Computational Prospects of Infinity II: Presented Talks (Lecture Notes Series, Institute for Mathematical Sciences, N)

COMPUTATIONAL PROSPECTS OF INFINITY Part II: Presented Talks LECTURE NOTES SERIES Institute for Mathematical Sciences...

Author: Chitat Chong | Qi Feng | Theodore A. Slaman | W. Hugh Woodin | Yue Yang

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COMPUTATIONAL PROSPECTS OF INFINITY Part II: Presented Talks

LECTURE NOTES SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Louis H. Y. Chen and Ka Hin Leung Institute for Mathematical Sciences National University of Singapore ISSN: 1793-0758 Published Vol. 4

An Introduction to Stein’s Method edited by A. D. Barbour & Louis H. Y. Chen

Vol. 5

Stein's Method and Applications edited by A. D. Barbour & Louis H. Y. Chen

Vol. 6

Computational Methods in Large Scale Simulation edited by K.-Y. Lam & H.-P. Lee

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Markov Chain Monte Carlo: Innovations and Applications edited by W. S. Kendall, F. Liang & J.-S. Wang

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Transition and Turbulence Control edited by Mohamed Gad-el-Hak & Her Mann Tsai

Vol. 9

Dynamics in Models of Coarsening, Coagulation, Condensation and Quantization edited by Weizhu Bao & Jian-Guo Liu

Vol. 10 Gabor and Wavelet Frames edited by Say Song Goh, Amos Ron & Zuowei Shen Vol. 11 Mathematics and Computation in Imaging Science and Information Processing edited by Say Song Goh, Amos Ron & Zuowei Shen Vol. 12 Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory — In Honor of Roger E. Howe edited by Jian-Shu Li, Eng-Chye Tan, Nolan Wallach & Chen-Bo Zhu Vol. 13 Econometric Forecasting and High-Frequency Data Analysis edited by Roberto S. Mariano & Yiu-Kuen Tse Vol. 14 Computational Prospects of Infinity — Part I: Tutorials edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Vol. 15 Computational Prospects of Infinity — Part II: Presented Talks edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang *For the complete list of titles in this series, please go to http://www.worldscibooks.com/series/lnsimsnus_series.shtml

LaiFun - Comp Prospects - Vol-15.pmd

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Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

Vol.

15

COMPUTATIONAL PROSPECTS OF INFINITY Part II: Presented Talks Editors

Chitat Chong

National University of Singapore, Singapore

Qi Feng

Chinese Academy of Sciences, China & National University of Singapore, Singapore

Theodore A. Slaman, W. Hugh Woodin University of California at Berkeley, USA

Yue Yang

National University of Singapore, Singapore

World Scientific NEW JERSEY

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BEIJING

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TA I P E I

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Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore – Vol. 15 COMPUTATIONAL PROSPECTS OF INFINITY Part II: Presented Talks Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-279-654-7 ISBN-10 981-279-654-1

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LaiFun - Comp Prospects - Vol-15.pmd

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Master Review Vol. 9in x 6in – (for Lecture Note Series, IMS, NUS)

contentsLNS-Vol15

CONTENTS

Foreword

vii

Preface

ix

Generating Sets for the Recursively Enumerable Turing Degrees Klaus Ambos-Spies, Steffen Lempp and Theodore A. Slaman

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Coding into H(ω2 ), Together (or Not) with Forcing Axioms. A Survey David Asper´ o

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Nonstandard Methods in Ramsey’s Theorem for Pairs Chi Tat Chong

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Prompt Simplicity, Array Computability and Cupping Rod Downey, Noam Greenberg, Joseph S. Miller and Rebecca Weber

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Lowness for Computable Machines Rod Downey, Noam Greenberg, Nenad Mihailović and André Nies

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A Simpler Short Extenders Forcing — Gap 3 Moti Gitik

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Limit Computability and Constructive Measure Denis R. Hirschfeldt and Sebastiaan A. Terwijn

