects) nonmechanical rules, a n d do not understand rules which (although mechanical) a r e too long or too detailed. We t h u s have to lean on analyses of t h e notion of mathematical constructive reasoning. T h e r e a r e many of t h e m , based on different approaches. Following t h e detailed treatments of (and, for more information, referring to) Godel [1958] a n d Kreisel [1965], [1966], we j u s t hint at t h e basic features of t h e most popular, increasingly more comprehensive ones: formalism (Frege, Russell) considers as constructive what can be done on physical objects (symbols), by purely combinatorial (hence mechanical) means (formal rules of derivation for symbols sequences).

finitism (Hilbert) accepts also what can be seen by p u r e intuition, provided only concrete (spatio-temporal) objects a r e used, a n d c l a i m t h a t any thinking process of t h i s kind must be finite (though not necessarily mechanical).

intuitionism (Brouwer) allows for whatever is mentally understandable, possibly using also abstract objects (like higher-type objects, or generalized inductive definitions). Nonconstructive reasoning enters only into platonism, which regards mathematical objects not as thoughts b u t as real objects, t h a t t h e mental process does not create, b u t only discovers. Their properties a r e t h u s perfectly defined as those of physical objects, a n d this justifies use of, for example, t e r t i u m non d a t u r a n d actual infinity. Formalism is directly related to Thesis C . On t h e o n e hand, t h e very formalistic program (of compressing mathematical knowledge into formal systems) rests on t h e belief t h a t some form of t h e Thesis holds ( t h a t t h i s knowledge can be mechanically reproduced); on t h e other hand, since everything computable in formal systems is recursive (by arithmetization, see I.7.7), each success of formalism is partial proof of t h e Thesis validity. Thesis C i s true when constructive is taken in the finitistic meaning as well: indeed, a finitistically defined arithmetical function is certainly (as shown by a n analysis of computations) finitely defined by a system of equations (Section a), a n d hence recursive (by 1.7.12). T h u s the whole problem of Thesis C lies in the intuitionistic meaning of constructive (law-like) function. Since allowing for abstract objects (which a r e not necessarily finitely representable) might make arithmetization of t h e involved mental processes troublesome, we will concentrate our discussion on formal systems capturing aspects of t h e intuitionistic (constructive) reasoning. T h e fact t h a t formal systems usually capture semantical notions of reasoning only extensionally is not important here, since t h e Thesis is precisely extensional.

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T h e technical advantage of formal systems is not only a m a t t e r of convenience (of having a syntactical, concrete version for semantical, a b s t r a c t notions): it could be instrumental in outright proving Thesis C. For this, since we know t h a t only recursive functions a r e representable in formal systems (by arithmetization), it would be enough to find a formal system equivalent to constwctiwe arithmetical intuition (by Kreisel [1972], even a recursive transfinite progression of formal systems would suffice). T h i s is certainly a delicate point: such a n equivalence proof might require unfamiliar principles of evidence, a n d would certainly provide a better insight into t h e notion of constructive validity. O n t h e other hand, failure of Thesis C would show t h e unfeasibility of t h i s reductionist program. Of course, no consistent formal system can be arithmetically complete in t h e classical sense, by 11.2.17 (and t h e s a m e holds for recursive progressions of formal systems, by Feferman a n d Spector [1962]). B u t this is not relevant to t h e reductionist program, since we do not expect constructive intuition to be itself classically complete: t h e problem would be to succeed in deriving w h a t is constructively valid, not to decide (let alone constructively) everything.' T h e realization of this reductionist program does not a p p e a r easy, also in light of a result of Kreisel (19621, [1965], by which Thesis C implies t h a t t h e set of constructively valid formulas of first-order logic is not recursively enumerable. In particular, i f Thesis C holds then there is no formal system capturing constructive logical validity: thus, if a formal system capturing constructive arithmetic validity exists, it cannot be obtained by j u s t extending (by means of arithmetical axioms) a logical system t h a t can be detached from it by recursive means. We t h u s have t h e amusing situation t h a t , in t h e process of searching for a complete formalization of constructive arithmetical reasoning, we might begin by a formulation of t h e purely logical constructive reasoning, a n d discover t h a t we lose t h e war by overwinning a battle: if we a r e completely successful with t h e logical formalization, t h e n Thesis C does not hold, a n d we a r e bound to fail in t h e arithmetical fornialization. Also, a n d t h i s is a situation with no analogue in classical mathematics, constructive validity for first-order logical formulas somehow depends o n what the constructive arithmetical functions are (in particular, on their being or not all recursive). Otherwise said, first-order constructive validity is actually a second-order notion. Short of proving Thesis C by t h e reductionist program, we may consider related questions, technically more manageable b u t , as we will see, more moderately interesting. We isolate two of t h e m . Given a n intuitionistic formal system F,we might see whether in F t h e recursive functions provide uniformization, in t h e sense of 11.1.13. T h i s is 'Note that, as G a e l himself has admitted (see Wang 119741, p. 324), it niight even be possible to find, or have already found, a formal system equivalent to full, not only constructive, mathematical intuition, although of mume, in this case, not provably so.

1.8 Church's Thesis

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expressed in a weak form by t h e following rule:

Church's Rule CR. If FF Vz 3 y R ( z ,y ) then, f o r some recursive function f and all z, FF R(?E,f(z)). By Kreisel [1972], C R is actually equivalent to t h e following:

Constructive 3-Rule. FF 3ycp(y)

* FF cp(?j), f o r some y .

To be s u r e (Kreisel [1972]), there a r e ad hoc intuitionistic systems for which CR fails. B u t t h i s does not automatically disprove Thesis C: it could merely be a s y m p t o m of incompleteness, since Vzcp(z, I(. might )) hold for s o m e recursive

f , b u t we might not be able to prove even its numerical instances in F. O n t h e other h a n d , no intuitionistic system is known to be incowistent with CR, something t h a t would disprove Thesis C . Moreover, for all current intuitionistic systems CR has actually been established, even in t h e stronger form:

t - ~V d y R ( z , y )

t-7 3eVz3z['&( e ,z , z ) A R(z,U(z))]

(see e.g. Kleene [1945], Kreisel a n d Troelstra [1970]). T h i s is however only a very indirect evidence in favor of Thesis C: it merely excludes t h e inconsistency of CR with these systems, a n d it thus shows t h a t they cannot be used to disprove t h e Thesis. We might be tempted (on t h e acceptable argument t h a t constructive validity of a n existential statement should exhibit explicit witnesses) to consider only those systems for which t h e Constructive 3-Rule holds. B u t , since we know t h a t t h e r e must be incompleteness (for any sufficiently strong arithmetical system, see I1.2.17), there is no reason to expect it to show u p necessarily soniewhere else t h a n in numerical instantiations of existential theorems. Only a formal system complete for constructive rawoning would automatically satisfy t h e Constructive 3-Rule (but, then not only C R would hold: Thesis C would indeed be true). Another property at least formally related to Thesis C , is its formal version:

CT1 VJf3eV'z3z['& ( e ,z, 2) A f(z)= U(z)] CT2 V z 3 y R ( r c , y ) 4 3eVz3z['&(e,z,z) A R(z,U(z))]. T h e former tells, via t h e Normal Form Theorem, t h a t every function is recursive a n d is suitable for second-order systems with functional variables. T h e latter is t h e axiom of choice (extracting a function from a V3 form), plus t h e fact t h a t every function is recursive a n d is also suitable for first-order systems. Of course, b o t h forms a r e false in usual classical systems, a n d t h u s CTl and CT2 are not provable in usual intuitionistic systems (in which t h e corresponding classical systems a r e interpretable).

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T h e relevance of t h e two principles t o our discussion is quite feeble. Even if we can disprove one of them in a formal system for constructive mathematics, this would not disprove Thesis C : it would simply mean t h a t it is absurd t h a t all functions can be proved recursive in t h e system (not t h a t some functions a r e not recursive). Kripke has given a formalization of Brouwer’s theory of the creative subject, and has shown t h a t it implies the negation of C T . However, for all current intuitionistic systems (not involving the concept of choice sequence) the consistency with C T has actually been established (see e.g. Kleene [1945], Kreisel and Troelstra [1970]). Once again this is not evidence in favor of Thesis C, not even indirect (as it was for CR): indeed, even a proof of C T would just show that every function we can talk about in the system is recursive and, once again, this would be interesting only for a system complete for constructive reasoning (since these functions would then be all t h e constructive functions). T h e reader will find more information on t h e philosophical analysis and (proofs of) t h e technical results of this subsection in Kreisel [1970], [1972], Troelstra [1973] and McCarty [1987].

To sum up, t h e arguments for Thesis C point out how it could be proved by a formal analysis of constructive reasoning (reductionist program), a n d disproved by showing - for any acceptable constructive arithmetical formal system - t h e inconsistency of closure under Church’s Rule. Both validity a n d failure of Thesis C have interesting consequences for constructive mathematics. Except for these methodological remarks, we have collected only very weak, and

certainly inconclusive, evidence in favor of Thesis C, whose validity must be retained as unproved (which is after all not surprising, since we do not even fully understand it: we still have only a partial grasp of what ‘constructive’ means).

Conclusion To recapitulate our discussion, recursiveness seems to be a model of discrete, deterministic processes general enough t o account for mechanical phenomena, according to classical physics. T h e notion certainly reaches beyond this, e.g. it takes care of probabilistic phenomena described by Markov’s chains, and of a wide variety of structurally stable systems. But we have no positive results, and actually some positive doubts, for what concerns subatomic phenomena governed by quantum mechanics. Turning t o biological computers, only very rough simplifications allow us t o look at the brain as a kind of machine, and we a r e still far from a coniplete theory. T h e analysis of human computations a n d reasoning produces a recursive description only under assumptions of routinnes a n d formal (at most finitistic) manipulation of symbols, respectively. It is a n open problem whether

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constructive reasoning in intuitionistic sense is recursive. Despite t h e weak evidence for some of them, t h e various theses have been proposed not out of empire-builder rashness (with t h e tacit ambition of convincing, short of proving, t h a t recursiveness is somehow a universally permeating concept), b u t rather out of experimenters circumspection (with t h e manifest hope of understanding t h e exact limits of t h e notion). T h e validity of Church’s Thesis (presently proved to some, but certainly not full, extent) is not what would give importance to Recursive Function Theory, although undoubtedly it adds to it (to t h e extent it holds). T h e notion of recursiveness has more than sufficient motivations (reviewed in t h e introduction to this section) to deserve a thorough mathematical study, disregarding its - certainly fascinating connections with mechanism, neurophysiology, and constructivism. But, independently of its practical relevance, work along the line of this section has a n abstract importance. To quote Kreisel [1970]: T h e principal interest is philosophical: not to confine oneself to what is necessary for (current) practice, but t o see what is possible by way of theoretical analysis.


Chapter I1

Basic Recursion Theory T h i s chapter contains t h e core of Recursion Theory, a n d introduces its basic notions, methods a n d results. We s t a r t , in Section 1, with a n extension of t h e not,ion of recursiveness, by dropping a weak point in t h e various definitions of Chapter I ( t h e request, not effectively verifiable, of totality for a n algorithm). T h i s leads to t h e class of partial recursive functions a n d their set-theoretical counterparts, t h e recursively enumerable sets. T h e elementary properties of these functions a n d sets a r e explored throughout t h e chapter, while a deeper structural analysis will begin in Chapter 111, a n d continue in Volume 11. T w o fundamental tools for nontrivial results are t h e method of diagonalization a n d t h e notion of degree. T h e former, one of t h e innovative inventions of Cantor, is a n extremely helpful technique which h a s become, in various disguises, a pervasive element of Recursion Theory. In section 2 we introduce t h e fundamentals of t h e method, including a codified version of it called t h e Fixed-Point Theorem. T h i s is a powerful a n d somewhat mysterious result which underlies t h e famous undecidability results of t h e Thirties, also treated a n d discussed in Section 2. T h e notion of degree is introduced in Section 3, which is devoted to relative computability as opposed t o t h e absolute computability dealt with so far. We generalize computations t h a t c a n be performed solely by machines, a n d allow t h e machine to stop, from time to time, a n d ask questions. T h e model still describes real computations, b u t t h e machine is not autonomous anymore, a n d may rely on interactions with t h e external world ( t h a t is, also during t h e computation a n d not only, as previously, in t h e input-output activity). T h e distinction between absolute a n d relative computations is t h e one between fully automatic a n d interactive (man-machine) behavior of computers, t h e latter being t h e common practice in sophisticated (not purely computational) projects, e.g in automatic theorem provers, or in Artificial Intelligence tasks. T h e deci-

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11. Basic Recursion Theory

sion to relax t h e autononiy of t h e machine still leaves various possibilities open in terms of t h e amount a n d t h e structure of t h e interaction with t h e outer world. In Section 3 we deal with Turing computability, t h e most general a n d fundamental case, imposing no limitation on t h e help given to t h e machine, except for a n obvious finiteness requirement. Other, more restrictive, notions of relative computability will be introduced in Chapter 111. A fundamental property of partial recursive functions is t h e possibility of enumerating their programs in a n effective way, a n d t h u s of assigning indices to t h e m , according to their place in t h e enumeration. Indices t h u s code descriptions of partial recursive functions, a n d can be used to refer to a function in a n oblique (intensional) way. Section 4 deals with t h e effective operations t h a t can be defined intensionally on t h e (partial) recursive functions (by working o n their indices), a n d their relations with t h e partial recursive functionals, which a r e their exteiisiorial analogues (working directly on functions). Section 5 considers various topics connected with indices. T h e results of this chapter collectively show t h a t the class of partial recursive funct.ions is very comprehensive as a result of i t s striking closure properties. First of all, t h e universal partial function (Theorem 11.1.8) provides a descriptional closure. Second, t h e recursive functions a r e closed under recursive diagonalization, with a two-fold escape from contradiction: for t o t a l recursive functions t h e r e is n o universal function (Theorem 11.2. l ) , hence a n y recursive class of total recursive functions is not exhaustive, a n d diagonalization j u s t produces another recursive function, which is not in t h e given class; for partial recursive functions, diagonalization simply produces particular undefined values (see p. 152). Finally, a n d this accounts for t h e name of t h e class, t h e Recursion Theorems 11.2.10 a n d 11.3.15 ensure closure under recursion of any kind (where ‘recursion’ can be taken to mean, in its greatest generality, t h e definition of a function in t e r m s o f itself a n d of known functions).

11.1

Partial Recursive Functions

We have introduced in Chapter I various independent approaches to t,he notion of effective cornputabilit,y, arid t h e methods of Section 1.7 showed t h e m to be all equivalent. We might t h u s be quite satisfied, b u t t h e r e is still a point t h a t seems a bit o u t of tune: we have been longing for a precise notion of effecLive computable function, a n d all our definitions have a strongly noneffective

element in t h e m , namely t h e infinitary restriction t h a t we consider devices computing only total functions. Having a device potentially computing a function, we did not accept it as an algorithm until we h a d somehow recognized t h a t it p r o d u c e answers for any input: since it is possible to prove (see p. 116) that, t,his cannot be done in general by any recursive means, t h e class of recursive

11.1 Partial Recursive Functions

127

functions seems to depend on something external to it, and it is even conceivable that it depends on the methods of proof allowed for the recognition of the totality of an algorithm. All this might sound quite discouraging, but the final solution to the problem of characterizing effective procedures is at, hand: we only have to set a missing brick, and the construction will be completed. Since it is the verifkation of totality that troubles us, we simply decide to drop it.

The notion of partial function A partial function is simply a function that may be undefined for some (and possibly all) arguments. The set of arguments for which it is defined is called its domain. Of course a partial function is total on its domain, but here we give a privileged status to the set of natural numbers, and consider a function whose domain is properly included in w as only partially defined. The step from total t o partial functions should be appreciated: it was a longstanding philosophical position that there cannot be precise logical laws for propositions about incompletely defined objects, from Aristotle (Metaph@ca, r 7, 1012a, 21-24) to this century. It was probably Brouwer [1919] (see also [1927]) who first corrected this position with his work on choice sequences. We use Greek letters to indicate partial functions, and an extended equality relation,',' meaning that both sides are equal as partial functions (i.e. their respective values are either both undefined, or both defined and with the same value). Also, ~ ( z5-)means that cp is defined (also said: it converges) for the arguments 2, while cp(.') t means the opposite (also said: it diverges). Finally, partial functions can be partially ordered by the inclusion relation naturally defined as: cy

G p e \dz[a(z)1 =+

p ( x ) = a(.)].

Thiis whenever a is defined so is p, and with the same value.

Partial recursive functions We adapt definition 1.1.7 to partial functions:

Definition 11.1.1 (Kleene [1938]) The class of partial recursive functions is the smallest class of functions 1. containing the initial functions 0, S and 2. closed under composition, i. e. the schema that given yl, . . . ,y7,,, $ produces 4 2 )= $ J ( T l ( q , . . . ? Y m ( q ) ,


128

where the left-hand side is undefined when at least one of the values of y l , . . . ,ymr11) for the given arguments is undefined 3. closed under primitive recursion, i.e. the schema that given @, y produces Cp(.’,O)

V(CY

+(Z) -d.’>Y,Cp(Z,Y))

2:

+ 1) =

4. closed under unrestricted p-recursion, i.e. the schema that given

I) pro-

duces cp(.’)

where

~p(.‘)

= pLY[(V. I Y ) ( I ) ( C

2)

1)A @(ZIY)= 01,

is undefined i f there is no such y.

At first sight, we may think t o define t h e p-recursion schema as: Cp)’(.

r?P Y ( + ( C Y )

= 0).

T h i s would mean t,o look for t h e least, y such t h a t +(Z,y) 2

2:

0 a n d , for every

terstill, sets, rather t h a n o n partial functions, a n d t h e associated reducibility riotion for sets is called

enumeration reducibility:

A 5, B e for some r.e. relation R, z E A e (3u)(D,C B A R(z,u)). T h e structure of degrees associated with this reducibility (called partial degrees) will be studied in Chapter XIV. An Enumeration Theorem for recursive operators (Rogers [1967])c a n easily be obtained by stepping from a n enumeration {Wz}eeuof all t h e r.e. ternary relations, to a n enumeration {Wf3(e)}eEu of t h e consistent ones, where W j c e ,is obtained from a n enumeration of Wz, by dropping t h e triple (2,u’, z’) whenever a triple (x,u,z ) with u , u’consistent has already been generated. T h e basic difference between recursive operators on the one hand, a n d partial recursive functionals on t h e other, is computational: t h e latter a r e serial, a n d a computation gets stuck if it tries to query t h e oracle for a n undefined value; t h e former a r e parallel, a n d c a n get around undefined values b y dovetailing computations (for general discussions of parallelism, see Elgot, Robinson a n d Rutledge [1967], Shepherdson [1975], Cook [1982]). We t h u s arrive at t h e notion of partial recursive functional by relativizing deterministic approaches to coniputability, like recursiveness (by adding a function to t h e initial ones, as we did in this section), Turing machine computability (by adding a n additional state ishat calls for t h e oracle, Turing [1939]) or flowchart computability (by adding assignment instructions of t h e kind


198

as in Ianov [1958]). O n t h e other hand, we arrive at t h e notion of (partial) recursive operator by relativizing nondeterministic approaches, like Herbrand-Giidel computability, or representability i n formal systems (by adding a functional letter to t h e constants of t h e language, Kleene [1943]). We t h e n have three notions of relative computability for partial functions: if a = F ( P ) , we say t h a t

a 0

so that f visits successive intervals. The converse is similar.) For more information on splinters see 111.7.10 and Myhill [1959], Ullian [1960],

Young [ 19651, [ 19661, [ 19671.

Results proved later (see 111.4.9) will show the existence of

11.6 Retraceable and Regressive Sets

*

245

1. a retraceable set which is not regressed b y total recursive functions 2. a set retraced b y a total recursive function, which is not regressed b y total

many-one recursive functions.

Existence theorems and nondeficiency sets We state our results in strong form, using the notion of degree introduced in 11.3.3.

Proposition 11.6.13 (Dekker and Myhill [1958]) Every T-degree contains a retraceable set. Proof. If A is finite then it is recursive and retraceable. Let A be infinite, and f be t h e enumeration of A in order of magnitude (f is not recursive, in general). Let B be enumerated (in order of magnitude) by the function

Then B is retraceable via the recursive function that chops off the last component of a sequence number of length greater than 1, and leaves unchanged t h e remaining numbers. Clearly B ST A by definition, and A 5~ B because z E A H z is a component of g(z),

since z

5 g(z).

Corollary 11.6.14 There are 2"O retraceable sets. T h e recursive sets are t h e simplest retraceable sets. R.e. nonrecursive sets cannot be retraceable (by 11.6.5 a nonrecursive, retraceable set must be immune), and thus the next level of complexity, and the first nontrivial one, for retraceable sets is being co-r.e. Exercise 11.6.16 Zf A is retraceable and co-r.e., it is retraced by a total recursive function. (Hint: given z, see if p(z)1 or z E 2.)

We now prove that such sets not only exist, but are as abundant as they can be.

Theorem 11.6.16 (Dekker and Myhill [1958]) Every r.e. T-degree contains a retraceable, co-r.e. set.


246

Proof. If A is recursive t h e n it is itself co-retraceable a n d r.e. Let t h e n A be r.e. nonrecursive, a n d let f be a recursive, one-one enumeration of i t . Let 5

EB

zE

B

*

(3Y > z)(f(y)< f(.)

(YY > X)(f(Y) > f(.)).

T h e elemeiits of B a r e called stages of nondeficiency, or true stages, in t h e enumeration of A given by f , because no new element of A smaller t h a n f(x) is generated by f in t h e future. Hence, for x E B,

B is clearly r.e. Moreover:

AFTB To see if z E A it is enough to find z E

z

E

B snch t h a t f(z)> z , a n d see if

{f(O),. . . ,f(z)).

And x exists because f is o n c o n e a n d B is infinite (given a n element b E B,a greater o n e c a n be obtained by taking first t h e smallest element a E A which is not in {f(O), . . . , f ( b ) } , a n d t h e n t h e stage in which a is generated by f ).

BFTA z E B if a n d only if there is some element in

B is co-retraceable Given x E B,we want to give a n effective procedure t o find t h e greatest element of B smaller t h a n z. Since for y > x is f ( y ) > f(z), i t is enough t o check t h e values o ff for arguments below z. In other words, for z

T h e n it is enough to define g(z) as t h e biggest z

(Y'y)(.

< x,

< z such t h a t

< Y 5 z * f(z) < f(Y))

if t h e r e is one, a n d z otherwise (so t h a t t h e first element of B is left fixed). 0


*

247

T h e idea of using nondeficiency stages is a n ingenious one, invented by Dekker [1954].It will show its usefulness time and again, in many different contexts (including infinite injury priority arguments, see Chapter X). As far as retraceable co-r.e. sets are concerned, this idea completely captured the heart of t h e matter:

Proposition 11.6.17 (Yates [1962]) A co-r.e. set A is retraceable if and only ii for some recursive function f, 5

EA

2

E

7i

* *

(YY > .)(f(Y) > fb)) (3> X)(f(Y) I f (z)).

Proof. If a function f as stated exists, the proof of 11.6.16 shows t h a t A is retraceable. If A is finite, it can easily be seen that a function f as stated exists. Let then A be a n infinite, retraceable and co-r.e. set: we want to find f . We have g recursive retracing A , and we may suppose t h a t g is total (11.6.15) a n d g(z) I x (11.6.3). Consider the recursive height function h ( z )= pz [g""(z)

= g(Z)(z)],

and t h e recursive height sets z E H , w h ( z ) = n.

Clearly t h e height sets are disjoint, cover w (since g is total and descending) a n d have exactly one element of A each (since A is infinite). T h e idea is t o define f on H , (ordered by magnitude), by letting it be n until t h e first element of A is hit, and greater than n afterwards. Thus, if E H , n A a n d y < x A y E H,, then y is a deficiency stage, while x becomes a nondeficiency one. Given x E H,, consider the set f = {y : y

z such t h a t f ( y ) I f(z). Note that f(z)= n 1 is not enough, since it might be that H,+1 contains no element y > z. And m > n such t h a t H , has a n element y > 2 is not enough either, since t h e unique element t E H , n A might be smaller than z, a n d setting f (z) = m would make f ( t ) = f(x), while t < X, and t has t o be a nondeficiency stage (since t E A ) . But then let f(z)= m for m > n such t h a t H , contains no element smaller than x. 0

x, x,

+


248

Exercises 11.6.18 Nondeficiency sets. A set A is a nondeficiency set if, for some

recursive function f,

zE

A

* (VY > Z)(f(Y)

> I(.)).

a) I f f is not finite-one, the nondeficiency set o f f is finite. b) A w-r.e. set is retruceable if and only i f it is the nondeficiency set of afinzte-one function. (Yates [1962]) (Hint: see the proof above.) c) Not every w-r.e. retmcenble set is the nondeficiency set of a one-one b n c tion. (Degtev (19701) (Hint: let A(g) be the deficiency set of g , and A,(g) be its approximation up to n, i.e. 5

E A n ( g ) * z < n A ( % ) ( z< Y

I ~ & A \ ( Y )I g ( z ) ) .

Define f as follows. Given f ( O ) , . . . ,f (n),let 1% = ( e , z) be the first stage in which cp, is total and one-one on (0,. . . ,z}, and An,,(f) Az(cpe),where no = m m { z I n : f ( z ) = e }

Then let / ( n + 1) = e. Otherwise, let f(z)= n. If cpe is one-one and total, then A(cpe) is infinite, and it cannot be A(cp,) = A( f ) , otherwise by construction f takes the value e infinitely often, and then A ( f ) is finite.)

Proposition 11.6.19 (Marchenkov [1976a]) T h e class of r.e. co-retruceable sets is r.e. without repetitions. Proof. Since every finite set is co-retraceable, it is enough (by 11.5.24) to show t h a t t h e class of r.e. co-retraceable sets is r.e. Let

T h e n { A e } e E wis a n r.e. class, a n d each A, is either finite (if peis not total) or co-retraceable (by 11.6.17). 0 T h e class of r.e. co-retraceable sets is not completely r.e. (by 11.4.2), since it is not closed under supersets (see II.6.21.b).

Regressive versus retraceable We now briefly come back to t h e original question of t h e extent of t h e analogy with recursive a n d r.e. sets, arid we show t h a t it fails q u i t e strongly. First we give a positive result, which is t h e analogue of 11.1.20. Proposition 11.6.20 (Dekker [1962]) Every infinite regressive set has a n infinite retraceable subset.


*

249

Proof. Let A be regressed by cp w.r.t. {ao,al , . . . }, a n d bo b,+l

= a0 = t h e first element in t he list of A greater than b,.

Then B = { b o , b l , . . . } is infinite, because A is. T o define t he retracing function for B on a given z, we first iterate p until we stop, which we must if z E A . Then we recreate t h e initial segment of B from bo t o z, by dropping t h e elements t h a t break t h e monotone growing, and take t h e biggest element obtained which is smaller than z. If z E B then we do recreate t h e initial segment of B , a n d we do choose t h e right element. 0 There a r e two properties t h a t we consider essential to claim a nontrivial analogy with recursive a n d r.e. sets, namely:

1. T h e recursive sets ar e closed under complementation.

2. A set which is r.e., together with its complement, is recursive (11.1.19) They both fail here:

1. There is a retraceable set with a nonretruceable complement. Take any retraceable, nonrecursive set: its complement is not retraceable, by 11.6.6.

2. There is u set regressive together with its complement, but not retruceable. Take any set A r.e. and nonrecursive, with a retraceable complement. Then b o th A (being r.e.) and ar e regressive, but if A were retraceable then it would also be recursive, again by 11.6.6. Exercises 11.6.21 (Dekker and Myhill [1958])a) If A and B are retraceable, so is A n B. (Hint: given z, consider the greatest element smaller than it on which 2 is sent by both functions retracing A and B.) Appel [1967] ha7 shown that this fails for regressive sets. b) There are retraceable sets A and B such that A U B is not regressive. (Hint: let A be an infinite recursive set, and B a nonrecursive, retraceable and cer.e. set. If A U B were regressive, it would be r.e. because A is an infinite recursive subset of it, and B would be recursive.)

Despite t h e failure of t h e analogy with r.e. and recursive sets, retraceable a n d regressive sets are interesting on their own, and are useful in some parts of Recursion Theory. See McLaughlin [1982] for a detailed study of them.


Chapter I11

Post’s Problem and Strong Reducibilities One theme of this chapter is relative computability. In Chapter I1 we introduced t h e most general and fundamental case: Turing reducibility. It will be recalled t h a t no limitation was imposed there on t h e help given t o t h e machine by t h e oracle, except for a n obvious finiteness requirement. Here we take a n opposite stand, and look at various possible limitations. Section 2 deals with t h e most restrictive case of m-reducibility, in which only one question is allowed to t h e machine during a computation, and only at t h e very end of it. Section 3 treats the case of Boolean combinations of atomic questions, called tt-reducibility, while a number of other, less fundamental, reducibilities are dealt with in Sections 4, 7 and 8. Relative computations induce equivalence classes, by identifying functions and sets which have t h e same degree of difficulty of computation. A second theme in t h e chapter is Post’s problem, introduced in Section 1, which asks whether there are only two such classes of r.e. sets. T h e solution, obtained in Section 5, will tell whether the r.e. sets can be distinguished, from a computational point of view, only between recursive and nonrecursive, o r whether instead this rough dichotomy can somehow be essentially refined. T h e strategy for a solution t o the problem is t o analyze the possible structure of r.e. sets (as opposed to giving direct constructions, a n alternative strategy pursued in Chapter X), and t h e tactic is t o solve t h e problem first for m-reducibility, in Section 2, a n d then gradually improve t h e solution for weaker and weaker reducibilities, in Sections 3 a n d 4, until we reach t h e one we are really interested in. T h e original motivation for the study of r.e. sets was t h a t they code (by 25 1

111. Post’s Problem and Strong Reducibilities

252

arithmetization) t h e sets of theorems of formal systems. A third theme of this chapter is the analysis of formal systems, from this abstract point of view. We make the relationship between formal systems a n d r.e. sets precise i n Section 10, where we also revisit some of t h e notions a n d results obtained in the chapter, and discuss their bearing on t h e subject of formal systems.

111.1 Post’s Problem We have so far encountered only two different kinds of r.e. sets, namely t h e recursive sets and K ,and they generate different degree.

Definition 111.1.1 A n r.e. T-degree is a degree containing at least one r.e. set. Two r.e. T-degrees are: 1. the T-degree 0 of the recursive sets 2. the T-degree 0’ of K. Note t h a t , because of Post’sTheorem and the fact t h a t a set a n d its complement are in t h e same degree (being obviously computable one from t h e other), a degree contains only r.e. sets if and only if it contains only recursive sets. This explains why we only require the existence of a n r e . set in a n r.e. degree. Recall t h a t there is a partial order on the degrees, induced by t h e relation ST. It is obvious t h a t 0 is the least degree with respect t o it, a n d t h e next result shows t h a t 0’ is t h e greatest r.e. degree.

Proposition 111.1.2 (Post [1944]) If A is any r.e. set then A 5~ K.

Proof. We prove t h a t there is a recursive function f such t h a t ZE

A

@

f(.)

E K: @ f(Z) E Wf(z),

where the last equivalence holds by definition of be a recursive function such t h a t

w

0

K. By the Sz-Theorem,

let

f

ifsEA otherwise.

Then: 0

* Wf(,) = w * f ( ~ E) Wf(,) + f ( ~E )K J(z)E K * f ( ~ E ) W f ( , )+ Wf,,) # 8 3 z E A.

5

EA

0

Thus, since A S T K: automatically holds for r.e. sets, an r.e. set A is in t h e greatest r.e. degree 0’ if and only if K 51A.

111.1 Post’s Problem

253

Definition 111.1.3 A set A is Turing complete (or T-complete) ij it is r.e. and its degree is O‘, i.e. K: ST A .

In general, given a reducibility S T ,we will call a n r.e. set A r-complete if A , a n d r-incomplete otherwise. As noted above, t h e r.e. sets we know at this point a r e all recursive or T-complete. It is natural t o ask whether there a r e others.

K

Post’s Problem (Post [1944]) Are there r.e. T-degrees different from 0 a n d O’? Equivalently, a r e there r.e. sets which a r e neither recursive nor T-complete? T h e reasons to isolate this natural problem a n d give it a name a r e many. First, despite its technical formulation, t h e problem was motivated by deep methodological questions, related to t h e undecidability results, a n d reviewed in t h e next subsection. Second, t h e solution to t h e problem escaped t h e r e searchers for many years a n d provided, as a by-product, new techniques a n d results, some of t h e m treated in this chapter. Finally, versions of t h e problem arise in different areas of Generalized Recursion Theory, a n d their solution is usually regarded as a proof of maturity for t h e new areas.

Origins of Post’s Problem

*

Post arrived at t h e formulation of his problem after a n exciting intellectual development, which is worth reviewing. I n his dissertation, completed in 1920, h e s t a r t e d by analyzing t h e system of Principia Matheinatica, a n d attacking t h e problem of i t s decidability. He was able to solve a particular case, namely t h e decision problem for propositional calculus, by proving a completeness theorem t h a t showed t h a t t h e theorems were exactly t h e tautologies. He published this in [1921]. In t h e academic year 1920-21, as a postgraduate, Post set down to generalize this decidability result, by attacking t h e general case. Trying to c a p t u r e t h e esscnce of formal systems h e considered, by successive abstractions, a sequence of notions, finally obtaining t h e canonical systems (see p. 143). By showing t h a t t h e system of Principia Mathernatica could be translated in a canonical system, h e convinced himself t h a t h e h a d a siifficiently general notion. Post then turned to t h e decidability problem for canonical systems, hoping t h a t t h e generality of t h e notion would make t h e proof simpler, because independent of details related to particular systems. He soon concentrated on a special problem, called t h e tag, of which he was able to handle some particular casm, but, t h a t turned o u t to be intractably complicated in general (with good reasons: it was undecidable, see Minsky [1961]).


254

At this point, t h e unsuccessful a t t e m p t s prompted a revision of t h e plan, a n d Post turned to undecidability. He defined a universal canonical system, which is j u s t a version of t h e set K O of p. 150, a n d showed its diagonal set, i.e. K , to be, in modern terms, r.e. b u t nonrecursive. To be able to deduce from this a general unsolvability result Post needed a version of Church’s Thesis, which h e stated in t h e form: every effectively generable set c a n be generated by a canonical system. From this t h e existence of incomplete formal systems followed easily. All this work, concluded in 1921, a n d anticipating a number of results by Gadel, Church, a n d Turing t h a t would follow much later, was left unpublished (see [1922]), because Post was not convinced of (his version of) Church’s Thesis. H e took it as a working hypothesis t h a t needed verification, in t h e form of psychological analysis of t h e computational process. A step toward such analysis was [1936], in which Post proposed a version of Turing machines, independently of Turing. T h e canonical systems were published only in [1943], a n d t h e form of Godel’s theorem based o n canonical systems only in [1944]. T h e mutual reductions among various notions of canonical systems, as well as particular formal systems like Principia M a t h e m a t i c a , led Post t o t h e concept of m-reducibility between sets. A n d t h e fact t h a t known undecidability proofs, by Post, Church, a n d others, were all obtained by appropriately reducing K to t h e problem in question, prompted t h e problem of whether t h e only undecidable systems were t h e universal ones, in which K could be interpreted. Post [1944] was able to disprove this for m-reducibility, a n d t h e n h e asked t h e s a m e question for t h e general notion of T-reducibility, introduced by Turing [1936]. Despite a good deal of intermediate work, h e could not reach a solution. Sections 2 to 4 a r e a report of Post’s work a n d of modern improvements, a n d Section 5 provides t h e missing brick.

Turing reducibility on r.e. sets Since t h e solution to Post’s Problem will require a detailed s t u d y of t h e struct u r e of t h e r.e. sets, we prove some technical results t h a t will facilitate t h e task.

Proposition 111.1.4 T-reducibility on r.e. sets (Rogers [1967])I f A and B are r.e. sets, t h e n A z)(y E We) fails.)

We can now show t h a t the simple sets constructed above are all effectively simple, and thus T-complete. We refer t o the proofs of 111.2.11 and 111.2.12, and use t h e same notations as there.

1. Post’s simple set. If We C 3 then IWel 5 2e 1, since otherwise We has more than 2e + 1 elements, so one of them is greater than 2e, and goes into S.

+

2. The set A built in 111.2.11. If We then ]We[5 e, since otherwise there is n 2. e such t h a t a, E W e . By going to a stage s after which t h e Wi’s with i < e do not contribute anymore elements to A , t h e elements of 2 up to a, have settled, and an E We,8,we would then put one element of We into A.

3. The set of nonrandom numbers. If Wecontains at least t ( g ( e ) )elements, then W g ( econtains ) at least g ( e ) elements by its definition. And if Wg(,) contains at least g ( e ) elements, then it contains a nonrandom number (because h ( g ( e ) , n ) 5 t(n)for any n 2 g ( e ) ) . This means that if We contains only nonrandom elements, then it must have at most t ( g ( e ) )elements, i.e. IWel 5 t ( g ( e ) ) . We consider now the simple sets provided by 111.2.14.

Proposition 111.2.20 (Smullyan [1964]) The deficiency set of K: is effectively simple. Proof. Let f be a oneone recursive function with range Ic, and

We want to find g recursive such that

Suppose then We C B. Since we already know t h a t B is simple, We is finite, say (21 < ... < x,}. Consider the behavior o f f on these elements.


266

“1

“2

5,-1

2,

Since each xi is a nondeficiency stage, there is no crossover. It is t hen enough t o find effectively a > f(z,):since f is one-one, a 2 IWel. To find a, we use t h e fact t hat

Since K is t h e range o f f , it is enough to choose W , containing all t h e elements of K: below f ( x n ) . This is possible because 2, E B,so

Then

Exercises 111.2.21 Strongly effectively simple sets. (McLaughlin [1965]) A is strongly effectively simple if it is a coinfinite r.e. set, and there is a partial recursive function d such that

We

2 =+ 4 ( e )1 A

(maxWe) < $(el.

An example is given by Post’s simple set. a) may always be supposed t o be total. b) There are effectively simple sets which are not strongly effectively simple. (Martin [1966]) (Hint: this follows from results in 111.4.24, since a strongly effectively simple set is not maximal, and there are maximal effectively simple sets. A direct construction, e.g. in the style of 111.2.19.a, is also possible.) McLaughlin [1973], and Cohen and Jockusch [1975], prove in different ways that the deficiency set of K is not strongly effectively simple.

Exercises 111.2.22 Immune sets. a) There are 2*O sets immune and coimmune, called bi-immune. (Hint: enumerate a set of pairs (zn,yn) such that 2, and yn are different elements of the n-th infinite r.e. set, not yet enumerated in the previous pairs. Any set with exactly one element of each pair is bi-immune.)

111.3 Hypersimple Sets and Tkuth-Table Degrees

267

b) A set A is effectively immune if it is infinite and, for some partial recursive function $, We C A =+ $ ( e l l A IW=II $ ( e ) . $ may always be supposed t o be total. Similarly for strongly effectively immune sets, where instead

We G A =+ $ ( e ) 1 A (maxWe) I $(e). (Hint: let g be a recursive function such that

{0

Wg(e)

we

otherwise. if $ ( e ) L

Then one of $ ( e ) and $ ( g ( e ) ) converges.) c) There are effectively bi-immune sets, i.e. sets effectively immune and effectively coimmune. (Ullian) (Hint: a natural construction, in which each We contributes a t most two elements, one for A and one for 2, gives a bi-immune set A such that if We 5 A or We C 2, then IW,l I 2e 1.) d) If A is strongly effectively immune, then 71 cannot be immune. (McLaughlin [1965]) (Hint: suppose We5 A max(W,) < f(e).

+

*

Let s, be the smallest stage in which x appears in K , if it ever does, and 0 otherwise. Define W t ( z )=

{

p) {sx)

otherwise.

ifx

Then x E K A sz 2 f ( t ( x ) )+ s, E 71. There are infinitely many such s,'s, otherwise for almost every x is x E K ($ 2 E K f ( q , ) ) , and K is recursive. But the condition 2 E K A s , 2 f(t(x)) is r.e., so 71 contains an infinite r.e. subset.) e) A set A is constructively immune if, for some partial recursive p,

W, infinite

+

p ( e ) 1 A p(e) E W, n 71.

If A is constructively immune, then 2 cannot be immune. (Li Xiang [1983]) (Hint: an infinite r.e. subset of 2 can be built, by starting with an r.e. index of w . ) f) The notions of effective and constructive immunity are independent. (Li Xiang [1983]) (Hint: every simple set is constructively coimmune; for the other directions, see c) and e) above.)

111.3 Hypersimple Sets and Truth-Table Degrees We have solved Post's Problem for m-reducibility by constructing a simple set, but have noticed t h a t this does not automatically imply a solution t o Post's Problem for T-reducibility, because there are simple sets which are T-complete


268

(actually, all t h e simple sets we built were such). T h e next step is to relax m-reducibility somewhat, a n d solve Post’s Problem for t h e weaker notion. In m-reducibility we only allowed o n e positive query to t h e oracle. T h e r e a r e many possible extensions, depending on t h e number of queries allowed (a fixed bounded number, o r unboundedly many), their n a t u r e (only positive, i.e. asking whether some elements a r e in t h e oracle, or also negative, asking whether some elements a r e not in i t ) , a n d t h e way they a r e combined (conjunctions, disjunctions, o r any possible combination). Some of t h e possible reducibilities, called respectively conjunctive, disjunctive, a n d positive (Jockusch [1966]) a r e t h e following:

A 5, B

H

for some recursive function f ,

zE A AId B

H

H

Df(.)

c B.


zE A

A SpB

H

D f ( z )n B #

0.


zE A

* ~ U ( UE Df(.)

A

D,

c B).

T h e consideration of these reducibilities makes good sense for the s t u d y of r.e. sets, d u e to t h e fact t h a t they only use positive information on t h e oracle, which is exactly what we may obtain from r.e. sets (note t h a t A is r.e. i f and only zf A S p K). We will not be too much concerned with t h e m , since t h e picture we get from m-reducibility is finer, b u t we will prove s o m e scattered results here a n d in Chapter X. Since our present concern is t h e solution to Post’s Problem, we might as well consider t h e strongest possible generalization along these lines: to allow for any (finite) number of questions, b o t h positive a n d negative, to t h e oracle. To define t h i s notion precisely, let {rs,},~w be a n effective enumeration of all t h e propositional formulas, built from t h e atomic ones ‘ m E X ’ , for m E w. These are also called truth-table conditions, since they c a n be arranged o,, means t h a t B satkfies on,i.e. t h a t in truth-tables. Given a set B , B t h e propositional formula rs,, becomes t r u e when X in t h e atomic formulas is interpreted as B .

Definition 111.3.1 (Post [1944]) A is tt-reducible to B (A Stt B ) iJ for some recursive function f , 2

EAH B

of(.).

A is tt-equivalent to B ( A E t t B ) if A Stt B and B Stt A . In terms of connectives, t h e various truth-table reducibilities correspond t o truth-table conditions built u p from t h e atomic ones by means of t h e following

111.3 Hypersimple Sets and Truth-Table Degrees

269

connectives:

while Srn correspond to using only atomic formulas. We will see in 111.8.4 that,, on t h e r.e. sets, Srn= {T}. Since 7 ,together with any one of A a n d V, generates all t h e propositional formulas, this would seem to t a k e care of all t h e possible t y p e s of truth-table-like reducibilities. Bulitko [1980] a n d Selivanov [1982] have shown t h a t it is almost so, in t h e sense t h a t only another o n e such rediicibility exists, called linear reducibility, corresponding to t h e logical s u m (i.e. addition modulo 2). Its definit,ion can be p u t as:

A

B

-++

for some recursive function f zEAH

IDf(.)fl BI

=1

(mod 2).

Truth-table degrees O u r first concern is to make t h e difference between truth-table a n d Turing reducibility clear. Recall t h a t a relative computation consists of two different kinds of actions, one purely computational (performed by a machine), a n d t h e other interactive (queries answered by t h e oracle). In Turing computations t h e two p a r t s c a n be strongly interwoven, a n d impossible to unravel: we may come to know t h e questions we need t o ask t h e oracle only during t h e computation itself, a n d t h e r e might even be no recursive bound on their number or size (as a function of t h e input). O n t h e other hand, truth-table computations clearly separate t h e two p a r t s of t h e relative computation, computing a h e a d of t i m e not only t h e elements which need to be queried, b u t also t h e outcome of t h e computation for any possible answer t h e oracle is going to provide for them. Another way t o see t h e difference is in terms of functionals. Recall t h a t A < T B if a n d only if CA N F ( c B ) ,for some partial recursive functional F .

Proposition 111.3.2 (Trakhtenbrot [1955], Nerode [1957]) A S t t B i f and only i f C A N F ( c s ) for some partial recursive, total functional F . Proof. Let 5

EA

* I= Of(.),

a n d define F ( a ,x) as follows. First see if a(.) 1 for every z such t h a t the atomic formula z E X occurs in af(.). If so, consider any set C such t h a t , for a n y z as j u s t said, z E C H a(.) = 1, a n d let

F(a,x)N

1 i f C k Of(.) 0 otherwise.

111. Post's Problem and Strong Reducibilities

270

F is partial recursive a n d total by definition, a n d CA N F(cB). Suppose now t h a t F is a partial recursive, total functional. For each x a n d X , F ( c x , x ) is defined, a n d for every branch of t h e space 2" of sets, t h e information on X needed to compute F ( c x , x) is bounded, by compactness. By Konig's Lemma, there is a bound which works for all branches. Another way to see this is to note t h a t F is continuous a n d total on 2", which is a compact space in t h e positive information topology: then F is uniformly continuous, a n d there is a modulus of continuity t h a t works for every member of 2". In any case, we may thus write down a truth-table t h a t gives F ( c x , z ) for any X , because only values of X up to t h e bound a r e needed, a n d there a r e only finitely many possible combinations. 0 By II.3.7.b we then know t h a t T-reducibility a n d tt-reducibility do not coincide. We now turn to a study of tt-reducibility. Exercises 111.3.3 a) If A is recursive, then A b) If A stt B and B is recursive, so is A . c) A Stt 2.

sitB for any set B .