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The Strength of Some Combinatorial Principles Related to Ramsey’s Theorem for Pairs Denis R. Hirschfeldt, Carl G. Jockusch, Jr., Bjørn Kjos-Hanssen, Steffen Lempp and Theodore A. Slaman

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contentsLNS-Vol15

Contents

Absoluteness for Universally Baire Sets and the Uncountable II Ilijas Farah, Richard Ketchersid, Paul Larson and Menachem Magidor

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Monadic Definability of Ordinals Itay Neeman

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A Cuppable Non-Bounding Degree Keng Meng Ng

207

Eliminating Concepts André Nies

225

A Lower Cone in the wtt Degrees of Non-Integral Effective Dimension André Nies and Jan Reimann

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A Minimal rK-Degree Alexander Raichev and Frank Stephan

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Diamonds on Pκ λ Masahiro Shioya

271

Rigidity and Biinterpretability in the Hyperdegrees Richard A. Shore

299

Some Fundamental Issues Concerning Degrees of Unsolvability Stephen G. Simpson

313

Weak Determinacy and Iterations of Inductive Definitions MedYahya Ould MedSalem and Kazuyuki Tanaka

333

A tt Version of the Posner-Robinson Theorem W. Hugh Woodin

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Cupping Computably Enumerable Degrees in the Ershov Hierarchy Guohua Wu

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April 10, 2008


foreword

FOREWORD

The Institute for Mathematical Sciences at the National University of Singapore was established on 1 July 2000. Its mission is to foster mathematical research, both fundamental and multidisciplinary, particularly research that links mathematics to other disciplines, to nurture the growth of mathematical expertise among research scientists, to train talent for research in the mathematical sciences, and to serve as a platform for research interaction between the scientific community in Singapore and the wider international community. The Institute organizes thematic programs which last from one month to six months. The theme or themes of a program will generally be of a multidisciplinary nature, chosen from areas at the forefront of current research in the mathematical sciences and their applications. Generally, for each program there will be tutorial lectures followed by workshops at research level. Notes on these lectures are usually made available to the participants for their immediate benefit during the program. The main objective of the Institute’s Lecture Notes Series is to bring these lectures to a wider audience. Occasionally, the Series may also include the proceedings of workshops and expository lectures organized by the Institute. The World Scientific Publishing Company has kindly agreed to publish the Lecture Notes Series. This Volume, “Computational Prospects of Infinity, Part II: Presented Talks”, is the fifteenth of this Series. We hope that through the regular publication of these lecture notes the Institute will achieve, in part, its objective of promoting research in the mathematical sciences and their applications. January 2008

Louis H.Y. Chen Ka Hin Leung Series Editors

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preface

PREFACE The Workshop on Computational Prospects of Infinity was held at the Institute for Mathematical Sciences, National University of Singapore, from 20 June to 15 August 2005. The focus in the first month of the Workshop was on set theory, with two tutorials running in parallel, by John Steel (“Derived Models Associated to Mice”) and W. Hugh Woodin (“Suitable Extender Sequences”). There were also 21 talks given during this period. The second half of the Workshop was devoted to recursion theory, with two tutorials given respectively by Rod Downey (“Five Lectures on Algorithmic Randomness”) and Ted Slaman (“Definability of the Turing Jump”). In addition, there were 42 talks delivered over four weeks. This volume constitutes Part II of the proceedings of the Workshop. It contains refereed articles, both contributed and invited, based on talks presented at the Workshop. Written versions of the tutorials are collected in Part I of the proceedings which appears as a separate volume. The workshop provided a platform for researchers in the logic community from many parts of the world to meet in Singapore, to discuss mathematics and to experience a city that is a meeting point of East and West. We thank the Department of Mathematics and the Institute for Mathematical Sciences at the National University of Singapore for their support and hospitality extended to all participants. November 2007