Note t h a t < t t is a reflexive and transitive relation, a n d thus alence relation.

Ett

is a n equiv-

Definition 111.3.4 The equivalence classes of sets w.r.t. tt-equivalence are called tt-degrees, a n d ( D t t ,5 ) is the structure of tt-degrees, with the partial ordering 5 induced on them by I t t . The tt-degrees containing r.e. sets a r e called r.e. tt-degrees, a n d two of thein are:

1.

Ott,

the tt-degree of the recursive sets

2. O i t , the tt-degree of

K

A n r.e. set A is tt-complete if its tt-degree is Ol,,, i.e. if

K SttA .

Note t h a t an r.e. tt-degree, being closed under complementation, contains only r.e. sets if a n d only if i t contains only recursive sets. Also, O t t a n d O:, are, respectively, t h e least a n d t h e greatest r.e. tt-degrees. T h e analogue of Post's Problem for tt-reducibility is: a r e there r.e. sets which a r e neither recursive, nor tt-complete? We already know t h a t simple sets are not m-complete, a n d we make sure t h a t they a r e not automatically tt-incomplete.

Proposition 111.3.5 (Post [1944]) There is a simple, tt-complete set.

111.3 Hypersimple Sets and 'Ikuth-Table Degrees

271

Proof. Consider Post's simple set S (111.2.11). Any coinfinite r.e. superset of it will still be simple (since any infinite r.e. subset of i t s complement would also be a subset of 3). We t h e n look for a n r.e. set S' such t h a t : 1.

K Stt S ' ,

2. S

i.e.

K is coded into S'

by t r u t h tables

S', a n d S' coinfinite.

be a strong array of disjoint finite sets intersecting 3. I t exists, Let {F,},,, because t h e construction of S ensures t h a t at most z elements of (0,. . . ,22} go into S , so 3 intersects each subset of (0,. . . , 2 z } with at least z 1 elements. It is then enough to let

+

F, = { n : 2" - 1 5 n < 2,'

- 1)

We can t h e n use t h e set F, to code t h e fact t h a t x E K ,by p u t t i n g it into S' if t h i s holds, a n d not otherwise. We also want S' to be a superset of S, a n d we t h u s let S'=SU F,.

u

Z € K

Then:

1. K

N o >PI > N1 > . . . T h e basic difference between t h e constructions of maximal a n d simple (or hypersimple) sets is t h a t the positive requirements P, are infinitary, a n d we cannot hope to satisfy them with a finite action. Actually, since t h e y only allow for finitely many exceptions, each requirement has to be considered cofinitely many times, a n d at any given stage we have to consider many positive requirements all together. But t h e requirements have different priorities, a n d e-states are a device to assign priorities not to single requirements, but t o groups of them. E.g., t h e order of priorities assigned by 2-states is

(Po,P I , Pi?)L (Po,Pl) L (Po,pz) 2 (Po) L (P,,P2) 2 (Pl)2 (P2). We might say t h a t locally this is a finite injury argument, since every element of moves at most finitely many times, b u t globally it is a n infinite injury argument, since a positive requirement may be injured infinitely often, each t i m e for different elements of 2. Exercises 111.4.20 a) Maximal sets are not closed under intersection. (Yates [ 19621) (Hint: take A maximal, and let z E B 2 1 E A.)

+

111.4 Hyperhypersimple Sets and Q-Degrees

293

b) There are hyperhypersimple sets which are not maximal. (Hint: the hyperhypersimple sets are closed under intersection.)

Maximal sets have thinnest possible complement. T h e existence of maximal T-complete sets would show t h a t a notion of thin complement alone is not sufficient t o solve Post’s problem. I n fact, t h e maximal set j u s t built is T-complete, as we now prove, as usual, by showing t h a t it is maximal in a n effective way.

Definition 111.4.21 (Lachlan [1968]) A is effectively maximal i f it is a coin$nite r. e. set, and there is a recursive function g such that, for every e, the sequence of 0’s and 1’s consisting of the values of the characteristic function of We on the elements of 2 in increasing order has at most g ( e ) alternations. Note t h a t having finitely many alternations simply means t h a t one of WJlA a n d W ,n 2 is finite.

Proposition 111.4.22 (Lachlan [1968]) Every effectively maximal set is T-complete. Proof. Let g witness t h a t A is effectively maximal, a n d define f ST A such that

Wf(,) = a finite set with g(e) + I alternations on

T h i s is possible because

2.

is infinite, e.g.

W,(,) = (a2i : i 5 g ( e )

+ 1).

T h e n f has n o fixed-points a n d , by t h e criterion for T-completeness 111.1.5,A is T-complete. 0

Corollary 111.4.23 (Yates [1965]) There exists a T-complete m a x i m a l set. Proof. T h e maximal set built in 111.4.18 is effectively rnaxin~al,since We c a n 0 have at most g ( e ) = 2e+1 alternations on

x.

Exercises 111.4.24

a)

For n o winfinite r.e. set A there is a recursive function g

such that Saniilarly f o r

W, n A infinite

* IW,n;il

-

+ Iw, nZ1 5 g ( e ) .

W , n;i infinite

5 g(e).

Thin these natural candidates for effective maximality fail. (Lachlan [1968]) (Hint: for the second property, put in Wh(,) the first g ( e ) 1 elements of A, by starting with ( 0 , . . . , g ( e ) } and, each time that one element goes into A, adding the first element

+


294

not yet in W,,(e), and not yet generated in A. By the Fixed-Point Theorem there is e such that We = Wh(=). Thus n x is infinite, but IW, fl= g ( e ) 1.) b) There is a maximal, eflectively simple set. (Cohen and Jockusch [1975]) (Hint: modify the construction of a maximal set given above by inserting steps to make A simple, as in the second proof of 111.2.11. Note that each a: may move more times than it did for maximality alone, but still only finitely often.) c) A maximal set is not strongly effectively simple. (Cohen and Jockusch [ 19751) (Hint: by 111.4.15 a maximal set is dense, and by 111.3.9 is not strongly effectively simple. See 111.6.21 for a different proof.) d) A strongly effective simple set is not contained in maximal sets. Thus Post’s simple set is not contained in maximal sets. (Cohen and Jockusch [1975]) (Hint: a coinfinite r.e. superset of a strongly effectively simple set is still strongly effectively simple.)

+

111.5

A Solution to Post’s Problem

Post concluded his paper [1944] by saying: we are left completely on t he fence as to whether there exists a recursively enumerable set of positive integers of absolut sly lower degree of unsolvability t han t h e complete set Ic, or whether, indeed, all recursively enumerable sets of positive integers with recursively unsolvable decision problems are absolutely of t h e same degree of unsolvability. On t he other hand, if this question can be answered, th at answer would seem to be not far off, if not in time, then in t h e number of special results t o be gotten on t h e way. This section can be seen as t h e missing conclusion t o Post’s paper, a n d shows t h a t he was indeed right, regarding t he number of special results needed to solve his problem.

Semirecursive sets We know t h a t hyperhypersimple sets are not Q-complete. We are t hen looking for a notion th a t, together with T-completeness, would imply Q-completeness. By coupling it with hyperhypersimplicity we would then have a notion implying T-incompleteness, and Post’s problem would be solved. Since th e difference between Q-reducibility and T-reducibility is t h a t we query t h e oracle on single elements in t h e first case, and on finite sets in t h e second, we need a notion t h a t would reduce a question of inclusion of a finite set t o t h e question of membership of a single element.

111.5 A Solution to Post’s Problem

295

Definition 111.5.1 (Jockusch [196Sa]) A set A is semirecursive i f there is a recursive finction f of two variables such that 1. f(., y ) =

.

Of

f(.,Y)

=Y

2. a:€ A V Y E A * ~ ( x , Y ) E A. Clearly a recursive set is semirecursive, since we can simply decide whether one of x a n d y is in t h e set. Also, the complement of a semirecursive set is semirecursive: i f f witnesses t h e semirecursiveness of A , then t h e function t h a t always chooses, between x a n d y , t h e element not chosen by f , witnesses t h e semirecursiveness of 71. T h e next result is not unexpected, a n d is actually t h e reason why we introduced t h e notion of semirecursiveness.

Proposition 111.5.2 (Marchenkov [1976]) If an r.e. set A is semirecursive and T-complete, then it is also Q-complete. Proof. We only have to show how to reduce finite questions of t h e form D, C 71 to single questions g ( u ) E 2,for some recursive g. Let f be as in t h e definition 111.5.1. Given D, = { x ~ , . . . , x C n } , l e t

Then we obviously have D, in A , then so is g ( u ) . 0

2 * g ( u ) E 71, because

if one element of D, is

Exercises 111.5.3 a) A semirecursive simple set is hypersimple. (Jockusch [1968a]) effectively (Hint: if A is semirecursive so is 2, and thus a finite set intersecting produces a n element of 2.) b) A semimcursive set is not p-complete. (Jockusch [1968a]) (Hint: a sernirecursive pcornplete set would be m-complete, and thus every r.e. set would be semirecursive, contradicting the existence of simple, nonhypersirnple sets.) We a r e obviously interested i n knowing which sets a r e semirecursive. Since t h e definition of semirecursive sets might appear somewhat a d hoc, we first give a n interesting alternative characterization.

Proposition 111.5.4 (Appel, McLaughlin) A set is semirecursive i f and only i f it is a cut of a recursive linear ordering of w.

111. Post's Problem and Strong Reducibilities

296

Proof. Let A be semirecursive via f . We define, b y induction, a recursive linear ordering 4 such t h a t

x 4 y ~y E A Let xo 4 x1 4 extend it to n

. . . 4 x, be

*x E A.

t h e ordering of t h e numbers u p to n. We want t o

+ 1. Three cases a r e possible:

1. f ( n + l , x o ) = n + l If zo E A then n 1 E A , by t h e properties of f , a n d we c a n t h e n let n+l+zo.

+

+

2. the first case fails,and f ( n 1,xCn) = 2, Similarly, if n + 1 E A then x,, E A , a n d we let x, 4 n

+ 1.

3. the first two cases fail Then f ( n 1,xo) = zo a n d f ( n 1 , ~=~n+ ) 1. T h e n t h e r e is i such that f ( n 1,xi) = X i A f ( n l,xi+i) = 7~ 1,

+

+

+

+

+

and thus ~ i +E l

T h e n we can let xi 4 n

A = + n + l E A * X ~E A.

+14

~ + 1 .

Let now A be t h e lower cut of a recursive linear ordering 4 . If

f(x,y) = least element of {x,y } w.r.t.

4

then f ( x , y ) = x or f ( x , y ) = y by definition, a n d if o n e of 2 a n d y is in A t h e n so is f ( x ,y), because A is closed downward w.r.t. 4. T h u s f satisfies t h e conditions of 111.5.1, a n d A is semirecursive. Exercises 111.5.5

a) Every ti-degree contains a semirecursive set. (Jockusch For n. [1968a]) (Hint: let A be infinite and coinfinite, and define r = x n E A 2 any z, let r, = 2-". Then z 4 y T, < r, is recursive. Let B be the lower cut determined by T . A 5~ B because if D, = (A n (0,.. . , n - 1)) U {n} then, by induction, nEA D, E A w r, < r x E B.

xnED,

*

*

*

And B c41ief

n such t h a t {n,. . . ,m - 1) # 8 (since this permits a n inductive generation of a strong array intersecting Given ao, . . . , a, we can find first of all a number p such t h a t they ar e deducible from PO,.. . , &. Then we can find a number m such t h a t &+I is deducible from ao, . . . ,a,. Now one of n, . . . , m - 1 is not in A otherwise, inductively, all of a,+l,. . . , a , would be deducible from (1~0,. . . ,a,, and hence from p0,. . . ,&. But then so would be &,+I, contradicting t h e fact t h a t th e /?’s are a n independent axiomatization. We now show th at if, for any axiomatization of 3,t he set

nx

x).

n E A@aoA

... Acr,

+antl

is not hypersimple, then 3 is independently axiomatizable. We d o this in two steps:

there is an effective procedure that produces, given any theorem y of 3, another theorem yl that is not deducible from it By hypothesis, we have a strong array { D f ( z ) } z E w t h a t intersects 2. Given any theorem y of 3,first find a n m such t h a t y is deducible from 0 0 , . . . , a m and r then an z such t h a t Df(.) contains only numbers greater than m. T h e conjunction yl of t h e a,+l’s such t h a t n is in D f ( = )is then a theorem of 3 (because a conjunction of axioms) t h a t cannot be

111.10 Formal Systems and R.E. Sets

*

359

deduced from y. Indeed, one number n in Df( is not in A , i.e. an+l ? cannot be deduced from t he a’s with smaller indices, and in particular is not deducible from y. But a,+1 is a conjunct of yfrand thus y’is not deducible from y either. there is a31 independent axiomatization of F Let { a n } n E w be a n enumeration of t h e theorems of F. We want t o generate this set with a set of independent axioms {Pn}nEw.T h e first part of th e proof produces, given any formula y, a formula y‘ not deducible from it. T h e opposite is not necessarily true, since y f could be t o o strong, an d imply y. To have independent formulas, we have t o relax y’a little, a n d this can be done by considering yy V y’,i.e. y + y’. T hi s is not deducible from y as before, since otherwise (by modus ponens) so would be y’.And if y is not valid, it is not deducible from y + y’, otherwise from +y (which is stronger than -y V y’) we would get y,a n d then y would be valid. More generally, t h e sequence

y

y + y’ y’4 y”

..

is independent. We still have t o make sure t h a t all t he an’s are going to be deducible from the axioms, and we simply add them one by one as conclusions, so t h a t an can be obtained from t he first n 1 axioms. Th u s our set of independent axioms for 7 is a set of formulas inductively defined as follows. First we let

+

po = ah. This makes sure t hat we start from a formula t hat is not valid. Then, if Pn = 6n ynr +

Exercise 111.10.14 For any hypersimple set A there is a consistent first-order formal of it, such that system F extending R , and a n miomatiration 7~

EA

* a0 A ... A an

a,+ 1

In particular, F is not independently axiomatizable. (Kreisel [1957])(Hint: let (xo be the conjunction of the finitely many axioms of Q, see p. 23, and of the formula Vx(cp(z) + P ( x ) ) , where P is a new predicate, and ‘p weakly represents A in R. Moreover, let on+] be P(5i). Then a,+] can be deduced from a0 only when n E A . )

Pour El [1968a] shows t h a t Q, and more generally every theory with effectively inseparable sets of theorems and of refutable formulas, has a consistent

360


extension which is not independently axiomatizable, a n d with t h e same language as t h e original theory. For more recursion-theoretical results a b o u t fornial systems see p. 510, as well as Smullyan [1961], Martin a n d Pour El [1970], a n d Downey [1987].

Chapter IV

Hierarchies and Weak Reducibilities T h e t h e m e of this chapter is definability in given languages, a n d a classification of sets a n d relations according to their best definition. Of course definability is not a n absolute notion, a n d it depends on t h e given language. Here we will consider three natural ones, t h e first two for Arithmetic, t h e last o n e for Set Theory. T h e first two will differ in t h a t we will allow only number quantifications in o n e case, b u t also function (or set) quantifications in t h e other, a n d will define, respectively, t h e arithmetical a n d analytical sets. T h e t h i r d approach will lead us to t h e constructible sets. Definability is a linguistical, more t h a n computational, notion. However, we already know t h a t it is possible to characterize t h e recursive sets in a purely linguistical way, see 1.3.6. T h i s suggests t h e possibility of considering definability as a n abstract version of computability, and we will see in later chapters t h a t this program is indeed feasible, for some of t h e definability notions t h a t will be introduced in this chapter.

We only scratch t h e surface of t h e subject in here, a n d refer to Volumes I1 a n d I11 for a detailed s t u d y of t h e arithmetical a n d t h e analytical sets, to which t h e two volumes a r e respectively dedicated. B u t we will prove a number of interesting results already in this chapter, providing some nontrivial characterizations of a number of classes. In particular, we deal with: t h e limit sets, t h a t can be obtained as limits of recursive functions; t h e hyperarithmetical sets, t h a t can be effectively computed modulo number-theoretical quantification; a n d t h e Xi sets, t h a t c a n be defined over t h e constructible universe in a particularly simple way.

36 1

362

IV. Hierarchies and Weak Reducibilities

A: = analytical I I I I I I

I

n: liyperarithmetical I

I I I I

I I

A: = Ah = arithnictical I I I I

I I I

Figure IV. 1 : The Aritlirnetical and Analytical Hierarchies

IV.l The Arithmetical Hierarchy

IV.l

363

The Arithmetical Hierarchy

We start by considering, as t h e simplest framework for t h e definition of sets of natural numbers, First-Order Arithmetic. We first introduce t h e notion of arithmetical definability, a n d then classify sets definable in Arithmetic by looking at their best possible definitions. We will t h u s obtain classes closely related, b o t h in computational content a n d in structure theory, to the classes of recursive a n d recursively enumerable sets.

The definition of truth

* Pilate said unto him, W h a t is t r u t h ? And when he had said this, he went o u t . (John, Gospel, XVIII, 38)

T h e idea for a definition of t r u t h comes from Aristotle (Metaphysica, r 7, 1011b, 25-27): it is false to say t h a t t h e being is not or t h a t t h e non-being is; it is t r u e to say t h a t t h e being is a n d t h a t t h e non-being is not. T h u s e.g. ‘it is raining’ is t r u e if a n d only if it is raining, where t h e quoted phrase is t h e one whose t r u t h we a r e trying to establish, a n d i t is considered as a purely syntactical object, while t h e unquoted phrase is taken semantically, for what it means. And t h e quoted phrase is t r u e if a n d only if it reflects what happens in t h e world, as expressed by t h e unquoted phrase. For natural languages this explains t h e meaning of t r u t h , b u t it is not particularly manageable. T h e advantage of formal languages is t h a t they a r e built by induction, a n d thus we can actually apply t h e ideas just introduced t o produce a n inductive definition of t r u t h . T h e original definition will be applied directly only to atomic formulas, while for compound formulas we will rely on induction, a n d will force only t h e interpretation of t h e new symbols introduced. T h i s was first done by Tarski [1936].

Truth in First-Order Arithmetic T h e r e a r e two natural first-order languages t h a t come to mind, for t h e purpose of classifying arithmetical definitions. They a r e in some sense extreme examples, allowing respectively for a minimal and a maximal set of nonlogical primitive functions a n d predicates. We consider both of them, a n d prove t h a t their definitional power is t h e same.

364


Definition IV.1.1 Definition of truth in Arithmetic (Tarski [1936]) Let C be the first-order language with equality, augmented with constants ?I for each number n, and binary function symbols and x, and let A be the intended structure for C, i.e. the natural numbers, with the usual s u m and product. cp) is inductively Given a closed formula cp of C , cp is true in A (A defined as follows:

+

+

A n n-ary relation P is definable in First-Order Arithmetic (briefly, is arithmetical) if, for some formula cp with n free variables,

As we have already noted, this definition explicitly defines the meaning of a n y syntactical first-order formula over Arithmetic by first reducing t h e nieaning of compound statements to t h e meaning of simpler ones, a n d t h e n by forcing t h e meaning of t h e function constants + a n d x t o agree with t h e usual standard meaning of sum a n d product. After this, being arithmetical is then defined in t h e same way as being representable (1.3.4),b u t with t h e notion of provability in a formal system replaced by t h e notion of t r u t h in First-Order Arithmetic. T h i s procedure of defining t,ruth for formulas of a given language over some structure is quite general, a n d in t h e future we will simply indicate t h e changes needed to extend t h e above definition of t r u t h to different languages a n d structures. Definition IV.1.2 (Kleene [1943],Mostowski [1947])Let L' be the firstorder language with equality, augmented with constants 7 i for each number n, and a relation symbol c p for ~ each recursive relation R, and let A* be the intended structure for C * , i.e. the natural numbers, with all the recursive relations. Given a closed formula cp of C*,A* cp is defined inductively as above, starting from

+

IV.1 The Arithmetical Hierarchy

365

A n n-ary relation P is in the Arithmetical Hierarchy if, for some formula cp of C' with n free variables,

P ( z ~ ,.. ,zn)H A*

+

~p(Z1,.. . ,?En).

I3riefly stated, the Arithmetical Hierarchy consists of the relations definable in First-Order Arithmetic, with the recursive relations as parameters. This seems natural from our point of view, since we want t o classify sets and relations according t o their noneffectiveness, as expressed by t h e complexity of their definition, a n d thus t h e recursive relations may be given for free. T h e next result shows t h a t we are classifying t h e same sets as before.

Theorem IV.1.3 (Godel [1931]) T h e Arithmetical Hierarchy contains exactly the arithmetical relations. Proof. One direction c o m e from t he fact t hat t h e relations represented by t h e atomic formulas of t h e language C are recursive, being built up from plus a n d times only. T h e other direction follows from t he fact, proved in 1.3.6, t h a t t h e recursive relations are representable in R,an d hence (since t h e axioms of R are tr u e in d)are arithmetical. 0

The Arithmetical Hierarchy Before we can classify relations according t o their definition in C*,we need to be able t o p u t these definitions in some kind of normal form. This is easily accomplished in a standard way, by manipulation of quantifiers.

Proposition IV.1.4 T h e following transformations of quantifiers are permissible (up to logical equivalence): 1. permutation of quantifiers of the same type 2. contraction of quantifiers of the same type

9. permutation of two quantifiers, one of which bounded

4. substitution of a bounded quantifier with a n unbounded one of the same type. Proof. P ar t 1 is obvious. P a r t 2 can be accomplished by codifying t h e various quantified variables into a single one. E.g., given a formula VZl . . .'JZ,R(ZI,.

. . ,Zn),

this is equivalent t o t h e formula ' J 4 ( 4 1 , .

where z = ( 2 1 , . . . ,xn).

. .,

b ) T L ) ,


366

Pa r t 3 is obvious when t h e two quantifiers are of t h e same type. For t h e remaining cases, consider e.g. (Vz

I a)(3y)R(z1y).

If we let ys be a number such t hat R(z,ys),for z 5 a , then y = (yo,. . . ,y,) witnesses th e tr u th of

(3Y)(V's I a)%

(Y)Z+d,

a n d thus t h e former formula implies t h e latter, while t h e converse implication holds trivially. T h e other case is treated similarly. P a r t 4 is standard:

(Vz

5 a)R(z)

H

(3z 5 a ) R ( z ) H

Vz(z I a + R(z))

3z(z I a A R(z)). 0

Proposition IV.1.5 Prenex Normal Form (Kuratowski and 'Igrski [1931]). A n y relation in the Arithmetical Hierarchy is equivalent to one with a list of alternated quantifiers in the prejk, and a recursive matrix. Proof. T h e previous transformations allow to contract quantifiers of t h e same type, without changing t h e recursiveness of t he matrix. T hus we only have t o show how to push quantifiers in front. This is accomplished by t h e following well-known transformations, read from left t o right, and which again work up t o logical equivalence. First of all note t hat bound variables can be renamed, according to t h e rules: 3 4 4 Yza(z)

* *

34Y) VW(Y),

where y is any variable t h a t does not occur free in a. T hus we can always suppose t h a t , in th e following rules, z does not occur free in p.

7(3z)a

*

-(Vz)a

e (3z)m

(Vz)-a

(3za)A @

H

3z(aA@)

(b'za)Ap

($

Vz(aA/?)