Chi Tat Chong National University of Singapore, Singapore Qi Feng Chinese Academy of Sciences, China and National University of Singapore, Singapore Theodore A. Slaman University of California at Berkeley, USA W. Hugh Woodin University of California at Berkeley, USA Yue Yang National University of Singapore, Singapore Editors ix


GENERATING SETS FOR THE RECURSIVELY ENUMERABLE TURING DEGREES Klaus Ambos-Spies Department of Mathematics and Computer Science University of Heidelberg D-69120 Heidelberg, Germany E-mail: [email protected] Steffen Lempp∗ Department of Mathematics University of Wisconsin–Madison Madison, WI 53706-1388, USA E-mail: [email protected] Theodore A. Slaman† Department of Mathematics University of California–Berkeley Berkeley, CA 94720-3840, USA E-mail: [email protected]

We give an example of a subset of the recursively enumerable Turing degrees which generates the recursively enumerable degrees using meet and join but does not generate them using join alone.

1. Introduction One of the recurrent themes in the area of the recursively enumerable (r.e.) degrees has been the study of the meet operator. While, trivially, the partial ordering of the r.e. degrees is an upper semi-lattice, i.e., the join ∗ Lempp was partially supported by NSF grant DMS-0140120 and a Mercator Guest Professorship of the Deutsche Forschungsgemeinschaft. † Slaman was partially supported by the Alexander von Humboldt Foundation and by National Science Foundation Grant DMS-9988644.

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K. Ambos-Spies, S. Lempp and T. A. Slaman

operator is total, the meet of two incomparable r.e. degrees may or may not exist (Lachlan (1966), Yates (1966)). The asymmetry between joins and meets is further illustrated by the fact that, by Sacks’ splitting theorem (Sacks (1963)), every nonzero r.e. degree is join-reducible, i.e., is the join of two lesser degrees, whereas there are both, meet-reducible (branching) and meet-irreducible (nonbranching), incomplete r.e. degrees (Lachlan (1966)). The existence of meets and the failure of meets are densely distributed in the partial ordering of the r.e. degrees. So Fejer (1983) showed that the nonbranching degrees are dense while Slaman (1991) showed that the branching degrees are dense. Similarly, every interval of the r.e. degrees contains an incomparable pair of degrees without meet (Ambos-Spies (1984)) and an incomparable pair of degrees with meet (Slaman (1991)). That, actually, the lack of meets is more common than the existence of meets has been demonstrated by Ambos-Spies (1984) and, independently, by Harrington (unpublished) who showed that, for any nonzero, incomplete r.e. degree a, there is an incomparable degree b such that the meet of a and b does not exist, but also that there is such a degree a such that, for any incomparable degree b, the meet of a and b does not exist. More evidence, that the failure of meets is more typical than their existence, was given by Jockusch (1985) who showed that, given r.e. degrees a, b and c such that a and b are incomparable and c is the meet of a and b, none of these degrees is e-generic. Another way to look at the join and meet operators in the r.e. degrees is to study generating sets, i.e., sets of r.e. degrees which generate all the recursively enumerable degrees under (finitely many applications of) join and meet. The question now arises naturally whether both the join operation and the meet operation are needed here. As observed in AmbosSpies (1985), the above results on nonbranching degrees easily imply that the join operation is indeed necessary, namely there is a subset of the recursively enumerable degrees which generates all recursively enumerable degrees using join and meet but not using meet alone. Ambos-Spies, however, left open the question of whether the meet operation is necessary (see Ambos-Spies (1985), Problem 1). The above mentioned negative results on meets by Fejer (1983), Ambos-Spies (1984) and Jockusch (1985) may suggest a negative answer to this question. More evidence in this direction has been obtained by Ambos-Spies (1985) who showed that any generating set intersects any notrivial initial segment of the r.e. degrees and, more recently, by Ambos-Spies, Ding and Fejer (unpublished) who showed that any generating set generates the high r.e. degrees using join alone. Despite

Generating Sets for the Recursively Enumerable Turing Degrees

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this negative evidence, in this paper, we answer Ambos-Spies’ question affirmatively by the following Theorem 1.1 There exists a subset G of the recursively enumerable Turing degrees which generates the recursively enumerable Turing degrees using meet and join but does not generate them using join alone. Proof. Our theorem follows by our technical result, Theorem 2.1, below, using a nonconstructive definition of the set G. Fix the recursively enumerable degree a from Theorem 2.1. Let {xn }n∈ω be a (noneffective) enumeration of all recursively enumerable degrees ≤ a. We now define a (noneffective) sequence of recursively enumerable degrees 0 = y0 ≤ y1 ≤ y2 ≤ · · · < a as follows: Set y0 = 0. Given yn < a, check whether yn ∪ xn = a. If not, then set yn+1 = yn ∪ xn . Otherwise, let b be the recursively enumerable degree given by Theorem 2.1 using x = xn and y = yn , and set yn+1 = yn ∪ b. Finally, we define G = {x | x ≤ a or ∃n (x ≤ yn )}. By Theorem 2.1, the degree a is clearly not the join of any finite set of degrees in G. On the other hand, fix any recursively enumerable degree x and assume x ∈ / G. Then x ≤ a, and so x = xn for some n ∈ ω. Since x∈ / G, we have x ≤ yn+1 and so x ∪ y = a for y = yn . Fix b, c, d, and e as in Theorem 2.1. Then x = b ∪ (d ∩ e) where all of b, d, and e are in G since b ≤ yn+1 and d, e ≤ a. 2. The technical theorem and some intuition for its proof Starting with this section, we will prove the technical theorem needed to establish Theorem 1.1: Theorem 2.1 There is a nonrecursive, recursively enumerable set A such that for every pair of recursively enumerable sets X and Y , if X and Y are recursive in A and A is recursive in XY then one of the following conditions holds. 1. A is recursive in Y . 2. There are recursively enumerable sets B, C, D, and E such that (a) X has the same Turing degree as BC,

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(b) D and E are not recursive in A and the degree of C is the infimum of the degrees of DC and EC, and (c) A is not recursive in BY . 2.1. Requirements and simple strategies We disassemble the statement of Theorem 2.1 into requirements as follows. First, A must be nonrecursive and so we must satisfy all the requirements Θ = A, where Θ is a recursive function. Second, for each X, Y , Λa,x , Λa,y , and Λxy,a , we associate the principal equations Λa,x (A) = X, Λa,y (A) = Y , and Λxy,a (XY ) = A. We can satisfy our requirement on X, Y , Λa,x , Λa,y , and Λxy,a in any of several ways. If the principal equations are not valid then our requirement is satisfied. Anticipating that the principal equations actually are valid, we enumerate the sets B, C, D, and E and recursive functionals Γx,b , Γx,c , and Γbc,x . We ensure that Γx,b (X) = B, Γx,c (X) = C, and Γbc,x (BC) = X. Now, our requirement is satisfied in one of two ways. For every recursive functional Θby , if Θby (BY ) = A then there is a ∆y,a , which we enumerate during our construction, such that ∆y,a (Y ) = A. If there is a ∆y,a such that ∆y,a (Y ) = A, then again our requirement is satisfied. Otherwise, we ensure that every instance of the following family of requirements is satisfied. 1. For all Θa , Θa (A) = D and Θa (A) = E. 2. For all Ψcd and Ψce , if Ψcd (CD) = Ψce (CE) then there is a Ξc such that Ξc (C) = Ψcd (CD) = Ψce (CE). 2.2. Strategies 2.2.1. Making Θ = A: σ0 (Θ) We ensure that Θ = A by choosing a number n, keeping n out of A until seeing Θ(n) = 0, and then enumerating n into A. This strategy σ0 is one of the standard methods to satisfy requirements of this form. 2.2.2. Measuring whether the equations hold: σ1 (X, Y, Λa,x , Λa,y , Λxy,a ) Now, we consider the more complicated requirements. Suppose that X, Y , Λa,x , Λa,y , and Λxy,a are given.