(3za)v g e 3z((Yvp) (Vxa)vp Vz(avp)

*

IV.l The Arithmetical Hierarchy (3zcr) ( k x )

--p --p

-p-(3sa)

p-

tj

@

@

367

vz(a+p) 3z(a--p) 34a-P)

(Vza) e. Vz(a+-p)

0

It should be noted that the rules t o bring a formula into prenex normal form can be applied in any order. In particular, prenex normal forms are not unique, and may have different numbers of quantifiers. However, since there are only finitely many possible manipulations, the prenex normal form with smallest number of quantifiers can always be found, starting from a given formula. Note t h a t we are not claiming that there is a n algorithm t h a t gives t h e best prenex normal form to which t h e formula is equivalent, since better prenex normal forms might be produced by equivalent formulas. T h e Prenex Normal Form suggests that we only have t o count t h e number of alternations of quantifiers in the prefix, t o measure t h e distance from recursiveness. And of course there are, for each such number, two possibilities, according to whether the first quantifier is existential or universal. It just remains t o give everything a name.

Definition IV.1.6 The Arithmetical Hierarchy (Kleene [1943],Mostowski [1947]) 1.

X : is the class of relations definable over A* by a formula of 13; in prenex form v i t h recursive matrix, and n quantifier alternations in the p r e f i , the outer quantifier being mktential.

2.

II: is defined similarly, with the outer quantifier being universal.

3.

A: is Z:na, forms.

i.e. the class of relations definable in both the n-quantifier

4. A: is the class of the arithmetical relations. By extension, we will call a formula E: or @, if it is in prenex normal form, with n quantifier alternations in t h e prefix, the outer one being, respectively, existential or universal. Note also that, by contraction of quantifiers, n quantifier alternations are equivalent to n alternated quantifiers.

The levels of the Arithmetical Hierarchy T h e first levels of the hierarchy are inhabited by old friends. First of all, t h e level 0 obviously consists of the recursive relations (because no quantifier is involved). More interestingly,


368

A:

= recursive

E:

= recursively enumerable.

This obviously follows from 11.1.10 and 11.1.19. If we wished to, we could define a similar hierarchy for formulas of C. If we count quantifier alternations, and ask the matrix t o be quantifier free (i.e. a Boolean combination of diophantine equations), we still get t h e r.e. sets at the first existential level, by Matiyasevitch result (see p. 135), a n d thus this hierarchy coincides, from t h e first level on, with t h e Arithmetical Hierarchy. By using C' we simply avoid the proof of this representation theorem for t h e r.e. sets on one side, and have more freedom in t h e computations on t h e other. If we a r e willing t o compromise a little, we may allow t h e matrix to contain bounded quantifiers. Then the proof of IV.1.3 would suffice t o show t h a t t h e r.e. sets a r e at the first existential level of this hierarchy (because of t h e form the formulas t h a t represent recursive relations in R have). T h e A: relations of this hierarchy (namely t h e sets definable with plus a n d times, by using connectives and bounded quantifiers) form the interesting class of rudimentary predicates (Sniullyan [1961]),t o which we will return in Chapter VIII. Exercises IV.1.7 The Bounded Arithmetical Hierarchy (Davis 119581, Harrow 119781) Let A: be the class of relations definable over C by using only connectives and bounded quantifiers. We can stratify it by counting the number of bounded quantifier alternations, getting classes E:,,, and @,n in the natural way. Because of the collapse in d) below, A;,,, is defined as the class of sets A such that both A and 3 are in the n-th level of the hierarchy. and E;,,, are closed under conjunction and disjunction. a) b) For n 2 1, II;,,, is closed under universal quantification bounded by a polynomial, and E&, is closed under exktential quantification bounded by a polynomial. (Hint: by induction, since e.g.

(32I P ( q + 9(q)P(z) (32i P ( q . S ( 4 ) P ( 2 )

* *

+

(3a I P(Z))(3bI 9 ( 3 ) P ( a b ) (3aI P ( q ) ( 3 b < 9 ( q ) q P ( q . b + a).)

c) The matrix can always be reduced to a diophantine equation. (Hint: diophantine equations p ( 2 ) = 0, where p is a polynomial with integral coefficients, are closed under Boolean combinat ions. Precisely,

P ( q =SF) P(3C3fO

* *

P(q=0 P(q_ 1, C;' is the class of sets of the f o r m (A1 - Az) U (As - A4) ... , with

Ai r.e. c) For each n 2 1, there are E,'-complete

sets. (Hint: let

(z,y) E A . B * z G A A ~ E B and ( z , y ) E A + B e z ~ A v y € B ,

and consider the sequence of sets

K

K C K

( K . K ) + K : *..)

d) For each n 2 1, E,' - I;' # 0, and I,' - C,' # 0. (Hint: use the sets given in part c.) e) For each 7~ 2 2, there is a A;' -complete set. (Hint: if A and B are, respectively, E;'-complete and II;'-complete, then A @ B is A,:,-complete.) c A;:.' (Hint: use the set given in part e.) f) For any n, C,' UII,' g) For n 2 1, a set is A* :; zf and only if it has a recursive appmximation that Then both A and changes at most n times. (Hint: suppose A is both C,' and I;'. its complement can be approximated by recursive functions g and h that change at most n+ 1 times, and with value 0 at stage 0. For any z, we then know that one of g(z, 0) and h(z,0) is wrong. Look for the first s such that g(z, s) # h ( z ,s), and let g(z, S) be the first value of a new function f approximating A. Let f change only when g does, and g and h differ. Then f changes at most n times.)

Note that there is nothing special about the second level: the characterization of A! obviously extends, by relativization, t o all levels.

Proposition IV.1.19 Shoenfield [1959]) For n 2 1, A is A!+l if and only if there is a n n + 1-ary recursive function g such that

Relativizations

*

We can relativize all the work done so far t o a given oracle X, defining the classes E:x, @" and A:x , and obtaining similar results, simply by substituting 'recursive' with 'recursive in X ' . In particular, we have

ASTB

* A E A:'".

We might thus think that a number of other reducibilities can be defined, by looking at higher levels. The next result shows that this is not the case.

IV.2 The Analytical Hierarchy

375

Proposition IV.1.20 T h e relation

A iao BwAE is transitive only f o r n = 1. Proof. T h e transitivity for n = 1 follows from the fact that in this case l a y is simply Turing reducibility. For any n > 1, we show a counterexample. Let A E E! -R, which exists by the Hierarchy Theorem. Then there is B E pn-l such that z E A w 3yB(z,y). By definition A is in E;lB, and hence in A:B (since n > 1). Since B is in A! by its choice, if 1, is that quantifiers simply sum up: if B has n quantifiers, and we stick n more in front of it, one might collapse (if the leftmost quantifier of t h e prefix of B is of the same type of the rightmost added in front of it), but the others are going t o remain. This ceases to be a problem, if we do not care anymore for a fixed number of quantifiers.

Definition IV.1.21 A is arithmetical in B ( A Sa B ) i f i t is in the Arithmetical Hierarchy relativized t o B. A is arithmetically equivalent to B ( A = a B ) if A l a B and B l a A. Exercises IV.1.22 a) I f A is arithmetical, then A B and B is arithmetical, so is A. b) If A

0 such that Ah+l i s the smallest indexed class of sets uniformly satisfying a Ah co nditio n. Proof. Let P ( z ,x , X ) be a A: formula meaning that X codes a class of sets indexed by numbers, as follows: z E G,

* ( z , z )E X .


392

Then the smallest class of sets 6 defined by P is strictly contained in Indeed, let A be the set coding it: (z,z) E

Then A is

A

H

( t I X ) ( P ( z , z , X+ ) (z,z) E X ) .

and hence, since n > 0,

T h e set B defined as

X E Be ( . , . ) @ A is thus too, but it cannot be in t h e class coded by A , otherwise there would be an e such that

xEB

@

(z,e)E A,

and for x = e we would get a contradiction.

0

Descriptive Set Theory * T h e Arithmetical and Analytical Hierarchies resemble very much, both in notions and results, classical hierarchies that were studied at t h e beginning of t h e century in Descriptive Set Theory, which started as a way of generating from below interesting and graspable classes of functions a n d sets (on T e a l s , not on natural numbers), as opposed t o t h e definition of the continuum as a whole, by the power axiom. Part of the motivation was t o prove t h e Continuum Hypothesis for larger and larger classes of sets, with t h e hope of finally getting t o a complete solution of the problem. T h e classical development has its first landmarks in Borel [1898], who introduces Borel sets (as t h e smallest class containing t h e open sets, a n d closed under complements and countable unions), Baire [1899], who introduces Baire sets (via Baire functions, defined as the smallest class containing t h e continuous functions, and closed under limits), and Lebesgue [1905],in which the classes of Borel and Bake sets are proved to coincide. Lebesgue also introduces the analytic sets (not t o be confused with t h e analytical sets dealt with in this section) as projections of Borel sets, and falsely claims t h a t a n analytic set is Borel. This is corrected by Suslin [1917], where it is proved t h a t a set is Borel if and only if it is both analytic and coanalytic. T h e Projective Hierarchy, obtained by iterating projections and complements, is defined and studied in Lusin [1925] and Sierpinski (1925). T h e classical period ends in the late Thirties, when a stumbling block is reached, with t h e impossibility of extending the theory beyond the second level of t h e Projective Hierarchy. This will be explained later, after the introduction of the methods t o prove t h e independence of the Continuum Hypothesis (Godel [1940], Cohen [1963]): the theory of higher projective classes is mostly independent of ZFC, and thus it cannot be pursued without additional set-theoretical hypothesis.

IV.2 The Analytical Hierarchy

393

A second period of development for the subject comes when the recursion theorists attempt t o classify sets of natural numbers, from their own point of view. Kleene [1943] and Mostowski [1947] introduce the Arithmetical Hierarchy, and while the former works without awareness of the classical work, the latter explicitly develops the hierarchy as an analogue of the Projective Hierarchy. After t h e introduction by Kleene [1955] of the Analytical Hierarchy, the analogies begin t o clarify. Addison [1954], [1959] not only makes them precise, but also sees that there is more: classical Descriptive Set Theory can be obtained b y relativization of the theory of recursion-theoretical hierarchies, using the following translations (in which the left-hand side is the effective version of the right-hand one): recursive function r.e. set hyperarithmetical set set analytical set

continuous function open set Bore1 set analytic set projective set.

In particular, the recursion-theoretical versions of classical results are stronger, and imply their classical counterparts. This explains the double assignment of credit t o results, in this section. After the classical and t h e effective periods, the subject has entered its modern era with the introduction of new set-theoretical axioms (which, as we have quoted, are necessary t o go beyond the first two levels of the Analytical Hierarchy). The first axiom t o imply results about analytical sets was t h e Axiom of Constructibility (IV.4.2): Gijdel [1940] and Addison [1959a] showed t h a t a coherent theory for all levels of the Analytical Hierarchy can be obtained from it. Although provably consistent with Z F C , the Axiom of Constructibility is however taken more as a useful technical tool than as a real additional axiom of Set Theory. The existence of measurable cardinals is taken more seriously, and it does provide for additional results about analytical sets (Solovay [1969], Martin and Solovay [1969]), but its influence does not seem t o extend much beyond the fourth level of the hierarchy. The most successful axiom t o date for the development of Descriptive Set Theory is the Axiom of Projective Determinacy, a restricted version of the full Axiom of Determinacy (V.1.15) saying that all projective games are determined. This can be seen as an axiom about the existence of very large cardinals, and it implies an extremely coherent picture of the Analytical Hierarchy (the first results in this direction have been obtained by Blackwell [1967], Addison and Moschovakis [1968] and Martin [1968]). We will come back t o this subject in the last chapter of our book. Classical references for Descriptive Set Theory are Hausdorff [1917], Lusin [193Oa],Sierpinski [1950],Kuratowski and Mostowski [1968]. The definitive test


394

about the subject, treating classical, effective and modern theory, is Moschovakis [1980].

Relativizations

*

As for the Arithmetical Hierarchy, the work done for t h e Analytical Hierarchy can also be relativized t o a given oracle X , defining t h e classes C > x ,f i x a n d A;x, and obtaining similar results. This time a number of new reducibilities do arise. Proposition IV.2.25 (Shoenfield [1962]) The relation

is transitive, for any n. Proof. For n = 0 this is obvious, since then 0 , A E A$B, and B E A$c. There are P E 1B R E E;-l such that z E A e 3f P ( z ,f ,B ) a V f R ( z f, , B ) .

To show t h a t A E C$c, note t h a t zEA

3 D [ D = B A 3 f P ( z ,f,D ) ] .

T h e expression B = D can be rewritten as

V x [ ( zE B

+

x E D ) A (z E D

+z

E B)].

By using t h e IIkC form for B in t h e first conjunct, a n d t h e E$c form in t h e second one we get, by manipulation of quantifiers, a E$c form for B = D which, substituted above, produces a CkC form for A. Similarly, noting that Z E A@VD[D=B +VfR(z,f,D)],

we get a II$c form for A, which is then A$C. T h e relation

1. C i -

IIt # 0, and hence A:

# 0, and C t u KAc A!+, .

2. IIt - E t 3.

c Ck hence A t c II$

Proof. By diagonalization and IV.3.16, as in IV.1.13.

I3

414


7l.F and the Arithmetical Hierarchy We have noted that the Axiom of Foundation allows a representation of the transitive closure of a set as a well-founded tree, describing t h e set-theoretical build-up of the set from the empty set. We now analyze t h e sets whose associated tree is finite.

Definition IV.3.19 X 3 is the set of hereditarily finite sets, i.e. the smallest class A of sets such that: 1. ~ 2. if

E A 51,.

. . ,x,

E

A then {XI7 . .. x,} E A.

Note the difference between being finite, i.e. having only finitely many elements (like {w}, that consists of only one element), and being hereditarily finite, i.e. having only finitely many elements, each of which is hereditarily finite (a definition by course-of-value recursion). In other words, a set is in XF if and only if its transitive closure is finite.

Proposition IV.3.20 XF is the smallest transitive model of G K P . Proof. We first verify that NF is a transitive model of G K P : 1. transitivity The tree representation of an element of x is a subtree of the representation of x,and it is finite if this is.

2. extensionality and foundation Automatic from transitivity.

3. pair T h e tree representation of {x,y} consists of a vertex on t o p of t h e tree representations of x and y 7and thus it is finite if these are.

4. union

T h e tree representation of u x consists of a vertex on top of t h e tree representation of t h e elements of t h e elements of x, and thus it is finite if there are only finitely many of these trees, each of them finite.

5. separation T h e tree representation of a subset of x is a subtree of the representation of x, and it is finite if this is.

IV.3 The Set-Theoretical Hierarchy

415

6. collection Suppose (b’xE a ) ( 3 y ) c p ( x , y )holds in X 3 . Since n is finite, it has only finitely marly elements ~ 1 , . . ,xn. For each of t h e m , t h e r e is a set y l in X 3 such t h a t cp(x,,yi)holds. By pair a n d union, which we have already verified, b = {yl,.. . , y n } is in ‘HF, a n d thus (Vx E n)(3y E b)cp(x,y) holds in ‘H3. We now verify t h a t ‘HFis t h e smallest transitive model of G K P . Suppose A is such a model: t h e n i t contains 8, a n d it is closed under pair a n d union. We want to show t h a t ‘HF5. A . T h i s is easily seen by induction, since each element z E ‘HF is obtained from t h e emptyset, by finitely many applications of pairing a n d union.

Note t h a t t h e natural numbers, represented in set-theoretical t e r m s (as finite ordinals), a r e all in ‘HF, by induction:

O=8

and

n+l=n~{n}.

Exercises rV.3.21 a) 7 - U = V,. b) 7-V is the smallest model of Z F C with the Axiom of Infinity replaced b y its own negation. (Hint: the power set of a finite set is finite. Choice is trivial, since the elements of W are all finite. Since w is not in ‘HF, the Axiom of Infinity does not hold. Its negation, that every set is finite, holds because if x E 7-V then any function from z to some natural number is already in W . )

T h e reason why we are particularly interested in ‘H3is t h a t , for sets of natural numbers, set-theoretical definability over it coincides ( a n d not only g l o l d l y , b u t level by level) with arithmetical definability. T h i s provides a n alternative, set-theoretical way of seeing Recursion Theory, a n d it is t h e s t a r t i n g point for some interesting generalizations (see p. 421).

Theorem IV.3.22 Set-theoretical definability of the Arithmetical Hierarchy (Ackermann [1937]) Let A C_ w. Then: 1. A E

‘HFi f and only

2. A is definable over f o r n 2 1:

if A is finite

‘H3i f and only i f A is arithmetical. More precisely, AEA;-’ AEC?’

AEA:

e AEC:.

Similarly f o r relations, of a n y niimber of variables.


416

Proof. T h e first assertion is easy to see: a n hereditarily finite set is, in particular, finite; a n d a finite set of natural numbers is hereditarily finite, because so a r e t h e natural numbers (as set-theoretical objects). T h e proof of t h e second assertion is more cumbersome, arid it a m o u n t s to show t h a t we can translate, by preserving t h e logical complexity, arithmetical assertions into set-theoretical ones, a n d conversely: 1. translation from Arithmetic t o Set Theory We already know how t o interpret natural numbers in set-theoretical terms. T h e n number quantifiers can be easily t u r n e d into set quantifiers:

(3x)(P(x) (Y.)(P(.)

--

( 3 x ) ( xE w A Cp(x))

(W. E w

--+

Y W .

T h e expression x E w is AFKp,hence A:=, a n d t h u s it does not increa3e t h e complexity of t h e rriatrix. I t only remains t o translate recursive matrices, i.e. graphs of recursive functions, into AFKr predicates. We refer to t h e characterization of recursive functions given in 1.1.8. S u m , product, a n d composition have already been dealt with in IV.3.14, while identities a n d equality a r e trivially AfK? For p-recursion, let

f ( q = CLYR(Z,Y). Then

f(z)= y

@

R(z,y) A (‘d~< y ) i R ( i ? , 2).

If R is A p K Pthen so is t h e graph o f f , by t h e closure properties of AFKP, a n d t h e fact t h a t bounded number quantifiers translate into bounded set quantifiers (because t h e order relation on ordinals is iiiduced by t h e membership relation). Note that, t h e whole argument does not require t h e exist,erice of w. If

w were present (i.e. if we worked with a model of G K P plus infinity), all t h e translations of arithmetical fornnilas would simply become Ao, because t h e number quantifiers would then be translated into bounded set quantifiers (see p. 419 for more on this point).

2. translation from Set Theory to Arithmetic First of all we have to interpret members of 7-t.F as natural numbers, a n d this can be done by induction 011 t h e construction of 7-t.F:


417

(we suppose all t h e z, distinct, since a set is determined solely by its elements). T h i s simply amounts to using canonical indices (11.5.13) hereditarily, by inductively decomposing t h e exponents in t h e binary decomposition of a number. Now set-theoretical quantifiers c a n be turned into number quantifiers:

(3z)(P(x) @

*

(VX)(P(.)

(wf4f-w (W(P(f-'(74).

I t remains to be proved t h a t t h e translations of AZF formulas a r e recursive. By t h e parallel closure properties of AFKP formulas a n d recursive relations, this reduces to show how to deal with membership a n d constants. For t h e former, note t h a t E

Y

* f(.)

E Df(v),

which is a recursive relation. To deal with constants, note t h a t a m o n g t h e values off there a r c some t h a t naturally correspond to t h e set-theoretical integers :

do) = 0 g ( n + 1) = g(n)+29(")

+

(because 0 = 0, a n d n 1 = n U {n}). Now g can be thought of as a function b o t h from 7l.F to w ,a n d from w to w. In t h e latter case, it is a recursive function. Arid g can be used to substitute occurrences of set-theoretical natural numbers with occurrences of t h e corresponding natural numbers coding t h e m as sets. 0

We have proved a correspondence between definability on Arithmetic a n d

X.F, for relations on natural numbers. T h i s suffices for our recursion-theoretical purposes, b u t there is more to it. It can actually b e shown t h a t , for s t a t e m e n t s a b o u t natural numbers, P A is equivalent to Z F C with the Axiom of Infinity replaced b y its negation (which is equivalent to V = 'H3) (Ackermann [1937]). I n other words, t h e translations provided in t h e proof above a r e actually faithful interpretations of t h e stated theories into one another (where interpretation means t h a t provable statements a r e translated into provable statements, a n d faithfulness t h a t no translation is provable unless its original version was already provable). Note t h a t a symmetric role is played by induction a n d foundation, which is t h e reason to consider Peano Arithmetic, a n d not weaker systems. T h e absence of t h e Axiom of Infinity is enough for t h e faithfulness of t h e translation of P A into Set Theory. T h e substitution of t h e Axiom of Infinity

418


with its negation is instead crucial to prove t h e faithfulness of t h e translation of Set Theory in P A (since otherwise ( V X ) ( XE XF),which is equivalent to t h e negation of t h e Axiom of Infinity, is not provable in Set Theory, while i t s translation, which amounts to ( V X ) ( XE u), is provable in P A ) .

Absoluteness and the Analytical Hierarchy Since there is no single privileged standard s t r u c t u r e for Set Theory, we will have to interpret t h e formulas on t h e various structures. T h e problem is t h a t t h e s a m e formula could be t r u e in some model of G K P a n d false in some other, t h u s not having a n absolute meaning. As a n example, consider t h e set zEbw

x E a A ( V z ) ' p ( ~z, ) ,

obtained from a by separation. Suppose 'p has no quantifier. For a n y A such t h a t a E A , there is a set b A obtained by interpreting t h e definition of b over

A: z E b A -35 E a f'

A A (b'z E A)cp(Z, Z ) .

If A is transitive then a n A = a , so X

E bA @ X E aA('dz E A ) c p ( ~ , z ) ,

Rut (Vz E A ) y ( z ,z ) could hold even if ( V Z ) ~ (zX) ,does not. T h u s b b A , a n d ' A believes b A is b', while this is not necessarily so. Similarly, by changing t h e universal quantifier into a n existential one, we could have b A b, while for more complicated formulas there is n o simple relationship between t h e t r u e b a n d t h e set b A , t h a t A believes to be b. T h i s is of course a n unpleasant situation, introducing a n element of relativity in Set Theory: on one side we have t h e 'real' sets, on t h e other their interpretations over given models, with no apparent connection between t h e m . T h e situation is not as disruptive as it might seem at first sight, since a number of formulas t u r n o u t t o have a n absolute meaning, in t h e sense of defining t h e s a m e set on every model.

Definition IV.3.23 (Godel [1940])A formula is absolute for a class of structures i f it has the same truth value in each structure of the given class. T h e next result isolates a class of formulas t h a t a r e absolute, a n d i t is q u i t e useful in applications. T h e notion of s t a n d a r d model is used in a crucial way, t h u s providing another reason to restrict our attention to such models.

Proposition IV.3.24 AT formulas are absolute f o r the standard models of T .