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Our strategy σ1 (X, Y, Λa,x , Λa,y , Λxy,a ) approximates if the principal equations hold for X, Y , Λa,x , Λa,y , and Λxy,a . We will abbreviate by σ1 the strategy σ1 (X, Y, Λa,x , Λa,y , Λxy,a ) and use similar conventions throughout this section. Essentially, σ1 measures expansionary stages in the approximation to these equalities. For technical reasons, explained below, σ1 waits for something more than simple expansion. In the following, a1 and a2 are variables of the strategy which enumerates pairs (a1 , a2 ) into a list of pairs of witnesses. 1. If a1 is undefined and it is possible to do so, choose a value for a1 that is larger than λa,x (A, x)[s] for every x previously mentioned in the construction during a σ1 -expansionary stage. Let x1 be the smallest number x such that λa,x (A, x)[s] is greater than a1 . Suspend the enumeration of any functionals associated with B, C, D or E. (We may assume that we have not enumerated any computations from BC of X at arguments greater than or equal to x1 .) Wait until the first stage s such that (Λxy,a (XY ) a1 + 1 = A a1 + 1)[s], and (Λa,x (A) = X)[s] and (Λa,y (A) = Y )[s] on all numbers less than or equal to the maximum of λxy,a (XY )[s] a1 + 1. At this stage, we let a2 equal the supremum of (λa,x (A) = X)[s] and (λa,y (A) = Y )[s] on all numbers less than or equal to the maximum of λxy,a (XY )[s] a1 + 1. We enumerate the pair (a1 , a2 ) into our list and let the strategies of lower priority resume the enumeration of any functionals associated with B, C, D or E. (The (a1 , a2 ) notation will be convenient below.) Go to Step 2. 2. At the next stage when σ1 is active, we say that a1 is undefined, and go to Step 1. Consider the possibilities. The strategy σ1 could reach a limit in Step 1. In this case, one of the principal equations fails and the requirement is satisfied. If σ1 does not reach a limit in Step 1 then it enumerates infinitely many stable pairs and has no other effect on the construction. For the remainder of this section, we assume that σ1 does not reach a finite limit and that all subsequent strategies act during the stages when σ1 enumerates a new pair. We call such stages σ1 -expansionary.

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2.2.3. Computations between B, C, and X: σ2 (X, Y, Λa,x , Λa,y , Λxy,a ) Our strategy σ2 builds functionals Γx,b and Γx,c and ensures that if the principal equations are valid then for each n there are infinitely many s such that (Γx,b (X, n) = B(n))[s] and (Γx,c (X, n) = C(n))[s]. This, combined with our preserving A, B, and C, will be sufficient to conclude that B and C are recursive in X. We ensure their correctness by imposing the constraint on all lower priority strategies τ that if Γx,b (X, n)[s] or Γx,c (X, n)[s] is defined while τ acts then τ cannot enumerate n into B or C, respectively, during that stage. Similarly, we ensure that X is recursive in BC by enumerating a functional Γbc,x and ensuring that if the principal equalities hold then for all n, Γbc,x (BC, n) = X(n) during infinitely many stages of the construction. We have complete freedom to define the uses of these functionals, but the construction does not require a subtle decision. During σ1 -expansionary stages, we enumerate new computations into Γbc,x . If n enters X during stage s and Γbc,x (BC, n) = 0[s] then we must enumerate a number less than or equal to γbc,x (BC, n)[s] into either B or C. We set the uses of these functions to be larger than any number previously used in the construction. In the case of maintaining Γbc = X, we also have the freedom to decide which of B and C to change when recording a change in X. The choice made is irrelevant to σ2 . In our construction, we will leave the decision to the highest priority strategy for which it is relevant. See the discussion of the strategies of type σ6 . 2.2.4. Making C the infimum of CD and CE: σ3 (X, Y, Λa,x , Λa,y , Λxy,a , Ψcd , Ψce) We will use the branching strategies from Fejer (1982) and attempt to make the degree of C equal to the infimum of the degrees of CD and CE. Suppose that Ψcd and Ψce are given and let σ3 denote our branching strategy associated with this pair. Then, σ3 enumerates a functional Ξc . Say that s is σ3 -expansionary if and only if the least number n such that (Ψcd (CD, n) = Ψce (CE, n))[s] is larger than at any earlier stage. First, during stage s, if there is an n such that Ξc (C, n)[s] is defined and a strategy of priority less than or equal to that of σ3 enumerates numbers into C, D, or E so that neither (Ψcd (CD, n) = Ξc (C, n))[s] nor (Ψce (CE, n) = Ξc (C, n))[s], then σ3 must enumerate a number less than or