419

Proof. Fix a transitive model A of T : we show t h a t t h e t r u t h value of any

AT formula interpreted over A is independent of A , a n d it coincides with t h e t r u t h value of t h e formula in t h e universe V of sets. First of all note t h a t , over elements of a transitive set A , membership is absolute: indeed, if z E A then z n A = z, a n d t h u s z E z has t h e s a m e meaning over A a n d over V . This shows t h a t , in particular, bounded quantifiers preserve absoluteness a n d thus, by induction on their complexity, A: formulas a r e absolute for standard models of T . Suppose now cp is AT. We want to show t h a t cp is t r u e over A if a n d only if it, is t r u e (over V ) . Let $1 a n d $2 be AT formulas such t h a t , in T , cp(.’)

-

P!/)@l(Z,Y)

-

(VY)+2(Z,Y).

Suppose cp(Z), i.e. (EIy)+l(Z,y), is t r u e over A . T h i s means t h a t , for some y E A , +l(Z,y) is t r u e over A . But this is a AT formula, which is absolute. T h e n @1(5,y), a n d hence (3y)@l(Z,y) a n d (p(Z),a r e true. Suppose now cp(Z), i.e. (Vy)@2(Z,y), is true. T h e n @2(?,y) is t r u e for all y , in particular for all y E A . T h u s (Vy)+2(Z,y), a n d hence cp(5),a r e t r u e over

A.

0

In t h e Arithmetical Hierarchy quantifiers range over w. For a n y model A of G K P a n d t h e Axiom of Infinity, w E A . T h e n number quantifiers can be interpreted as bounded set-theoretical quantifiers, a n d arithmetical relations are absolute, being translated into A; formulas (when s u m a n d product a r e replaced by their A f K pdefinitions, see IV.3.14.a). In t h e Analytical Hierarchy quantifiers range also over P(w), which is only a I I ? ~ ‘ object: z = P(w) H Vy(y E z H y w).

c

P(w) i s n o t absolute, a n d hence not A:”‘: by absoluteness of y C w , i t s interpretation over a transitive model A of G K P is P(w) n A , a n d t h u s it varies with A (see also IV.4.27.c). T h i s means t h a t function quantifiers do not automatically translate into bounded quantifiers, a n d analytical relations a r e not, automatically absolute. But relations in t h e first two levels a r e , as we now see.

Proposition IV.3.25 (Mostowski [1949])II: relations are absolute for standard models of G K P containing w (i.e. models of Z F - ) . Proof. By t h e First Representation Theorem for IIi sets (IV.2.15), A is II: if a n d only if t h e r e is a recursive sequence {Tz},E,of recursive trees, such t h a t z E

A ts T, is well-founded.


420

Rut we have already noted t h a t recursive relations (being arithmetical) a n d well-foundedness (being A f K P )a r e absolute. T h e n so is A . 0 Note t h a t t h e proof shows t h a t every ni relation is actually A, over G K P plus infinity. Similarly for Ci relations, by taking negations. Exercise IV.3.26 n: formulas are n o t absolute f o r standard models of Z F C minus infinity. (Hint: W is a model for it, but the relations over w definable over it are all arithmetical.)

Theorem IV.3.27 (Shoenfield [196la])Ci relations are absolute f o r standard models of G K P containing all countable ordinals. Proof. By t h e relativized version of t h e First Representation T h e o r e m for ni sets, A is C; if a n d only if there is a recursive sequence { T , , f } z c wof trees uniformly recursive in f , such t h a t zE

A

H

(3f)(T,,f is well-founded).

Since well-foundedness of T,,f is equivalent to t h e existence of a n order-preserving m a p from T,,f t o t h e countable ordinals (because t h e trees a r e countable), zE

A

tj

(3f)(3g)(g : T,,f

+ w1

is order-preserving)

Now we can reproduce t h e proof of t h e First Representation T h e o r e m IV.2.15 (by looking at t h e first places in which t h e matrix fails), a n d get uniformly absolute trees R,,f,s on w x w1 such t h a t

x E A tj R,,f,, is not well-founded (since A is now defined b y existential quantifiers, in place of universal ones). T h u s , if M contains all countable ordinals, A is absolute for M . 0 Note t h a t t h e proof shows t h a t every relation is actually of t h e form (Y ranging over countable ordinals, a n d ‘p A, over G K P plus infinity.

(3a countable)cp, with

Exercise IV.3.28 Ci formulas are not absolute f o r standard models of Z F C . (Hint: the formula translating ‘the set X w codes a countable transitive model of ZFC’ is A:, by arithmetization. ‘There exists a set X coding a transitive model of ZFC’ is thus a true Ci formula, which is not true in the least countable transitive model of Z F C . Note that this reasoning requires the existence of a standard model of Z F C , and thus it is not formalizable in Z F C . For the weaker result relative to standard models of Z F C - only, i.e. without the Power Set Axiom, such a n assumption is not necessary.)


Admissible sets

42 1

*

We have worked with the theory G K P because we wanted t o have structural results for all levels of t h e Set-Theoretical Hierarchy. But the full power of G K P is needed only for the full results, and it is possible t o refine G K P , and isolate what is needed to get the structural results for t h e first (or, more generally, t h e n-th) level only. T h e Kripke-Platek system K P (Kripke [1964], Platek [1966]) has t h e same axioms of G K P , with separation and collection limited t o A0 formulas, and it can be seen as a kind of constructive Set Theory. This theory is strong enough to prove (separation for A, formulas, collection for C1 formulas, and) t h e closure properties of t h e first level of the Levy’s Hierarchy. T h e transitive sets which are models of K P (i.e. the standard models) are called admissible sets, and can be seen as domains suitable for a theory of C1 relations and functions analogous t o that of r.e. relations and partial recursive functions, and hence for a Generalized Recursion Theory on sets. Note that XF is the smallest admissible set, and thus the usual notion of recursiveness is a special case of recursion on a n admissible set (by IV.3.22). Moreover, Gordon [1968] has proved that on a n admissible set the notion of recursiveness coincides with t h a t of search computability (see p. 204). This should not be taken t o mean that the notion of admissibility is either sufficient or necessary for a n abstract analogue of all parts of Recursion Theory: Simpson [1974] and Harrington (see Chong [1984]) have shown t h a t there are admissible sets which do not admit a positive solution to the analogue of Post’s Problem, while F’riedman and Sacks [1977] have extended a good deal of Recursion Theory, including a positive solution t o the analogue of Post’s Problem, t o special nonadmissible sets. T h e full power of G K P is not always avoidable, even in the study of t h e first level of t h e Set-Theoretical Hierarchy. A crucial example is the notion of well-foundedness, which is ApKP but not Afp, and thus is not absolute for admissible sets. What fails here, since well-foundedness is IIfp by definition, is the possibility of carrying on recursion on well-founded relations (the so called @-property,Mostowski [1959]), and thus t o provide the C r p form: we have noted t h a t t h e justification of recursion on well-founded relations requires some form of separation. A,-separation, provided by admissibility, is not enough, although C1-separation is. In particular, the Collapsing Lemma IV.3.5 is not provable in K P , although its corollary is (because its proof requires only a course-of-value recursion on E). For a development of t h e theory of admissibility see Barwise [1975] and Fenstad [1980]. T h e implications of the notion of admissibility for t h e study of P(w) will be dealt with in Volume 111.

422

IV.4


The Constructible Hierarchy

T h e Analytical Hierarchy is immensely extended a n d i t contains, already at low levels, all sets of natural numbers naturally occurring in practical considerations. Nevertheless it is still countable, a n d t h i s h a s to be t r u e of all t h e hierarchies t h a t simply stratify t h e relations definable in s o m e fixed countable language, including t h e universal language of Set Theory. If we want to overcome this defect, we have to allow for a n extension of t h e notion of definability. O n e way to do this is by transfinitely iterating definitions, a n d we pursue t h i s p a t h here.

The Constructible Hierarchy At t h e t u r n of t h e century various mathematicians began t o feel uncomfortable with t h e development of Set Theory. T h e center of t h e dispute was t h e Power Set Axiom t h a t , in one of its simplest applications, allowed consideration of t h e class of all sets of natural numbers as a completed totality. T h e discovery of paradoxes added ground to t h e objections, a n d one possible way o u t was seen in a strong form of t h e vicious circle principle, according to which a totality cannot contain members t h a t a r e only definable in t e r m s of it (PoincarC [1906], Russell [1908]). As a consequence, only predicative definitions, t h a t define objects without referring to sets already containing t h e m , would be accepted. Contrary to t h e form of t h e vicious circle principle considered o n p. 399, which is consistent with usual mathematical practice, a n d is actually a consequence of t h e Axiom of Foundation, this strong form would permit only a limited development of mathematics. See e.g. Weyl [1918] for what c a n be saved of analysis, a n d Godel [1944], [1964] for discussion. T h e idea of a predicative iterative approach, in which o n e would s t a r t with easily definable a n d graspable sets, a n d would add at each step only t h o s e sets t h a t were definable by using t h e previously obtained ones, is however worth pursuing. T h e decision of when (i.e. at which ordinal) to stop t h e iteration process is crucial. If we really were interested only in t h e predicatively definable sets, t h e n we should allow only for predicatively definable ordinals: t h i s requires a n analysis of t h e notion of predicativity, a n d will be considered in Volume 111. Here we t a k e a more generous s t a n d , a n d allow for a n y number of iterations. By so doing we lose t h e property of predicativity, a n d we can think of the constructible hierarchy as consisting of those sets which are predicatively definable modulo the ordinals. T h i s hierarchy should not be taken as exhausting t h e whole universe, b u t rather as a kind of minimal model (see IV.4.7). T h e constructible hierarchy also extends ideas of Hilbert [1926], w h o tried to prove t h e Continuum Hypothesis by considering t h e generalized recursions (using higher-type objects) needed t o generate all functions of natural num-

IV.4 The Constructible Hierarchy

423

bers, and attempted a proof to show t h a t they could be reduced t o transfinite recursions on ordinals up t o w1. Gbdel’s improvements on Hilbert’s tentative are of two kinds: he uses all ordinals, instead of only countable ones, and firstorder definitions, instead of recursions (which correspond only t o onequantifier definitions).

Definition IV.4.1 (Godel [1939])T h e Constructible Hierarchy is described, in terms of ordinals, as follows:

Lo La+1 Lp L

= 0 = def(La) = Ua 0 t h e n follow by adding quantifiers, because P(w)C

‘He.

To generalize t h e result for Ci to a n y level we have used in a crucial way t h e fact t h a t P(w) IFIC. T h e analogue could not be proved for L , a n d t h i s was t h e reason why we had to content ourselves with t h e absolute result for X i , in IV.4.20. Of course, t h e full result can be obtained also for L if we assume V = L (Takeuti a n d Kin0 [1962]), by t h e s a m e proof (or by IV.4.25.a). As it was already t8hecase for 1V.3.22, embedded in t h e proof of IV.4.26 a r e translations of Second-Order Arithmetic a n d Set T h e o r y on IFIC into o n e another. T h e translation of Second-Order Arithmetic to Set T h e o r y is stand a r d , while t h e other consists of seeing hereditarily countable sets as countable trees (and t h e n coding t h e m as subsets of w),a n d by interpreting equality as tree isomorphism. Peano Arithmetic was equivalent to Z F C with t h e Axiom of Infinity replaced by V = ‘H3:we now have t h a t Second-Order Arithmetic is equzualent to Z F C with the Power Set Axiom replaced b y V = ‘HC (Kreisel [1968], Zbierski [1971]). It should be noted t h a t by Second-Order Arithmetic we mean here t h e second-order version of PA plus t h e Axiom of Comprehension for analytical formulas (asserting t h e existence of t h e analytical sets), a n d a n Axiom of Choice which asserts t h a t if (Vx)(ElA)(p(x,A)t h e n there is a subset A of w x w such

IV.4 The Constructible Hierarchy

443

T h i s obviously t h a t , if A , is t h e section of A w.r.t. 2, then (b'z)cp(z,A,). corresponds to collection for sets, a n d it is needed ( G a n d y [1967a]) to model t h e Axiom of Collection. Second-Order Arithmetic, as well as subsystems of it obtained by variously restricting t h e Axioms of Comprehension a n d Choice, will be studied in Volume

111. Exercises W.4.27 a ) Levy Absoluteness Lemma. A Ef" formula with parameters in 'HC true in V is already true in 'HC. (Levy [l965]) (Hint: by LowenheimSkolem, Collapsing Lemma, and absoluteness, as in IV.4.13.) h) P ( w ) n L G L,, follows from Levy's Absoluteness Lemma and Af"-definability of L, (Karp 119671). This provides a slightly different and easier proof of IV.4.13. (Hint: if A G w and A E L the CP" formula (+)(A E L a ) with parameter A E 'HC is true in V. By Levy Absoluteness it is true for some countable ordinal.) c) P ( w ) is not A?". (Hint: otherwise the formula asserting its existence would be true in 'HC.)

Recursion Theory on the ordinals

*

T h e whole idea of constructibility rests o n t h e fact t h a t L is definable by recursion on t h e ordinals. Since usually t h e ordinals a r e defined within Set Theory, so is L. While investigating t h e theory of ordinals, Takeuti [1957] discovered t h a t it could be developed independently of Set Theory. It t h u s became clear that, if o n e could also develop independently a theory of recursion o n t h e ordinals, t h i s would allow a different approach to L . Takeuti [1960] carried o u t t h e task, by defining t h e notion of recursive function on t h e ordinals by schemata, in i i way similar to recursiveness on t h e integers. He then discovered t h a t Giitlel's result IV.4.13 could be recast in recursion-theoretical t e r m s by saying t h a t w, (and, more generally, any uncountable ordinal) was stable, in t h e sense of being closed under t h e recursive functions on all t h e ordinals. W i t h this, Recursion Theory was generalized both t,o t h e class of all ordinals, a n d to cardinals. Independently, a n d motivated by needs related to t h e theory of infinitary languages, Machover [1961] developed a n equivalent approach to recursion on cardinals, using systems of equations. T h e circle was closed by Takeuti a n d Kino [1962], when it was realized t h a t recursion on t h e ordinals was actually equivalent to E: definability. After discovering t h a t cardinals were appropriate domains for Recursion T h m r y , it was natural to wonder whether t h e strong closure properties of cardinals were somehow needed. Kripke [1964] a n d Platek [1966] answered t h e question by reversing t h e attack. T h e y relativized t h e previous approaches to any ordinal a , by defining t h e a-recursive functions (e.g. by schemata, using a scm-c.11 operator on ordinals less t h a n a ) . T h e n t h e y defined admissible


444

ordinals as t h e a’s closed under t h e a-recursive functions, a n d showed t h a t for t h e m all approaches a r e equivalent. In particular, for a n admissible ordinal a , &-recursiveness means ,Efa definability, a n d L, is t h e smallest s t a n d a r d model of K P of ordinal a . This relates effective Set Theory to Recursion Theory on t h e ordinals, a n d provides finer versions of various results of t h i s section. T h e first two admissible ordinals a r e w a n d wfk (see p. 385). While admissible sets t u r n out to be nice domains only for elementary Recursion Theory (see p. 421), niany deeper p a r t s of Recursion T h e o r y carry over to any admissible ordinal (see Chong [1984] for a detailed t r e a t m e n t ) . In particular, Post’s Problem always a d m i t s a positive solution (Sacks a n d Simpson [1972]). Admissible ordinals will provide, in Volume 111, a uniform way of describing a number of classes of subsets of w ,a n d will also be useful from a methodological point of view, for a better understanding of which properties of w a r e used in proofs of single results.

Relativizations

*

As for t h e Arithmetical a n d Analytical Hierarchies, t h e work d o n e for t h e Constructible Hierarchy c a n also be relativized to a given set A . T h e r e a r e two natural ways of doing t h k , corresponding to adding A as a constant or as a predicate.

Definition IV.4.28 (Hajnal [1956], Levy [196Oa]) 1. The class L[A] is defined like L, by starting with A U T c ( A ) in place of

0. 2. T h e class L(A) is defined like L by allowing, at successor stages, also parameters over A . From a set-theoretical point of view, t h e two ways a r e not equivalent. T h e first produces t h e smallest standard model of Z F containing A a n d all t h e ordinals, a n d i t does not necessarily satisfy t h e Axiom of Choice, without further assumptions on A . T h e second produces t h e smallest s t a n d a r d model M of Z F C containing M n A a n d all t h e ordinals. From o u r point of view, however, they a r e equivalent, since we only consider relativizations to sets A w.

Definition IV.4.29 A is constructible from B ( A ILB ) zf i t i s in L [ B ] . A i s constructibly equivalent to B (A B ) i f A 5~ B and B st A . Exercises W.4.30 a) If A is constructible, then A b) I J A < L B and B is constructible, so is A.

5~ B

for any

B.

IV.4 The Constructible Hierarchy Note t h a t relation.

[(/,

h) E

A].

Now C(n,ul is open (because union of basic open sets) a n d dense (because so is

A ) .If

n

c = c(n,ul n,o

t h e n C is comeager a n d , as above,

d, is comeager whenever g E C.

0

Exercises V.3.12 A has the Baire property if there is an open set U such that ( A- U ) U (U - A ) is meager, i.e. A differs from an open set by a meager set. Ifd has the Baire property, then the converse implications in the Kuratowski-Ulani Theorem hold. (Hint: suppose A is not meager, but {g : A, meager} is comeager. Since Ag differs from the open set U, by a meager set, U, must be meager for a comeager set of g's. But U is not empty, and it contains a basic open set: hence there is also a comeager set of g's such that U, contains a n open set, and thus is not meager. The intersection of these two comeager sets is not empty by the Baire Category Theorem,

contradiction.)

Baire Category and Degree Theory Going back t o t h e proofs of t h e results in Section 2, we notice t h a t t h e y all shared t h e following characteristic features. We had a set of requirements R, to satisfy, which may be identified with t h e class of sets satisfying t h e m . T h e general p a t t e r n of t h e proofs was to show t h a t , for each n,

T h u s we were actually showing t h a t , for each n, t,here is a n open dense set contained in h!,, (by V.3.6). T h e Baire Category Theorem could t h e n be applied to claim t h a t t h e intersection of requirements is not empty, a n d hence t h a t t h e r e is a set satisfying all t h e requirements, without further constructions. More precisely:

Proposition V.3.13 The Finite Extension Method (Myhill [1961], Sacks [1963]) Given a countable collection of requirements R.,, such that (Yo)(+

the set

2 a ) ( V A 2 T ) ( A E &),

nnEwR,, is comeager (and hence nonempty).

V. Turing Degrees

478

T h e categorical approach is useful for a methodological analysis of t h e finite extension method. Precisely, from it we obtain t h at : 1. requirements can be taken care of separately, b y showing that each of them

is dense 2. we can freely combine constructions known to be performable separately, as long as the global list of requirements remains countable For example, suppose we want t o prove t ha t , given a countable sequence {Ao,A l , .. .} of nonrecursive sets, there is a set B incomparable with all of them. Then we only have t o prove t hat , for a given e a n d a given nonrecursive set C ,

(Vo)(+)(VA 2 7 ) ( A74 {e}") (Vo)(+)(VA 2 .)(C $ {e}"). We know th a t we can d o this, by t he work done in Section 2. It thus follows th at, for a fixed nonrecursive set C , t h e set

is comeager, and hence so is

being a countable intersection of comeager sets. In particular, t h e set is not empty by Baire Category Theorem. Note t h a t we could even avoid t h e consideration of t he condition A $T C , because there are only countably many sets recursive in C: t h e set { A : A $T C} is thus automatically comeager.

3. we cannot use the finite extension method t o produce sets belonging to a meager class This follows from t h e fact t hat t h e finite extension method produces sets in a comeager class, and t h e intersection of two comeager classes is nonempty, by t he Baire Category Theorem. In particular, if we can use th e finite extension method to build a set satisfying certain requirements, then we cannot use t h e same method t o build a set not satisfying t h e same requirements. 4. a game-theoretical approach to Degree Theory is possible, via Banach-

Mazur games This follows from V.3.10, and has been developed by Yates (19761. See also p. 495.

V.3 Baire Category *

479

We can reforniulate a number of results proved before in terms of category notions, with t h e convention t hat a set of degrees A is comeager or meager if such is t h e class of sets whose degree is in A. This makes sense because a degree contains only countably many sets, and thus it is a meager class.

Proposition V.3.14 The following sets of degrees are comeager: 1. { a : a is incomparable with a &ed

nonrecursive degree}

2. { a : a is the 1.u.b. of two incomparable degrees}

3. { a : a is the 1.u.b. of a minimal pair}

4. { a : a

realizes the least possible jump}.

The following sets o f degrees are meager:

5. { a : a is comparable with a fied nonrecursive degree} 6. { a : a is a minimal degree}.

Proof. T h e proof of V.2.2 builds two sets A and B with incomparable degree: this can be seen as t h e construction of a single set A @ B which is t h e least upper bound of two incomparable sets, a n d proves 2. Part 6 follows from this, since a minimal degree cannot be the 1.u.b. of two incomparable degrees. Similarly, t h e other parts follow from V.2.13, V.2.16 and V.2.24. 0 Note t h a t this shows, in particular, t hat degrees comparable with a given nonrecursive degree, as well as minimal degrees, cannot be built by the finite extension method. T h e work done so far also allows t o decide whether any quantifier-free question about jumps, I.u.b.’s a n d g.l.b.’s of degrees holds for a comeager set of degrees or not.

Proposition V.3.15 (Stillwell [1972])The theory of degrees with subformulas containing a term t l n t2 thought of as pre&ed by ‘tin t2 exists’, and with the quantifiers V and 3 interpreted, respectively, as meaning ‘for a comeager set of degrees’ and ‘-b’, is decidable. Proof. For simplicity, we will say ‘almost always’ t o mean ‘for a comeager set of n-tuples of degrees’. Note t h a t terms are obtained from variables and 0 by using n, U, a n d ’. By induction, we show t h a t for every term t there are degrees ai occurring in t such th at, for some m,

t = al u

-.- u a,,u o ( ~ )

V. Turing Degrees

480

is almost always true. T h i s reduces every t e r m to a normal form. 1. ( ~ ~ U . . . U U , U O ( P ) ) U ( ~ ~ U . . . U ~ , U O ( ~ ) a1 U U a, U b l U * . U b, U 0

-

..-

/=

max(p9q)).

T h i s always holds.

-

-

2. (alU- .Ua, U O ( P ) ) n ( b l U. -Ub , U O ( Q ) ) = c l U - .U c, almost always, where f

UO(m'n(P9Q))

{ c i , . . . ,~,}={ai,...,~,}n{bi,...,bn}.

First note t h a t it is always possible to rearrange terms, a n d possibly introduce a new O(m'n(P*q)) (which has no influence on U) in such a way to have, for some d a n d e, a1 u bl U

... lJa, . - u bn

u o(p) = U O(Q)

.

u . .u c, u o(m'n(p,q)) I u d

(cl

= (c1 U

0

-

U c, U O("'"(p~Q)))

U e.

It is t h e n enough to show t h a t almost always, given a a n d d, is ( a u d) n ( a u e)

= a.

And this is j u s t t h e relativization to a of t h e fact t h a t almost every degree d is p a r t of a minimal pair (V.2.16).