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equal to ξc (C, n)[s] into C. (We will have to argue that this enumeration is compatible with C’s being recursive relative to X.) Second, if s is σ3 -expansionary then for the least n such that ξc (C, n) is not defined, we choose a value for ξc (C, n)[s] which is larger than any number previously mentioned in the construction and enumerate a computation into Ξc setting Ξc (C[s], n) = Ψcd (CD[s], n) with use ξc (C, n)[s]. If Ψcd (CD) = Ψce (CE) then there will be infinitely many σ3 expansionary stages. Since we will be preserving computations from CD and CE, the converse will also be true. So, if Ψcd (CD) = Ψce (CE), then σ3 will act finitely often. Otherwise, it produces a functional Ξc from C which is defined infinitely often to agree with the common value of Ψcd (CD) and Ψce (CE). Again, since we are preserving the sets that we construct, this will be sufficient to ensure that Ξc (C) is equal to this common value. We will assume that there are infinitely many σ3 -expansionary stages and describe the appropriate strategies to follow. These strategies act only during σ3 -expansionary stages. Instability in C and compatibility between σ2 and σ3 . The strategies σ3 introduce an instability to the initial segments of C. Namely, suppose that a strategy τ enumerates a number c into C. Then, c enters both CD and CE and could change the common value of Ψcd (CD, c1 ) and Ψce (CE, c1 ). In response, τ enumerates ξc (C, c1 ) into C, possibly changing C at a number less than c, and the effect can propagate. We call the set of numbers that enter C in this way the cascade initiated by c. When combined with the strategy to ensure that C is recursive in X, the branching strategies make it difficult to enumerate any number at all into C. If Γx,c (X, m) is defined then we cannot enumerate any c into C unless we can be sure that the instability in C will not propagate to the point of requiring that m enter C. We will use some of the ideas of Slaman (1991) to work within this constraint. Definition 2.2 A number c is σ3 -stable at stage s if for all m, if ξc (C, m)[s] < c then either ψcd (CD, m) < c or ψce (CE, m) < c. We note that if c is σ3 -stable at stage s then any cascade initiated by a number greater than or equal to c during stage s does not include any number less than c. To prove this claim, consider the recursive propagation of a cascade initiated by a number greater than or equal to c, and let m be the first number less than c to appear in the cascade. Earlier in the

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propagation of the cascade, C would have to change below the minimum of ψcd (CD, m)[s] and ψce (CE, m)[s]. By the stability of c, this minimum is less than c and we have contradicted m’s being first. Thus, if c is stable and Γx,c (X, c)[s] is not defined, then we can enumerate c into C and respect both σ2 and σ3 . We will design all the strategies to follow so that they enumerate only stable numbers into C. Of course, there is no such constraint on B, since B is not constructed to be branching. 2.2.5. Making Θa (A) = D and Θa (A) = E: σ4 (X, Y, Λa,x , Λa,y , Λxy,a , Θa ) and σ5 (X, Y, Λa,x , Λa,y , Λxy,a , Θa ) We use a variation, σ4 , on the basic Friedberg strategy to ensure that Θa (A) = D. (The strategy σ5 for E is similar.) We choose n larger than any number previously mentioned in the construction and constrain n from entering D. We wait for a stage s such that (Θa (A, n) = 0)[s]. By our assumption, s will be σ1 and σ3 -expansionary. Then, we enumerate n into D and constrain any number less than s from entering any set under construction other than D. We note that these actions are consistent. Since we did not enumerate anything into A and A’s computation of X exists on a longer interval than ever before, X cannot change at any number m such that Γbc,x (BC, m)[s] is defined. So, σ2 will not require any change in B or C. Since s is σ3 -expansionary, both Ψcd (CD)[s] and Ψce (CE)[s] were defined (before we changed D), agreed on a longer interval than ever before, and agreed with Ξc (C)[s] where Ξc (C)[s] was defined. Since we did not change E, Ψce (CE)[s] is still equal to Ξc (C)[s] where the latter is defined and σ3 does not require any change in C. 2.2.6. If Θby (BY ) = A then ∆y,a (Y ) = A: σ6 (X, Y, Λa,x , Λa,y , Λxy,a , Θby ) Now we come to the crux of the proof of Theorem 2.1. Our strategy σ6 must either diagonalize Θby (BY ) against A, or it must determine that A is recursive in Y . Since Y is an arbitrary set below A, both cases are possible. In the context of the construction, σ6 can assume that every stage is σ1 and σ2 -expansionary. Now, this implies that if we enumerate a number a into A during stage s, there is a later stage t during which A recomputes X and Y and either X[s] = X[t] or Y [s] = Y [t] and the least m at which the inequality occurs is less than λxy,a (XY, a)[s]. In other words, if we change A then one of X or Y must change in order to correct Λxy,a (XY ). To establish Θby (BY ) = A, at least once we would have to change A without having to change B and without Y ’s having changed. This could


9

happen, since the change in A could be recorded in X and we could record the change in X (for the sake of σ3 ) in C. If, on the other hand, this diagonalization is not possible then we must conclude that A is recursive in Y . The conclusion is not unreasonable since every change in A results in a change in Y . But, if even one change in A results in a change in X, then we must be able to record that change in C. We have reached the technical problem to be solved to prove the theorem. For every A-change allowed by σ6 , if it results in a change in X, then we must be able to record that change in X and in C. Now remember that we are only able to enumerate numbers into C which are σ3 -stable during their stage of enumeration. So we must ensure that changes in X can be recorded in C by the enumeration of such numbers. This is the purpose in our strategy σ6 . Configurations. Consider a possible stage-s situation as depicted in Figure 1. In this picture, we illustrate a number a1 which we intend to enumerate into A; x1 is the least number x such that a1 is less than λa,x (A, x)[s] and hence the least number which might enter X when a1 enters A; x2 is equal to λxy,a (XY, a1 )[s] and a1 ’s entering A would cause a change in XY below x2 ; a2 is the supremum of λa,x (A, x2 )[s] and λa,y (A, x2 )[s], so our preserving A on numbers less than a2 will preserve the relationship between a1 , x1 , and x2 ; c is γbc,x (BC, x1 )[s] and so enumerating c into C would correct the computation of X from BC on every argument at which X might change; we intend that c be σ3 -stable at stage s and so, for any m, if ξc (m) is less than c then one of ψcd (CD, m)[s] or ψce (CE, m)[s] is also less than c; finally, x3 is γx,c (X, c), the use of X’s computation of C at argument c. Note that we can preserve the relationships between these numbers by preserving A up to a2 , and B,C,D, and E up to c. (We can keep x3 above x2 by enumerating our functions so that the uses of new computations are at least as great as the uses of earlier computations at the same argument.) Suppose that, in this situation, we were to enumerate a1 into A and Y did not change below x2 to record that fact (for example, if Y ≥ A). Then X would change below x2 , allowing us to enumerate the σ3 -stable number c into C and thereby correct BC’s computations for any change in X allowed by a1 ’s entering A. We say that the situation depicted in Figure 1 is a σ1 -configuration for a1 . Anticipating that Θby (BY ) = A, we must ensure that for all but a