--

-

3. (al u * u a, u o(P))' = al u * u a, u o(P+') almost always. ' = aUO' a n d , by relativization to b, almost always First, almost always a ( a U b)' = a U b'. T h e n , by induction, almost always a u O(P) = a ( P ) . Finally, almost always a1U...Ua,

uo(p)

=(alU...Ua,)(P),

a n d hence

T h e decision procedure now follows easily, since every formula with free variables is 0,l-valued, being satisfied either by a comeager set of degrees or by a meager one. Precisely: 1. t l 5 t2, with t l a n d t 2 terms After putting t h e two terms in normal form, t l 5 t 2 holds if a n d only if all t h e variables of t l appear in t2, a n d t h e exponent of 0 in t l is not bigger t h a n t h e exponent of 0 in tz.

V.3 Baire Category

2.

*

481

7+

T h e complement of a meager set is comeager, a n d t h e complement of a comeager set is meager.

3. + A v T h e intersection of two comeager sets is comeager, and t h e intersection of any set with a meager set is meager. 4. Vx+(x) This follows from t h e Kuratowski-Ulam Theorem, by t h e interpretation of t h e universal quantifier. 0

We should not expect too much from this decision procedure, since practically every interesting question we may ask will involve real quantifiers as well. E.g., it is tr u e t hat almost every degree has no minimal predecessor (see V.3.17), but we cannot express this sentence in t he above language, since t h e notion of minimal degree requires a true universal quantifier.

Meager sets of degrees T h e results quoted above were simply old facts rephrased in categorical terms. We now prove a theorem whose very statement genuinely requires t h e notions introduced in this section. T h e plan of t he proof can be read independently, but its implementation relies on methods and notations t hat will be introduced in Section 5.

Theorem V.3.16 (Martin [1967]) If A is a downward closed, meager class of degrees then the upward closure of A - ( 0 ) is still meager. Proof. We want to build A by finite extensions, in such a way t o have

This condition can be broken down in t h e requirements

Re : (e)" total Let

00 =

0. At stage e + 1, let u,

(e)" @ A - ( 0 ) .

be given: we attack Re.

1 . If there is u 2 ue such t hat (3Z)(VT 2 m e ) W

t)

then we let oe+l = 0. This ensures t hat if A 2 total.

ue+l

then (e)" is not

V. Turing Degrees

482

2. If there is a (T 2 oe with n o e-splitting extensions, again let This ensures t h a t if A 2 0,+1 then { e } A is recursive.

ue+l

=

(T"

3. Suppose now t h a t :

0

(yo2 ae)(yX)(37 2 g)({e}T(z>1) for all (T 2 (T, t h e r e a r e e-splitting extensions of

0.

It will be useful in t h e following to picture {e}' as a finite string, with t h e property t h a t (T 0' {elu c { e l u ' .

*

T h e proof goes roughly as follows. We build a n order-preserving recursive m a p Q, from strings to strings such t h a t

( y 2~ @ ~ ( { e } ~ ~ )2) (m3e )0[ . = @({e}')]. This gives a homeomorphisin between t h e set {CP({e}") : (T 2 (T,} a n d t h e whole space of strings. Since A is meager, there is a n open set contained in 3, hence there is u 2 u, such t h a t

2 4(@WA) @ 4. But CP is recursive, a n d so Q , ( { e } A ) downward, hence

ST

{e}A.

Moreover, A is closed

( y A 2 ~ ) ( { e@}4 ~, and

Re is satisfied.

T o finish t h e proof, it remains to build Q,. We first build a n admissible triple (i.e. a uniform tree, see V.6.2) with t h e following properties: if TO(.) is t h e set of strings extending oe,of length g(n+ l ) , a n d agreeing with f~ on t h e g ( n + l ) ) ,a n d T,(n)is defined similarly using f ~t h e, n whenever interval [g(n), (TO E To(n) a n d 01 E Tl(n),(TO a n d (TI e-split. T h e construction is t h e s a m e as in V.6.5, only we have to consider every string extending u, a n d of length g ( n ) , when we build t h e n 1 level (instead of only t h e strings of length g ( n ) which a r e on t h e tree, as we did in V.6.5). We now define as follows. If CP(p)(m)converges for all nz < n a n d n < Ipl, then:

+

undefined

if p 2 {e}. for some o E To(n) if p 2 {e}' for some 0 E Tl(n) if pl{e}' for every u E To(n)u .I'I(n) otherwise,

V.3 Baire Category *

483

where pl(e}" means t h a t Ihe two strings a r e incompatible. By definition @ ( p ) is a n initial segment, a n d @ is order-preserving a n d singlevalued (if o E TO(.) a n d 0' E Tl(n)t h e n {e}" a n d {e}"' a r e incompatible). T h e third condition in t h e definition ensures t h a t @ ( p ) ( n )is defined whenever p is long enough. More precisely, if @ ( p ) ( n )converges for all m < n , then so does @ ( p ) ( n )whenever ,

Hence, in general, @ ( p ) ( n )converges if IPI 2 sup(l{e}"l

:0E

(J(To(m)u T1(m))}. mT u)(VA 2 7 ) ( A fZ A), and by V.3.6 2 is comeager. Otherwise, ( b ) ( V A 2 a ) ( A E A), and one such A is recursive.)

x.

Since t h e recursive sets do not provide, in general, witnesses for branches of a n y infinite binary tree, we introduce t h e following notion:

q

Definition V.5.29 A class of sets A is a basis for classes i f every rr(: class has a n element in A. A class of degrees is a basis i f so is the class of sets with degrees in it.

rr(: classes. First we show t h a t it will be

We now s t a r t a search of bases for impossible to find a single best answer.

Proposition V.5.30 The intersection of all bases for IIy classes is the class of recursive sets, which is not a basis. I n particular, there is no least basis. Proof. Every recursive set must be in every basis, because given B recursive t h e condition A = B defines a lIy class with B as t h e only member. To show that t h e intersection of all bases is t h e class of recursive sets, it is enough to show t h a t given A nonrecursive there is a basis B not containing A . We build 23 by putting a n element of each @ class in it. Given a class, there a r e two cases: either it has recursive elements, in which case we can p u t one in D,or (by t h e proposition above) it has 2N0 elements, a n d t h u s we can choose one different from A . 0 .

A simple analysis of t h e proof of Konig's Lemma provides t h e first result. Proposition V.5.31 Kreisel's Basis Lemma (Kreisel [1950]) An infinite recursive binary tree has a A! irlfinite branch.

508

V. Turing Degrees

Proof. Let T be a given infinite recursive binary tree. Given un E T , we have t o decide whether t o choose u,+1 as u, * (0) or an * (1). We see if (b'm

> n)(3r E T)(IrI= m

A

r 2 g,II.* (0)).

If yes, we let c,+1 = on * (0). Otherwise, on+l = u, * (1). Let A = UnEwun. Then A E T . Since the quantifier on r is bounded, because there are only finitely many strings of length m, and T is recursive, the question is @. It can thus be 0 answered recursively in K. Then A ST K ,and A E A!. Kreisel's Basis Lemma has been improved by Shoenfield [1960a], who proved t h a t there is always a branch of degree less than 0'. T h e next result is much stronger.

Theorem V.5.32 Low Basis Theorem (Jockusch and Soare [1972]) The degrees a such that a' = 0', called low degrees, f o r m a basis for @ classes.

Proof. We proceed as in V.2.21, on trees. Let T be a given infinite recursive binary tree. We want t o build A on T such t h a t e E A' can be decided recursively in K. Let To = T . Given T,, to decide whether e E A' we consider t h e set Ue = { u E T e : { e } " ( e ) t } . There a r e two cases:

1. We is infinite Then we can let Te+l = U,, and for every A on Te+l we will have e $ A'. Note t h a t U, is indeed a tree, being closed under subsequences (since if a string does not decide a computation, neither does any substring of it). 2. U, is finite Then we can let Te+l = T,. For every A on Te+l we will have e E A', since { e } " ( e ) is undefined for at most finitely many strings on Te+l,a n d thus it must converge for any string which is big enough. Since the case distinction is recursive in Ec, if A E neEwTe then A' ST And clearly A E T , because TO= T . 0

K.

Exercises V.5.33 (Jockusch and Soare [1972], [1972a])a) The r.e. degrees form Q b a s i s for IIy classes. (Hint: given an infinite recursive binary tree T , consider its leftmost infinite branch A. If

u E B w u E T A u is on the left of A

V.5 The n e e Method

509

then A ET B. Moreover, B is r.e. because if u E T is in B then we discover it by generating all strings of T of length up to n, for n big enough.) b) The r.e. d e p e s strictly below 0' do not form a b a s i s for II: classes. (Hint: the example given in V.5.26 consists only of effectively immune sets. They, by the proof of 111.2.18, cannot have r.e. degree strictly below d.) c) If b 2 0' then 0 and the degrees with jump b f o r m a b a s i s for rI: classes. (Hint: let T be an infinite recursive binary tree without infinite recursive branches. Since it has 2N0branches, it is possible to build a total subtree Q LT K all of whose branches force the jump, as in V.5.32. If B 5~ K then the branch determined by B, i.e. turning right at level n if n E B and left otherwise, has jump B.)

Proposition V.5.34 (Jockusch and Soare [1972])The hyperimmune-free degrees f o r m a basis for @ classes. Proof. We proceed as in V.5.6. Let T be a given infinite recursive binary tree. We want to build A on T such t h a t { e } A ,whenever is total, is majorized by a recursive function. Let To = T . Given Te,consider

U,Z = {o E T, : { e } ' ( x ) t } . There a r e two cases: 1. U z is infinite for some x Then we can let Te+l = U z for one such x , and for every A on Te+l we will not total. Note t h a t U z is indeed a tree, being closed under have subsequences (since if a string does not decide a computation, neither does a n y substring of it).

2. U z is finite for every x Then we can let Te+l = T,. For every A on T,+1 it is enough to find recursively a level n such t h a t , for all strings o of length n, {e}'(x) 1. Then

T h e results just proved for binary recursive trees can easily be seen to hold also for recursively bounded recursive trees, i.e. recursive trees such t h a t t h e size (i.e., t h e greatest component) of nodes on T of length n is bounded by f(n), for some recursive f . E.g., a binary tree is recursively bounded by 2. T h e results for recursively bounded recursive trees can be extended to finitely branching recursive trees, i.e. recursive trees such t h a t t h e number of nodes o n T of length n is finite. T h e reason is t h a t such trees a r e bounded by a function recursive in 0': to know a bound on t h e size of strings of level

V. Turing Degrees

510

n on T we can ask, for a n y m, whether t h e size of every string of level n on T is bounded by m. T h i s can be answered recursively in Ic, a n d we can inductively determine t h e smallest m for which such a sentence holds (which exists, because T is finitely branching). T h u s the results of this subsection hold for finitely branching recursive trees, when relatiuized to O f . E.g., V.5.32 shows t h a t a finitely branching infinite recursive tree has a n infinite branch with j u m p recursive in 0”. If t h e condition of being finitely branching is dropped, t h e n t h e situation changes radically (e.g., V.5.23 fails). T h e theory of infinite recursive trees with infinite branches can be developed in a way largely parallel to t h e o n e for binary recursive trees, when t h e notions of recursive set a n d Turing degree a r e replaced by those of hyperarithmetical set a n d hyperdegree. See Volume I11 for a treatment.

Complete extensions of Peano Arithmetic

A

Since t h e class of sets separating two disjoint r.e. sets is a nonempty so are:

class,

1. the set of consistent extensions of a given consistent theory 2. the set of complete extensions of a given consistent theory. Jockusch a n d Soare [1972] a n d Hanf [1975] provide converses to these examples in t h e spirit of 111.10.3, showing t h a t the class of degrees of members of a given IIy class coincides with the class of degrees of complete extensions of some (finitely axiomatizable) first-order theory. In t h e following we will restrict o u r attention to complete extensions of Peano Arithmetic, because they provide a possible description of t h e arithmetical world in accord with t h e partial b u t fundamental picture given by PA. In particular, we will t h u s be able to t a k e advantage of t h e power of P A in proofs, including t h e Induction Principle. T h e basic link with t h e subject of t h e last subsection is another basis result.

Theorem V.5.35 Scott Basis Theorem (Scott [1962]) If F is a consistent extension of PA, the sets recursive i n 3 form a basis for classes. Proof. Let T be an infinite recursive tree. To be able to choose a n infinite branch recursively in 3, we proceed inductively. Let 00 = 8. Given u, of length n, consider all its extensions of length n 1 o n T . Given a n y two of them, say TO and 7-1, we have to decide which one looks better as a n initial segment of a n infinite branch of T . T h e statement

+

&

(3m)(T0has

a n extension of length in on T , b u t

7-1

does not)

V.5 The Tree Method

511

is Ey (note t h a t the quantifiers on strings are restricted t o strings of a given length). If it is true then, for some m, so is TO

has an extension of length m on T , but

TI

does not.

But this is a true recursive sentence, which is then provable in in 3. Then so is $0. Similarly for $1

($

(3m)(-r1 has an extension of length m on T , but

PA, a n d hence

TO does

not).

Now $0 and cannot both be provable, because 3 is consistent (otherwise there would be a number m such that one of TO and TI has at the same time a n extension of length m, and no extension of length m). Thus T, looks better if $i is provable in .F, and otherwise TO and TI look the same. We can now compare all pairs of strings on T of length n 1 extending cn. Since there is an infinite branch on T , all strings which do not extend to an infinite branch are eliminated (when compared t o a string that does). All the remaining ones do extend to an infinite branch, and we can choose any of them

+

=cn+1.

0

T h e next result provides a converse to Scott Basis Theorem.

Theorem V.5.36 Characterization of the degrees of complete extensions of P.4 (Solovay) T h e following conditions are equivalent: 1.

a i s the degree of a consistent extension of P A

2. a is the degree of a complete extension

3. D ( < a ) is a basis for

ofPA

classes.

Proof. Clearly, 2 implies 1. Scott Basis Theorem shows that 1 implies 3. If 3 holds then there is a complete extension of P A recursive in a (because t h e set of complete extensions of P A is a class): that 3 implies 2 thus follows from t h e upward closure of the degrees of complete extensions of PA, which we now prove. Let 3 be a complete extension of P A recursive in a set C . It is enough t o build a tree of complete extensions of P A , recursively in F. Then t h e branch determined by C has the same degree of C . Let { ( P ~be}an~ enumeration ~ ~ of the sentences in t h e language of arithmetic. We start with F0 = PA. Given .Fv,we proceed i n two steps:

1. completeness Given vn,with n = ((71, we decide how t o consistently add t o 30one of

V. Turing Degrees

512 pn a n d -cpn.

As in V.5.35, we let

71,o

e

71,1

(3m)(nzcodes a proof of cpn in Fu, b u t no smaller m’ codes a proof of -cp,) (3m)(mcodes a proof of -pn in FU, b u t no smaller m’ codes a proof of cp,).

Since Fo is a finite extension of P A , it is a formal system, a n d t h u s $0 a n d $1 a r e Ey, a n d hence provable in F if true. Moreover, by consistency of F,they a r e not b o t h provable. Recursively in F we c a n see which of them, if any, is provable.

If $1 is provable in 3 t h e n we know t h a t $0 is not provable in hence t h a t -cpn is consistent with it. We can t h u s let

FA

Fu,a n d

= Fuu{-cp,}.

Otherwise, we can let

2. branching Since t h e sets of provable a n d refutable formulas of P A a r e a n effectively inseparable pair of r.e. sets (see III.10.11), t h e r e is a recursive function t h a t produces, given a disjoint pair ( A , B ) of r.e. sets extending t h e m , an element not in A U B . T h i s applies in particular to t h e sets of provable a n d refutable formulas of FA which is, by construction, a finite extension of PA. In other words, there is a n effective way to find a sentence 71, which is neither provable nor refutable from Fi.T h e n we c a n let FU*(O) FV*(l)

If

w-J{71,)

= = %J{-@}.

F ST C then Uucc3uis a complete extension of P A of t h e s a m e degree as

c.

~

0

T h e theorem shows t h a t t h e degrees of consistent or complete extensions of P A describe particularly simple bases for classes. A complete degreetheoretical characterization of them is not known, b u t from t h e basis results we already have we can easily derive a number of consequences, b o t h positive a n d negative. T h e latter show t h a t a complete extension of P A cannot be too simple, in various ways. T h e y generalize 11.2.17, which stated only t h a t a consistent extension of P A cannot be recursive.

V.5 The D e e Method

513

Proposition V.5.37 (Scott and Tennenbaum [1960], Jockusch and Soare [1972], [1972a]) A consistent extension of P A can have neither incomplete r.e. degree, nor minimal degree. B u t there are complete extensions of P A of low degree, as well as of degree 0'.

Proof. If t h e r e were a consistent extension of P A of r.e. incomplete degree, by Scott Basis Theorem t h e r.e. incomplete degrees would be a basis, contradicting V.5.33.b. Similarly, if there were a consistent extension of P A of minimal degree t h e n t h e minimal degrees together with t h e recursive sets would be a basis, contradicting t h e fact t h a t t h e members of t h e following nonempty @ class a r e neither recursive nor of minimal degree:

A@B E C

tj

( V e ) ( B ( e )$ { e } ( e ) )A ( V e ) ( A ( 2 e )74 (el(2e)) A ( V e ) ( A ( 2 e 1) $ { e } B ( 2 e

+

+ 1)).

T h e first t w o conditions imply t h a t both A arid B a r e not recursive, since they diagonalize against every recursive function (on fixed arguments). T h e last condition implies t h a t A $T B . T h u s A @ B cannot be recursive or of minimal degree. A n d C is fl because, e.g.,

B(e)

+ {eHe)

.3

M e )t

"

f {eHe>,

a n d to diverge on a given argument is a @ condition. T h e existence results follow from V.5.32 a n d V.5.33.a: t h e latter implies t h e existence of a complete extension of P A of r.e. degree, which must be 0' because all t h e r.e. incomplete degrees a r e ruled out by t h e first p a r t of t h e proof. 0 Jockusch a n d Soare [1972a] show t h a t the complete extensions of a n essentially undecidable formal system can have both incomplete r.e. degree and minimal degree. T h u s t h e role of P A (through t h e fact, used in t h e proof of V.5.36, t h a t t h e sets of t h e o r e m a n d of refutable formulas a r e not only recursively, b u t also effectively inseparable) is crucial. An i n t e r g t i n g way to analyze t h e behavior of consistent extensions of P A is to see which sets a r e weakly representable in them. Clearly, in t h e s t a n d a r d model exactly t h e arithmetical sets are. By 11.2.16, in every consistent formal system extending R exactly t h e r.e. sets a r e weakly representable. B u t for consistent extensions F of R which a r e not fornial systems t h e following might happen. If y weakly represents a set A in R, cp(Z) might become provable in F for some z E a n d t h u s cp represents only a superset of A in F,possibly a cofinite (and hence recursive) one. Moreover, since F is not a formal system, t h e set represented by y is not necessarily r.e.

x,

V. Turing Degrees

514

Proposition V.5.38 For every consistent extension 3 of PA, the class of sets weakly representable in 3 properly includes the recursive sets. Proof. Since, by 11.2.16, t h e recursive sets are actually strongly representable

in R,they remain strongly representable in every extension of it, a n d hence weakly representable in every consistent extension. Given a recursive enumeration {(P,},~~of t h e sentences in t h e language of arithmetic let, as in V.5.35 and V.5.36,

$o(n)

*

$l(n)

(3m)(mcodes a proof of cpn in PA, but no smaller m' codes a proof of -cp,) (3m)(mcodes a proof of 7 c p n in PA, but no smaller m' codes a proof of cp,)

By consistency of 3,$0 weakly represents a set A . If p n is a theorem of P A then $o(n) is a true Ey formula, which is then provable in 3. If -pn is a theorem of Pd then $ ~ ~ (isnprovable ) in F and hence, by consistency of F , $o(n) is not provable. Thus A separates theorems and refutable formulas of PA, an d thus it cannot be a recursive set (by 111.10.11). 0 We now provide some examples of complete extensions of PA, pathological from th e point of view of weakly representable sets.

Proposition V.5.39 (Jockusch and Soare [1972]) 1. There is a complete extension of P A in which no hypersimple set is weakly represent ab 1e. 2. There is a complete extension of

P A in which only A; sets are weakly

representable. 3. For any n 2 2 there is a complete extension of P A an which only A:+l sets are weakly representable, but no nonrecursive Cz or rr", set is.

4.

There is a complete extension of P A in which no arithmetical nonrecursive set is weakly representable.

Proof. By V.5.34 there is a complete extension 3 of P A of hyperimmune free degree. If A is weakly representable in 3,it is recursive in it. Thus it has hyperimmune-free degree, because t h e hyperimmune-free degrees are downward

closed (V.5.3). Thus 2 cannot be hyperimmune, and A is not hypersimple. Similarly, 2 follows from the Low Basis Theorem, which provides a complete extension of P A recursive in K , in which all weakly representable sets must then be A;.

V.5 The D e e Method

515

For t h e remaining results, we first show t h a t given countably many nonrecursive sets A , and an infinite binary recursive tree, there is A on it in which no A , is recursive. T h i s c a n be obtained inductively as usual, once we know how to get, given e , n, a n d a n infinite recursive tree T , a n infinite recursive subtree T’ of T such t h a t A , $ { e } A , for every A on T . T h e r e a r e t h r e e cases: 1. for some z, there are infinitely many u E T such that {e}‘(z) T h e n t h e set (0E

T:

1

(4T)

is a n infinite subtree of T (being closed under subsequences), a n d we can let it be T’. If A E T’ then { e } A is not total, a n d hence is different from

A,. 2. for some 5 , there is u E 7’ such that {e}.(x) is defined and different from A , ( z ) , and u has infinitely many extensions on T T h e n we can let T’ be t h e full subtree of T above 0.If A E T’ t h e n is different from A,. 3. otherwise T h e n we let TI = T , a n d show t h a t { e } A is recursive for any A on T , a n d hence different from A,. Given z, go t o a level n of T such t h a t {e}‘(z) 1 for every u E T of length 11: this is possible because we a r e not in t h e first case. If two strings give different values on z, then they cannot b o t h have infinitely many extensions on T , otherwise one of t h e m would give a value different from A,, a n d we would be in t h e second case. We c a n t h u s generate enough of t h e tree to discover which of t h e s t r i n g belong to finite branches, until only strings with t h e s a m e value remain. T h i s must be t h e value of { e } A ( z ) ,for any A E T . By letting { A n } n E w be a list of t h e arithmetical nonrecursive sets, we get p a r t 3. For p a r t 2 we only have to let { A n } n E be w a list of E: and to compute t h e complexity of t h e construction. T h e division in cases is based o n two-quantifier questions, a n d thus it is recursive in which accounts for t h e b o u n d n 2 2. 0

Ue,

a’’,

KuEera (19861 (see X.4.8) has shown t h a t part 2 cannot be improved as in p a r t 3, because there is no complete extension of P A in which only A: sets are

weakly representable, but no nonrecursive r.e. set is. For more on I l: classes a n d their applications to complete extensions of P A , see Jockusch a n d Soare [1971], [1972], [1972a], Jockusch [1974], [1989], KuEera

(19851, [1986], [1988], [1989].

V. Turing Degrees

516

V.6

Initial Segments *

Initial segments more complicated t h a n minimal degrees have been used in t h e original proofs of many of t h e global results of t h e next section. T h e y a r e now obsolete in this respect, since much simpler proofs have been obtained. Initial segments a r e still necessary for a complete algebraic characterization of t h e algebraic structure of D , as well as for some advanced p a r t s of Degree Theory (like t h e results quoted on p. 492). T h e techniques involved in t h e proofs a r e however mostly not recursion-theoretical, a n d outside t h e scope of our book. T h e reader is referred to Epstein [1979] a n d Lerman [1983] for detailed treatments. We will introduce only techniques which have other uses as well.

Uniform trees To obtain initial segments we need trees t h a t have more flexibility t h a n t h e simple ones used in t h e previous section. T h e following notion introduces some uniformities in our trees. Definition V.6.1 T is a uniform tree if, f o r every z = 0 , l a n d u,

1. IT(u)l depends only on 2. there is a unique

101

~ i ,depending

only on 1aI7s u c h t h a t T(o* i) = T ( a )* 7,.

A uniform tree is nothing more t h a n a tree in which, at each level, t h e strings immediately following each node a r e independent of t h e node itself, a n d have t h e s a m e length. A useful way of representing t h e situation is t h e following: t h e r e a r e three functions g (strictly increasing), a n d f ~f~, (left a n d right functions) such t h a t f~ a n d f~ t a k e values in (0,l) a n d are incompatible (i.e. they differ o n at least one argument) in every interval [ g ( n ) ,g(n 1)). T h u s t h e levels of t h e tree a r e determined by g, a n d a branch of t h e tree is simply a p a t h which at every node follows one of fL a n d fR, u p to t h e next node.

+

Definition V.6.2 (Spector [1956]) A n admissible triple is a triple g7 I,,,

f~ o f f ~ n c t i o n from s w into (0, l}, such that:

V.6 Initial Segments *

517

level n

+1

fR

level n

Our present task is t o build minimal degrees by uniform trees. By doing so we will actually reproduce Spector's original proof. What we do is to reprove the lemmas of the last section, this time building uniform trees instead of simple trees.

Proposition V.6.3 Diagonalization Lemma for uniform trees. Given e and a recursive uniform tree T , there is a recursive unaform tree Q C T such that, for every A on Q, A { e } . Proof. T h e same proof of t h e Diagonalization Lemma works, because any full subtree of a uniform tree is still uniform. 0 Proposition V.6.4 Totality Lemma for uniform trees (Martin and Miller [1968]) Given e and a recursive uniform tree T , there is a recursive uniform tree Q T such that one of the following holds: 1. for every A on Q, { e } A is not total 2. for every A on Q , { e } A is total and

(vn)(vO)(lal = n where

101

3

{e}Q(%)

11,

is the length of u

Proof. Seeif

(30 E T)(3z)(V.r2 O ) ( T

ET

{e}T(z)T).

V. Turing Degrees

518

If so, choose such a u: as in t he Totality Lemma, we let Q be t h e full subtree of T above u, which is uniform since T was. Then case 1 holds. Otherwise, we define Q by induction as follows. Q(0) = least Given Q(ai),for 1 5 i

T

E T such t h a t { e } 7 ( 0 )1 .

5 2n and ui string of length n, take:

Q(u1)* 7 1 E T Q(o2)* 71 * 7 2 E T etc.

such t h a t such t h a t

{e}Q("l)*T1 (n11 {e}Q(uz)*T1*Tz (41

Let T = T~ * . . . * 7 2 " : for each i we then have { e } Q ( u t ) * T (1.n It ) is now enough t o take two incomparable extensions po a n d p1 of T with t h e same length a n d such th at &(a,)* /.LO and Q(ui)* pl ar e on T for every i, which is possible because T is uniform, and let

Q(ai * 0 ) = Q(ai)* po

and

Q(u~ * 1) = Q ( 0 i ) * pi.

By definition Q is then uniform, since we extend all strings of a given level in t h e same way. 0

At this point we know how t o build hyperimmune-free degrees by uniform trees. To obtain th e same result for minimal degrees it is enough t o show t h a t we can handle, by uniform trees, the only case of t h e Minimality Lemma in which we a r e not taking full subtrees. Proposition V.6.5 (Spector [1956]) Given e and a recursive u n i f o m tree

T , if

1. every u on T has e-splitting extensions on T 2. (Vo E T)(V'z)(3.r2

D)(T

E T A (e}'('z)

1)

then T has a recursive e-splitting uniform subtree Q. Proof. Define Q inductively by t he following procedure. Suppose &(a,)is given, for any 1 5 i 2" a n d ai string of length n.

V.6 Initial Segments

*

519

By 1 there are TI and 7-2 such t h a t Q(u1)* 7-1 and Q(a1) * T2 are on T and e-split. Reproduce them above Q(u2):since T is uniform, the new strings are still on T . By 1 again, there are 73 and 74 such that Q(a2)* 7 1 * T3 and Q(a2)* T~ * 7 4 are on T and e-split, say on 5 . By 2 there is some ~5 such that {e}Q(u2)*T2*T6(x)Thus, for i = 3 or i = 4, Q(a2)* TI * ~i and Q(a2)* ~2 * 75 e-split on x . Reproduce TI * ~i and 7 2 * T5 above Q(a3),and so on. We thus get, at t h e end, two big strings T and T’ such that, for each i, Q(ai) * T and Q(ui)* T ’ a r e on T and e-split. By possibly extending one of them t o a string on T , we can actually find two such strings of the same length, and they provide the new level in Q. 0

I.

Theorem V.6.6 (Spector [1956]) I t is possible t o build minimal degrees by recursive uniform trees. Proof. We start with

TO= identity tree, which is clearly a recursive uniform tree. Given Tze,we let

T2e+l = the Q of the Diagonalization Lemma, for T = Tze. T o define T2e+2,let T = Tze+1and see if the condition 2 of the above proposition holds. If not, choose a string u on T for which it fails, and let

Tze+2= full subtree of T above u. Then {e}” is not total, for any A on it. If condition 2 holds, see if 1 does. If not, there is a string a with no e-splitting extensions, and we let

T2e+2= full subtree of T above u. Then {e}A is recursive if total, for any A on it. Otherwise, let

T2e+2= the Q of the proposition above. Then A ST {e}” if {e}” is total, for any A on it. If A is on all the T,’s, then it has minimal degree. 0 One might wonder why we should want t o build minimal degrees by uniform t r e e , since t h e proof is more complicated than the one given in V.5.11. There are two independent answers to this. T h e first is that this is a first step toward the construction of minimal degrees by recursive coinfinite conditions. T h e second is t h a t t h e proof just given can be modified and, taking advantage of t h e uniformities, turned into a proof of the existence of more complicated initial segments. These two applications are treated in the next subsections.

V. Turing Degrees

520

Minimal degrees by recursive coinfinite extensions W e close now t h e circle s t a r t e d in Section 4, by showing how coinfinite conditions a r e nothing else t h a n particular uniform trees.

Definition V.6.7 (Lachlan [1971]) T is a strongly uniform tree (or a 1-tree) i f it is uniform and, for all u, T ( u1: 0 ) and T ( u* 1) differ only on one argument (we say they are adjacent). Equivalently, we could define strongly uniform trees as admissible triples (V.6.2) satisfying t h e stronger condition t h a t t h e functions fL a n d fR differ, at each level, on exactly o n e argument. We can see t h e arguments on which t h e two sides differ as t h e uncommitted ones, a n d t h u s a strongly uniform t r e e defines a coinfinite condition

undefined

)O if ~L(z) = ~ R ( z= if f ~ ( x = ) fR(2) = 1 if f ~ ( x#) f ~ ( 2 )

Of course t h e s a m e translation would work for uniform trees as well. W h a t makes strongly uniform trees special is t h a t any set A extending 8 is o n t h e t r e e (since in each interval there is only one uncommitted point, t h e t w o branches f~ a n d f~ take care of all t h e possibilities). T h i s is not t r u e if t h e t r e e is only uniform (if in a n interval there a r e n uncommitted points t h e n t h e r e a r e 2n possible extensions, b u t only two of t h e m a r e o n t h e tree). Conversely, a coinfinite condition 8 defines a strongly uniform t r e e as follows. Let g enumerate, in increasing order, t h e elements on which 8 is undefined, a n d

Moreover, t h e translations j u s t given preserve recursiveness (recall t h a t a coinfinite condition always has recursive domain, a n d so g is always recursive), a n d t h u s recursive strongly uniform trees and recursive coinfinite conditions are

interchangeable. It is immediate t o note t h a t t h e Totality Lemma proved for uniform trees also works for strongly uniform ones (since we j u s t split a string at t h e very end), a n d t h u s it is possible to build hyperimmune-free degrees by recursive coinfinite extensions. To prove t h e analogue of t h e Minimality L e m m a requires instead much more work.

Proposition V.6.8 (Lachlan [1971)) Given e and a strongly uniform tree T , if 1. every u E T has e-splitting extensions on T

V.6 Initial Segments *

52 1

2. ('doE T ) ( ' d z ) ( 32~ O ) ( T E T A {e}.(z)

1)

3. T does not have strongly uniform subtrees without e-splittings then T has a strongly uniform e-splitting subtree Q.

Proof. Clearly condition 1 is redundant, and it follows from 3. We make it explicit only t o show where the new hypothesis is used. Since 2 holds, we may suppose t h a t ('d4(W(bl= 71. 3 {e}T'"'(4

L),

otherwise we apply V.6.4 first. Again we proceed by induction, showing t h e first two steps. Given u , by 1 there a r e T and T' extending it and e-splitting, say on z. We may suppose they are of the same length (otherwise extend t h e shortest), and t h a t for all strings p of that length, {e}p(z) 1 (otherwise go to a level high enough, by t h e initial observation). Since T is strongly uniform, there is a sequence T O , . . . ,T, of strings on it of t h e same length, each adjacent t o t h e . {e}TJ (z)1 for all following in t h e list, and such that T = TO and T' = ~ i Since j 5 i, and {e).O(z) $?{e}.l(z), two of these adjacent strings e-split on z. So u E T has e-splitting adjacent extensions. This shows how t o build t h e first level of t h e e-splitting uniform subtree. Now let 01 and up be given. We set up t o build t h e second level.

v'

xv

1. We first build a strongly uniform subtree of T above 02, as follows. Take a pair of e-splitting adjacent extensions of 02, and let it be t h e first level. Then consider t h e leftmost branch, take a pair of e-splitting adjacent extensions of it, and reproduce them on the rightmost one, and this is t h e second level. We go on by considering the leftmost branch at each level, finding an e-splitting adjacent pair above it, and reproducing it on every node of the same level.

V. Turing Degrees

522

2. Then we take this subtree and reproduce it, as it is, above o l . There must be a n e-splitting on this strongly uniform subtree of T, by condition 3. We want t o find one such t h a t t he two branches only differ on one element (since we are building a strongly uniform tree), a n d they also e-split above 02. Note t hat we know only t h a t t h e leftmost branches e-split above 02, a n d we thus make our search by staying on t h e leftmost branch of our subtree above 01. By methods we know, we may choose an e-splitting v a n d v‘ such t h a t )vI = /v‘I and: 0

v’ is on t he leftmost branch of t h e strong uniform subtree (given any e-splitting, say on x, it is enough t o wai; until, for a long enough segment v’ on t h e leftmost branch, {e}” (x)1: one of t h e original branches and v’ must e-split on x.)

0

0

v goes right on t he tree as late as possible (i.e. t h e common part of v and v’ is maximal) if x is t he element on which v and v’ e-split, a n d p E T , then

14 = I4 = I4 =+

{e}%)l.

Now take X which is like v , only it goes right one level after v does: X a n d v are obviously adjacent, and {e}’(x) N {e}”(x) by t h e choice of v (since th e common part of v and v’ is maximal). T hen {e}”(x) $ { e } ” ( x ) ,a n d X and v are an adjacent e-splitting above 01. By definition they are also a n (adjacent) e-splitting above 0 2 (since v’ lies on t h e leftmost branch). This shows how t o build t he second level of t h e e-splitting uniform subtree. T h e remaining levels can be built inductively, in t he same way. 0

Theorem V.6.9 (Lachlan [1971])It is possible t o build minimal degrees by recursive coinfinite extensions.

Proof. We use strongly uniform recursive trees, which we know t o be interchangeable with recursive coinfinite conditions. Let

TO= identity tree, which is clearly a strongly uniform recursive tree. Given T2e+l=

the

Q of t he

T2e,let

Diagonalization Lemma, for

T = Tze.

To define Tze+2, let T = Tze+1,and see if condition 2 of t h e above proposition holds. If not, choose a string u on T for which it fails, and let

7be+2= full subtree of T above o.

V.6 Initial Segments * Then { e } A is not total, for any not, let

523

A on it. If condition 2 holds, see if 3 holds. If

Tze+2 = a strongly uniform subtree of T without e-splittings.

Then { e } A is recursive if total, for any A on it. Otherwise, let T2e+2

= t h e Q of t h e proposition above.

Then A 0 is definable i n 2) by a %-formula. Of course the definitions provided by t h e proof above are not very natural from a recursion-theoretical point of view, because they simply translate definitions from Second-Order Arithmetic. In the next section we will start a path that will be pursued all along the rest of t h e book, by finding natural recursion theoretical definitions of particular classes of (and relations on) degrees.

Homogeneity T h e fact t h a t every particular result about 2) seems t o relativize above any given degree, led to the following conjectures:

V. Turing Degrees

544

1. strong homogeneity (Rogers [1967]) For every degree a, t h e structures 2) a n d P(2a ) a r e isomorphic.

2. homogeneity (Yates [1970]) For every degree a, t h e structures D a n d D ( 2 a) a r e elementarily equivalent, i.e. they satisfy t h e same first-order formulas. T h e same relativization phenomenon which led to t h e homogeneity conjectures is t h e key to their disproval.

Theorem V.7.13 Failure of homogeneity (Shore [1982], Harrington

and Shore [1981], Jockusch and Shore [1984]) I f P ( 2 a ) is elementarily equivalent t o P,then a is arithmetical. Proof. Consider t h e formula p(z) defining t h e arithmetical degrees: when interpreted in 73 it defines t h e set A of t h e arithmetical degrees, while when interpreted in Do(?a ) it defines (by relativization of t h e proof of V.7.9) t h e set A a of t h e degrees arithmetical in a a n d above it. If P a n d P ( L a) a r e elementarily equivalent, then so a r e d a n d d a a n d hence, by V.7.6, t h e theories of Second-Order Arithmetic with set quantifiers restricted, respectively, to sets arithmetical a n d arithmetical in a. T h e n a is arithmetical, otherwise t h e sentence saying t h a t there is a nonarithmetical set would distinguish them (being false in t h e former a n d t r u e in t h e latter). 0

Corollary V.7.14 P and V ( 2 0 c w are ) ) not elementarily equivalent. Since homogeneity fails, relativized versions of results t h a t hold in 2) a r e not automatically true, and require proofs. Also, there a r e results t h a t simply fail to relativize. However this very result (saying t h a t not everything relativizes) does relativize to degrees b which a r e definable in Second-Order Arithmetic (because then we can define in Second-Order Arithmetic t h e formula saying t h a t all degrees satisfying cp a r e arithmetical in b ) . T h e exercises show t h a t it is consistent, b u t unlikely, t h a t t h e relativization holds in general. Exercises V.7.15 A cane of elementarily equivalent cones is a cone such that, for any a and b in it, the cones D ( 2 a) and D ( 2 b) are elementarily equivalent. The existence of such a cone would provide a homogeneous substructure of the degrees. a) If Projective Determinacy holds then there is a cone of elementarily equivalent cones. (Martin [ 19681) (Hint: as in V.7.18, using V.1.16 and cones instead of comeager sets. Note that the set of degrees a such that 'D(2a) satisfies cp is an analytical set, and thus only Projective Determinacy is needed, in place of full Determinacy.) b) If V = L then there is no cone of elementarily eguivalent cones. (Shore [1982]) (Hint: if V = L then the degrees above a and b are elementarily equivalent only if a and b are arithmetically equivalent. Indeed, with notations as in V.7.13, an exact

V.7 Global Properties

545

pair for the set of degrees satisfying ‘p defines d c in D(> c ) . The least such exact pair w.r.t. S L , which is definable in Second-Order Arithmetic, defines uniformly the same set in elementarily equivalent structures.) c) In a cone of elementarily equivalent cones, no degree can be definable an SecondOrder Arithmetic. (Hint: since V.7.13 relativizes to degrees definable in Second-Order Arithmetic, in any cone with such a base there is a cone not elementarily equivalent to it.)

To get a result t h a t fully relativizes we need to look at isomorphism, rather t h a n elementary equivalence.

Theorem V.7.16 Failure of strong homogeneity (Shore [1979], [1981], Harrington and Shore [1981], Jockusch and Shore [1984]) I f P ( > a ) i s isomorphic to P(2b ) , then a and b are arithmetically equivalent. Proof. Consider a n isomorphism carrying D ( 2 a) into D(>b ) . T h e image of a copy of t h e standard model of Arithmetic defined in t h e degrees arithmetical in a n d above a is carried into a structure isomorphic t o it, in t h e degree arithmetic in a n d above b. Indeed, a n isomorphism preserves definable properties, a n d thus degrees satisfying t h e formula defining t h e arithmetical degrees in D(2 a) a r e sent into degrees satisfying t h e same formula in P(2 b ) . Now t h e sets arithmetical in a a r e coded by degrees arithmetical in a,a n d their images in P(>b ) must code t h e same set (by t h e isomorphism), which is now arithmetical in b. T h u s a must be arithmetical in b. T h e converse holds similarly, a n d thus a a n d b have t h e same arithmetical degree. 0

A cone of isomorphic cones is a cone such t h a t , for any a a n d b in it, t h e cones P(>a) a n d P(> b ) a r e isomorphic. T h e existence of such a cone would provide a strongly homogeneous substructure of t h e degrees. Corollary V.7.17 There is no cone of isomorphic cones.

Proof. By t h e previous result, two cones can be isomorphic only if their bases a r e arithmetical one in t h e other. T h e n there a r e at most countably many cones isomorphic to a given one, a n d there cannot be any cone of isomorphic cones (because a cone has uncountably many elements). 0 We t u r n now to positive cases of homogeneity. We do not know whether there is a cone with nontrivial base which is elementarily equivalent to D ,or whether there a r e two isomorphic cones, b u t certainly there a r e lots of elementarily equivalent cones.

546

V. Turing Degrees

Proposition V.7.18 (Jockusch [1981]) There is a comeager set of degrees which are bases of elementarily equivalent cones, i.e. a comeager set such that if a and b are in it, the cones P(>a ) and P(>b) are elementarily equivalent.

Proof. Consider t h e first-order sentences of t h e language of partial o r d e r i n e . For each such sentence 9,consider t h e set of degrees a such t h a t P(> a) satisfies p. It is easy to prove (see V.3.15) t h a t this set is either meager or comeager. Let 4 be this set if i t is comeager, a n d its complement otherwise. Then is always comeager, a n d cp holds either in every P ( 2 a) or in none of them, for a in A,+,.Since there a r e only countably many sentences cp, t h e intersection A of all t h e 4 is still comeager. Moreover, t h e truth-value of cp in P(>a) is independent of a in A, for every cp. T h i s means t h a t t h e first-order theory of P( a) is independent of a in d. 0

4

>

Exercises V.7.19 a) There is a corneager set of degrees of elementarily equivalent principal ideals D(

Automorphisms T h e questions of t h e last subsection, about homogeneity a n d strong homogeneity, can be asked about t h e relationships of P not only with some cone, b u t also with itself. T h e relevant notions a r e t h e following.

Definition V.7.20 A map f : 2) + ?) is called:

1. an automorphism if it is an isomorphism that preserves the order, i.e.


547

2. an elementary map if it is preserves the first-order formulas, i.e. for any first-order formula cp,

T h e analogues of homogeneity and strong homogeneity then ask about t h e existence of nontrivial elementary maps and automorphisms. First of all, t h e two questions are equivalent.

Proposition V.7.21 (Slaman and Woodin [1986]) A map from P to is elementary if and only if it is an automorphism.

2)

Proof. An automorphism obviously is a n elementary map. For t h e converse, a n elementary map automatically preserves the order, and it is one-one because it preserves t h e equality relation. It remains t o prove t h a t any elementary map f is onto, i.e. that for any y there is x such that y = f(z). T h e proof of part 2 of the next theorem will show t h a t f is t h e identity on t h e cone above a degree a (e.g. O w ) . By V.7.2 there are degrees I coding a standard model of arithmetic, and 2 coding the graph of a function g enumerating (on t h e natural numbers of t h e model) t h e degrees below yU a. Moreover, there + is a first-order sentence with parameters y and a stating that I and d have t h e desired properties. Since f is elementary, this statement is true of f ( y ) , f(a), + fee), and f@). In particular, f(c) code a standard model of arithmetic, a n d frd) code a function t h a t enumerates (on the natural numbers of this model) t h e degrees below f ( y U a). But f ( y U a ) = y U a because f is t h e identity above a, a n d hence y is one of the degrees enumerated by t h e function coded by f@), say t h e n-th in t h e enumeration. Since f is elementary, it preserves all t h e relevant properties. In particular, it must be t h a t y$ t h e image via f of t h e n-th degree x enumerated by t h e function coded by d . This shows t h a t 0 y is in t h e range of f , as wanted. We can formulate t h e analogue of (strong) homogeneity as follows: t h e only automorphism of P is t h e identity. Algebraic structures without nontrivial automorphisms are called rigid. Although t h e rigidity of P has not been proved, all known results point in t h a t direction, and at least show t h a t the automorphisms of P are severely restricted.

Theorem V.7.22 Restrictions on automorphisms (Nerode and Shore [198Oa], Harrington and Shore [1981], Shore [1981], Jockusch and Shore [1984]) Every automorphism of P : 1. sends any degree into a degree which is arithmetically equivalent to it

548

V. Turing Degrees

2. is the identity on every degree above all the arithmetical ones, in particular on the cone above 0 ” .

Proof. Consider a n automorphism f : P 4 2). T h e first part follows from V.7.16 and t h e fact that f induces an isomorphism between t h e cones above x and f(x), for any x. For t h e second part, note that there is? copy of t h e standard model of arithmetic such t h a t t h e relation ‘the degrees b code a set of degree z’is analytical, and hence first-order definable for degrees x above all t h e arithmetical ones, by V.7.10. Then this relation is preserved by f and hence, for any coding a -+ set of degree 3 above all the arithmetical ones, f(b) code a set of degree f(x). But f is an automorphism, and hence t h e degrees 7; and f( b) must actually code t h e same set. Then x = f(x). 0 4

Actually something better can be achieved, by more direct calculations: any automorphism of 2, is the identity on a cone hawing a n arithmetical degree as a base (Nerode and Shore [1980a], Harrington and Shore [1981], Shore [1981], Jockusch and Shore [1984]). We now introduce a useful tool for t h e study of automorphism.

Definition V.7.23 An automorphism basis for 2) i s a n y set of degrees A such that the behavior of a n y automorphism is completely determined by its behavior on elements of A. Producing many automorphism bases is one way t o show t h a t there are few automorphisms. Another one is t o show that there a r e small bases. We will exhibit results in both directions. T h e first way t o obtain automorphism bases is to consider sets of degrees t h a t generate P under U a n d n.

Definition V.7.24 Given a set of degrees d, the set generated by A in P is the smallest set: 1. containing A 2. closed under joins

3. closed under g.l.b. ’s, whenever they exist.

Any automorphism of a partially ordered structure must preserve I.u.b.’s and g.l.b.’s, whenever they exist, and thus t h e behavior of a n automorphism on a set A completely determines its behavior on t h e set generated by it. In particular, if A generates P then A is a n automorphism basis.


549

Proposition V.7.25 (Jockusch and Posner [198l]) If A is a corneager set of degrees, then A generates P under U and n. More precisely, a n y degree can be represented in the f o r m (a1 u a2) n (a3

u a4),

with ai E A.

Proof. Let a be a given degree: by relativization of the minimal pair construction (V.2.16), given b we can get c such that ( a U b) f l ( a u c )

= a.

Fix b E A: then t h e set of such degrees c is comeager, and hence so is t h e intersection of this set with A. Thus we have b and c in A such t h a t t h e above equation holds. We only have t o represent any degree of t h e form a U d, with d in A, as t h e join of two degrees in A. Note t h a t , in general, A@D=TD@(AAD) for any A a n d D , where A A D is t h e symmetric difference (A - D ) U ( D - A ) . Moreover, if A is comeager then so is t h e set

AA = { A A D : D has degree in d}. We may then suppose t h a t the degrees b and c above are not only in A, but also in AA, a n d then t h e result follows. 13 Clearly a countable set cannot generate P,which is uncountable. But there are uncountable meager sets of degrees that d o generate P,as t h e next exercise shows. Exercise V.7.26 The minimal degrees generate 2). (Jockusch and Posner 119811) (Hint: given two sets A and B such that A ST B, we show that there are sets of minimal degree A41 and M2 such that B and A41 @ MZ have A as g.1.b. We then apply this to any B above A and 6”,which by V.6.10.b is the join of two minimal degrees, and have that any degree is generated by four minimal degrees. We extend the proof of V.6.10.b and build M I and M2 by recursive coinfinite conditions. There are two additional requirements:

A 0, , P, ( 5 a) is isomorphic t o a countable distributive uppersemilattice I . L = I x I is still a countable distributive uppersemilattice, containing two distinct copies of I as ideals. Extend P, ( 5 a) to a n ideal isomorphic to L, by VI.3.4, and let b # a be t h e t o p m-degree corresponding to t h e second copy of I in L. Then P, ( 5a) a nd P, ( 5b ) are isomorphic, and there is a n automorphism of P, carrying a into b. T hen a is not fixed

VI.4 Global Properties

577

under every automorphism of P,, and in particular it cannot be definable. Thus O,, is t h e only m-degree fixed under every automorphism, a n d t h e only definable m-degree. 0

Corollary VI.4.5 Every definable set of m-degrees different from (0, } has power of the continuum. Proof. Given a set S # (Om} of m-degrees of power less than t h e continuum, choose a E S - {Om}. T h e m-degree b obtained as in t h e previous corollary can be taken t o be not i n S, because S has power less than the continuum, while there a r e (by VI.3.5) 2N0possible choices for b. Since there is a n automorphism of P, carrying a into b, S is not closed under automorphisms a n d hence it cannot be definable. 0 It follows t h a t many natural classes of m-degrees a r e not definable, e.g. t h e r.e. a n d t h e arithmetical m-degrees. O n the other hand, there a r e nontrivial definable sets of m-degrees, e.g. t h e minimal m-degrees, and thus t h e result is t h e best possible. Fkom VI.4.3 it easily follows t h a t there are 2N0 automorphisms of P, : given a minimal m-degree a,for any other minimal m-degree b there is a n automorphism carrying a into b (because P, ( 5 a) and P, ( 5 b) a r e isomorphic), and there a r e 2N0 minimal m-degrees. This is not t h e best possible result, since there a r e 22N0possible maps from V, to P, . We now show t h a t this bound is attained.

Proposition VI.4.6 (Shore) There are 22Noautomorphisms of 23,. Proof. Note t h a t VI.4.2 produces a n automorphism of V, , if both cones a r e Dm itself. Moreover, t h e back-and-forth argument of VI.4.1, on which t h e proof of VI.4.2 relies, takes 2N0 steps (we have t o ensure t h a t each nz-degree is in both t h e domain a n d the range). T h e only new step here is to actually build a tree of height 2N0 of automorphisms of P, , by extending every partial automorphism in two different ways (by using VI.3.5) at successor stages, and taking unions at limit stages. Each branch of t h e tree is now a n automorphism of P m , and different branches produce different automorphisms by construction. Thus there are 22Noautomorphisms. 0

The complexity of the theory of many-one degrees We have characterized t h e complexity of t h e first-order theory of P in V.7.3. If we t r y t o adapt t h e proof used there t o V, we run into trouble. T h e main point is t h a t we are unable t o prove t h e analogue of V.7.1, because its proof uses in a n essential way t h e fact that every Turing degree contains a n introreducible

VI. Many-One and Other Degrees

578

A 1L

al;

U

a;

Figure VI.1: Coding by graphs

21


579

set (11.6.7), and this is false for m-degrees: a nonrecursive, not immune set is not recursive in each of its infinite subsets (since some of them are recursive), and there are m-degrees containing only nonrecursive, not immune sets (e.g. any m-degree above the m-degree of see III.6.lO.b). T h e next result comes to t h e rescue and provides a different, slightly less direct way of coding arithmetic. We state it in more generality t h a n needed because of its interest.

c,

Theorem VI.4.7 (Nerode and Shore [1980]) Let ( P ,C,U) be an uppersemilattice with least element such that: 1. every countable ideal is the intersection of two principal ideals 2. every countable distributive lattice is isomorphic to an initial segment of

P. Then the theory of Second-Order Arithmetic is 1-reducible to the first-order theory of P. Proof. There are many steps toward t h e result: 1. translate Second-Order Arithmetic into second-order logic on countable sets This is a standard and well-known procedure, based on t h e fact t h a t Peano Axioms actually define w up t o isomorphism, in second-order logic (see p. 22).

2. translate second-order logic on countable sets into the theory o f countable distributive lattices with quantification over ideals This is the crucial step, which we split into two parts. We refer t o the various parts of Figure 1. 0

code relations by graphs (Lavrov [1963], Rabin and Scott) We start with a binary relation R. Recall that a graph is a symmetric, irreflexive, binary relation, which we may picture as a set of points related by lines. First of all we have t o put down t h e elements of the domain. A simple way is t h e following: for each element u add two points a;l and a;, and relate them as in part a) of t h e picture. Thus in t h e graph the elements of the domain of t h e given relation are t h e points in which two lines coming from end points arrive. We now have t o relate points u and u when R(u,u)holds. A simpleminded solution as in part b) of the picture is not enough, since R might be in general not symmetric, while t h e proposal is. So we add two elements, a n d relate them as in part c) to show t h a t R ( u , u ) ,

580

VI. Many-One and Other Degrees as opposed t o R(w,u),holds. This is still not enough since if, e.g., R(u,u)holds then we would have t he situation of part d). But this is ambiguous, because we might then think t h a t R(u,u)holds whenever we see a triangle with a vertex in u,while t h e triangle might come from t he coding of R(u,w) for some u # u. Our final choice is then t h e following: given u and w we add three new elements, a n d relate them as in part e), t o show t h a t R(u,w)holds. If R(u,u) holds then we get t he unambiguous situation of part f). This technique codes binary relations. But n-ary relations are easily reduced t o binary ones, and thus to graphs as above. For example, if R is ternary we can introduce a nonsymmetric binary relation as follows. When R ( z ,y, z ) holds introduce four new elements, a n d relate them as in part g). An arrow u -+ 21 shows t h a t t h e new binary relation holds for (u,u). T h e old elements are those from which no arrow comes out. If there is more than one relation, we simply have t o arrange for their domains to coincide. code graphs by ideals of a distributive lattice Given graphs on a countable domain we build a countable distributive lattice as follows. T h e atoms of t h e lattice correspond to t h e elements of t he domain. Add 1.u.b.k x U y for every pair of atoms z and y, and on t op of x U y add a n element c(x,y) as a code for { x , y } ,see part h) of t h e picture (the reason why we d o not simply take c(z,y) = 2 U y is t hat we want t h e codes to be indecomposable elements, for reason t o be explained shortly). T hen add t h e necessary elements t o get a distributive lattice. A graph on t h e elements is simply a set of unordered pairs, a n d can be translated as a set of codes. There is thus a natural c o m e spondence between ideals of t h e lattice a n d graphs on t h e atoms, as follows: a n ideal I defines a graph R as

a n d a graph R defines t he ideal generated by t h e codes c ( z , y ) ,for every x and y such t hat R ( z , y ) holds. T h e crucial fact is t hat t h e correspondence is one-one: if R1 a n d R2 ar e different graphs, they generate different ideals. Indeed, t h e only obstacle t o this could be t hat , given a graph R, t h e ideal generated by R as above also contains codes c ( x , y ) for z a n d y such t h a t R ( x , y )does not hold, so t hat decoding t h e ideal would not produce


581

t h e original graph. That this is impossible follows from VI.1.6 and t h e fact that t h e codes are indecomposable elements.

3. translate the theory of countable distributive lattices with quantification over ideals into the first-order theory of P A formula cp with quantification over elements and ideals can be translated into a formula cp* by replacing the ideals by exact pairs coding them, and quantification over ideals by quantification over exact pairs. By the initial segment assumption on PI cp is satisfiable in the theory of countable distributive lattices if and only if there is an element a E P such t h a t the initial segment determined by a in P is a distributive lattice, and cp* holds in it. 0

Corollary VI.4.8 (Nerode and Shore [1980]) The first-order theory of

P, is recursively isomorphic to the theory of Second-Order Arithmetic.

Proof. We prove that the two theories have the same mdegree by interpreting each in the other, thus providing faithful translations that will preserve theorems. Since t h e translations will actually be one-one, the theories will have the same 1-degree,and hence will be recursively isomorphic by 111.7.13. One direction is clear, since every formula about the ordering of m-degrees can be interpreted] in the natural way, as a formula about sets of integers. Thus t h e theory of m-degrees is interpretable in Second-Order Arithmetic. For the converse, we want to show that Second-Order Arithmetic is interpretable in P, . It is enough t o show that V, satisfies t h e conditions of t h e theorem, which it does: the required initial segments exist by V1.2.6, while the existence of exact pairs follows from VI.3.4, although it is also essentially implied by t h e proof of Spector’s Theorem for Turing degrees (since m-degrees are closed under finite differences). 0 Corollary VI.4.9 (Lachlan [1970) The firsborder theory o f V,,,is undecidable and not axiomatizable. For what concerns the extent of decidability, as for Turing degrees (see p. 490) we have that the two-quantifier theory of P, is decidable (Degtev [1979]), and the three-quantifier theory of V,,,is undecidable (Nies [1996]). Both proofs exploit the distributivity of the many-one degrees, and are thus quite different from t h e proofs of the same results for t h e Turing degrees given in Lerman [1983]. T h e next result can now be proved as in Section V.7 (using the present coding for arithmetic), and has the same consequences as there.


582

Theorem VI.4.10 (Nerode and Shore [198Oa]) If C is an ideal 0123closed under jump, the first-order theory of C has the same degree (and actually the same isomorphism type) as the theory o f Second-Order Arithmetic with set quantifiers restricted to sets with degree in C .

Comparison of Degree Theories 3r

VI.5

In this section we consider other notions of degree introduced in Chapter 111, namely 1-degrees, tt-degrees, a n d wtt-degrees. We will mostly quote results about t h e m , a n d will content ourselves to develop their theories only to t h e point needed to show t h a t they a r e not elementarily equivalent a m o n g thernselves a n d with Turing a n d m-degrees (with t h e only exception of tt-degrees a n d wtt-degrees, for which it is not known whether this holds).

1-degrees We have already seen in Section 111.7 t h a t 231 is L special case a m o n g all t h e degree structures we have introduced, because we cannot talk of a least 1-degree in a n y natural way (111.7.4). T h e next result shows t h a t t h e differences a r e even deeper.

Proposition VI.5.1 (Young [1964]) 23, is neither a n upper nor a lower semilattice, i.e. 1.u.b. and g.l.b. d o not always exist. Proof. By 111.2.14 every nonrecursive r.e. T-degree contains a simple set. We will prove in X.1.7 t h a t there a r e incomparable r.e. T-degrees, a n d t h u s t h e r e a r e two incomparable simple sets A a n d B. Suppose they have a 1.u.b. D w.r.t. 1-reducibility: 1. for some 2 E 0, A 51 D U { z } and B 51D U { z } Since D is a n upper bound for A a n d B , t h e r e a r e recursive one-one functions f a n d g such t h a t zE

A ts f(x) E D

and

zEB

@

g(z) E D.

We show t h a t D fl range of f # 0 a n d D f l range of g # 0. Suppose, e.g., t h a t D n range of f = 0: we show D 51 A a n d t h u s B 5 D 51 A , contradicting t h e fact t h a t A and B a r e incomparable. Simultaneously enumerate D a n d t h e range of f , a n d define h by induction as follows: 0

if z shows u p first in D , let h ( z ) be t h e smallest element of A not yet in {h(O),. . . , h(z - I)}, so t h a t h ( z ) E A .

VI.5 Comparison of Degree Theories -k 0

583

if x shows u p first in t h e range of f , let y be t h e unique element such t h a t f(y) = x. If y $ {h(O),. . . , h(x - 1)) let h(x) = y, so t h a t

x

E

D

@

f ( y ) E D ts y E A e+h(x) E A .

Otherwise, y must have been defined by t h e first clause, so let h(x) be t h e smallest element of A not yet in {h(O),. . . , h(x - 1)). T h u s D n range of f # 8, a n d D n range of g # 8 can be shown similarly. Let t h e n z E -d n range of f a n d z* E D n range of g. A 51 D U { z } via f itself (since z is not in t h e range of f), a n d B 51 D U { z } via 9' so defined: ifg(x) # g'(x) = otherwise.

{ 4:"'

Note that g* is still o n e o n e , because z* is not in t h e range of g. 2. D U { z } < I D , arid hence D is not the 1.u.b. of A and B First note t h a t A , B 51 A @ B , so if D is t h e 1.u.b. of A a n d B t h e n D 51 A @ R . B u t A @ B is simple because so a r e A a n d B , a n d t h e n so is D . Suppose D 51 D U { z } . T h e r e is f recursive such t h a t zED

e f(x) E D U { z } .

_ _

T h e n (since z E D ) D has a n infinite r.e. subset { z , f ( ~ ) , f ( ~ ) ( z. .).}, , contradiction. T h u s DU { z } < 1 D , since DU { z } 51D clearly holds. By a symmetrical argument (using z E D a n d D a n d B have n o g.1.b. 0

-

{ z } ) one can show t h a t A

Even if t h e r e is 110 least 1-degree, one can consider segments above O 1 . Lachlan [ 19691 proves t h a t every distributive uppersemilattice which is the direct

limit of an ascending sequence of finite distributive lattices is isomorphic to a segment of Dl above 01. T h e proof consists in forcing all t h e m-degrees of t h e initial segment of D, built in VI.2.5 to contain only cylinders (see VI.S.l), so t h a t they a r e actually 1-degrees. Exercise VI.6.2 I f A is a set of minimal niring degwe coilstructed b y u s i n g strongly uniform trees which is neither immune nor wimmune, then A hay minimal 1-degree. (Hint: as in VI.2.8, using recursive subsets of A and 2 t o make the mreductions one-one.)

A complete characterization of t h e segments of 1-degrees above 01 is not known, even for t h e finite ones, a n d t h e following results of Lachlan [1969] show t h a t it might be complicated:

584


1. every finite segment of ZY1 is a lattice (this is not as trivial as VI.l.lO, since PI is not a n uppersemilattice) 2. some finite segment of ZY1 is nondistributiue

3. not all finite lattices are isomorphic to a finite segment of ZY1. W h at is known is however enough for t he analogue of Simpson’s Theorem, proved by Nerode and Shore [1980]: the first-order theory of ZY1 is recursively isomorphic to the theory of Second-Order Arithmetic. T h e proof uses VI.4.7, once some problems are solved. T h e first problem is t h a t ZY1 is not a n uppersemilattice, a n d thus VI.4.7 h a s t o be rephrased for directed sets, in which every pair of elements has a n upper bound. T h e second problem is t hat t he segments we have for ZY1 are only above O1. This does not introduce complications, because O1 is definable in PI: (8) and {w} are th e only minimal l-degrees (in t h e sense of being degrees with no smaller degree), and O1 is t he srnallest degree above bot h of them. T hus we can work only above 01. T h e final problem is t h e existence of exact pairs. T h e same proof of Spector’s Theorem V.4.3 shows t hat for any countable set of l-degrees in which every pair of elements is bounded there is a pair such t h a t every set l-reducible t o it is also l-reducible t o t h e disjoint union of finitely many finite modifications of representatives of t he given l-degrees. By the restriction above we only work with l-degrees above 01,which are closed under finite modifications because a set A whose l-degree is above 01 is neither immune nor coimmune, and thus infinite recursive subsets of A and 2 can be used to patch u p finite modifications. Closure under disjoint union is needed only for t h e ideals generated by subsets of t h e codes of t h e distributive lattices used for VI.4.7, and it is proved in Nerode and Shore (19801.

Truth-table degrees and weak truth-table degrees We will treat the two structures P t t a n d Dwtt simultaneously, in t h e sense t h a t we will state our results f o r tt-degrees only, but note here that they all hold for wtt-degrees as well, either by t h e same proofs or by minor changes t h a t we will indicate when needed. To get a n elementary difference between ZY an d ZYtt we must develop some theory of t h e latter. We use for it the usual notation for t h e j u m p operator, which is well-defined on tt-degrees by V.1.6. T h e next result provides t he analogue of th e J u m p Inversion Theorem V.2.24.

Theorem VI.5.3 Jump Inversion Theorem for Dtt (Mohrherr [1984]) The range of the jump operator on ZYtt is the cone ZYtt(>O:,).

VI.5 Comparison of Degree Theories *

585

Proof. By V.1.6, for any tt-degree a we have a ' 2 O t t . To get t h e converse, let C be a set such t hat K < t t C : we want t o get A such t h a t A' - t t C . Consider th e construction of V.2.24, with t h e understanding t hat 'the least string u 2 us such t hat {e}"(e) 1' means 'the least string u 2 us such t h a t t h e search of a pair (u,t) (in a n exhaustive recursive list of them) for which {e}:(e)l succeeds'. Then t o be such a u is a n r.e. predicate. 1. A' < t t C By induction on e we want t o determine a truth-table which is satisfied by C if an d only if e E A'. Recall t h a t e E A' is decided at stage 2e 1 of th e construction, since

+

e E A' e {e}"ze+l(e)l

@ (37

2 ~z,)((e}"(e)

I),

where us+l is inductively defined as follows:

us+l

=

{ 1:

pu(u

ifs=2i A i#A' A i E A' if s = 2 i + 1 .

2 usA { i } " ( i ) J )if s = 22

* (C(i))

Since we only need to determine uze, we only have t o use i < e. We can thus fix two initial segments TO and 7 1 of A' and C of length e. Recalling t h e initial observation, t he following is a n r.e. predicate: there is a string 132, that satisfies t he above inductive definition with r O a nd T~ in place of A' a n d C , and a string u 2 u2, such t h a t {e}.(e) 1.

It can thus be reduced t o a question on K and hence, using t h e fact C, t o a tt-condition which is satisfied by C if a n d only if t h e that K itt predicate is true. We still have t o express t h e fact t hat O r and TI really are, respectively, initial segments of A' and C. By induction hypothesis, for each i < e we already have a tt-condition which is satisfied by C if and only if i E A'. Th u s there is a tt-condition t hat is satisfied by C if and only if TO of length e is a n initial segment of A'. And it is trivial t o find a tt-condition satisfied by C if and only if 7 1 of length e is a n initial segment of C itself. Th u s we find a tt-condition, depending on TO and T ] of length e, which is satisfied by C if and only if e E A'. We still have t o eliminate t h e reference t o TO a n d T ~which , is easily done by considering all possible pairs of strings TO a n d T ] of length e, and the disjunction of t he tt-conditions relative to them.


586

2. C A' By induction on e we want to determine a truth-table which is satisfied by A' if a n d only if e E C. Recall t h a t e E C is decided at stage 2e 2 of th e construction, since

+

eEC

*

We thus have t o find

g ~ e + 2 ( 1 ~ ~ 2 e += l ( )1

1uZe+l

1,

102e+1(

E A.

which can be explicitly defined as

where

use (2, A ) =

i E A' other wise.

\ p ( a 2 u2, A {2}"(i)1)1 if

+

Indeed, at stage 22 1 we see if we can make { 2 } A ( i ) converge, a n d if so we take a string t hat does it, otherwise we leave 0 2 i + l = u 2 i . T hus use (2, A ) determines t he length increase due t o forcing t h e j um p, while coding C always produces a one-point extension, and this accounts for th e factor e in t h e expression for J u 2 e + l ) . Unraveling th e definition of 1u2e+ll (and using t h e first e values of C , which we suppose to know by induction hypothesis) we can write down 1 0 2 ~ + l ) , and thus a tt-condition on A' whose truth-value is equivalent to e E C. T h e trouble is t h a t we have used A' explicitly, while tt-reducibility allows only recursive procedures. T h e first step is t o consider, as above, approximations TO a n d TI of A' an d C , respectively of length e 1 and e. This however might cause a problem when computing use ( i ,A ) , since we might look for a string u such t h a t {i}"(i)1 because t he approximation to A' tells us t h a t 2 E A', and we thus believe t hat such a string exists, while this might not be t h e case. But we only need t o look for a string u of length bounded by t he true u s e (2, A ) , since we may ask whether (30 A ) ( { i } " ( i1): ) this is a question r.e. in A , which can be translated into a tt-condition on A'.

+

T h e fact th a t TO and 7 1 are, respectively, initial segments of A' and C can be dealt with as above, this time using t h e induction hypothesis on C. And reference to TO and 7 1 can also be eliminated as above. 0

The Jump Inversion Theorem actually holds for any reducibility 5, between < T , because if K 5, C then K S t t K @ C E, C. By t h e theorem there is A such t h a t A' -tt K C, and hence A' z,. C.

S t t and

VI.5 Comparison of Degree Theories *

587

Kallibekov [1973] showed that O:, is not a minimal cover in the r.e. tt-degrees, a n d a simple modification of his proof actually shows that Oit is not a minimal cover in the tt-degrees. This result relativizes, a n d shows t h a t no tt-degree which contains a jump is a minimal cover. By the Jump Inversion Theorem we then get t h e next result, which provides a n elementary difference between P a n d Dtt. T h e proof uses methods and notations typical of t h e study of r .e. degrees (priority, coding, and Sacks agreement strategy), and will be best understood with some knowledge of Chapter X.

Theorem VI.5.4 (Mohrherr [1984])P t t ( > O i , ) is dense. Proof. By t h e J u m p Inversion Theorem for Ptt, it is enough to show t h a t given sets A and C such t h a t A e is contained in B. But then B E t t A , since t h e @th column codes A , the (i 1)-th column for i 5 e is finite, and the other columns are contained in B. Then C' S t t B I t t A , again contradicting the hypothesis A

Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers

Classical recursion theory: the theory of functions and sets of natural numbers

Classical recursion theory: The theory of functions and sets of natural numbers