10


A

X Y C B x3 Z Z Z ZΓx,c Z Z Z c ~ Z a2 Z

Z Λa,x

Z

Z Z ~ x2 Z

ξc (m)

Γ Λxy,a

bc,x = a1

Z

Z Λa,x Z

Z

Z x1 ~

Z

D

E

ψce (m)

Figure 1: Configuration for a1 . recursive set of numbers a, if a enters A then it does so in the role of a1 with a configuration as above. Generating configurations. Though configurations seem artificial at first, they are very common. In fact, a new configuration can be produced during every σ1 -expansionary stage. Note that, by the constraint imposed by σ3 , at the beginning of every stage t, for every m, if Ξc (C, m)[t] is defined then one of Ψcd (CD, n)[t] and Ψce (CE, n)[t] is also defined with the same value. Now consider a σ1 -expansionary stage s. Let c be the least strict upper bound on the range of ξc (C)[s]. By the observation above, c is σ3 -stable. Since s is σ1 -expansionary, σ1 enumerated a pair (a1 , a2 ) related as in Figure 1. Further, by the choice of a1 and a2 , a1 is greater than λa,x (A, x)[s] for every x such that Γbc,x (BC, x) has ever been defined. Consequently, there is no computation in Γbc,x [s] which applies to the argument x1 . Then, we can use σ2 to enumerate computations into Γx,b , Γx,c , and Γbc,x so that γbc,x (BC, x1 ) > c, and γx,c (X, c) > λxy,a (X, a1 ). In Figure 1, λxy,a (X, a1 )


11

would be x2 and γx,c (X, c) would be x3 . In short, at the beginning of each stage s, the whole of ξc [s] is stable and at a σ2 -expansionary stage, we can use the pair (a1 , a2 ) enumerated by σ1 and enumerate new computations into our functionals to extend ξc [s] to a configuration for a1 . Restricting to configured numbers.

We satisfy the requirement

Θby (BY ) = A =⇒ ∆y,a (Y ) = A as follows. 1. If a is undefined and we are not preserving an inequality between Θby (BY ) and A, then choose a value for a such that a is larger than any value previously chosen and such that there is a configuration for a. We restrain a from entering A and preserve the configuration for a until we find a stage s such that (Θby (BY, a) = 0)[s] and that Λa,y (A)[s] is equal to Y [s] on all numbers less than or equal to θby (BY, a)[s]. At stage s, we go to Step 2. 2. Let a0 be the largest number that we have previously enumerated as allowed to enter A with a certified configuration (or 0 if there is no such number). For each number n between a0 and a, if n ∈ A[s] then we restrain n from ever entering A. We enumerate a into the set of numbers still allowed to enter A and we say that the current configuration for a together with the current computation (Θby (BY, a) = 0)[s] is the certified configuration associated with a during stage s. For all strategies τ of lower priority, require that if τ enumerates a into A during stage t then the certified configuration associated with a during stage s must also exist during stage t. That is, the initial segments of the sets involved in the configuration for a and the computation from BY must not have changed. Further, if during the next σ1 -expansionary stage u it happens that Y θby (BY )[t] is equal to Y θby (BY )[u] (i.e., Y did not change) then the change in X is recorded in C and we preserve the inequality Θby (BY, a) = A(a) by preserving the appropriate initial segments of the sets under construction. We say that a is now undefined and go to Step 1. Clearly, if for every Θby we can conclude that Θby (BY ) = A then we have satisfied our requirement. Further, each of the strategies will act at most finitely often and cause little trouble to the rest of the construction.

12


Suppose this is not the case and consider the effects of the above strategy σ6 when Θby (BY ) = A. Assume that the strategy is never injured (or be willing to accept finitely many exceptions). Then σ6 enumerates an infinite increasing sequence of numbers a as still being allowed to enter A. Call this sequence the σ6 -stream of numbers. For each number n, if n is not an element of the σ6 -stream then n is an element of A if and only if n is enumerated into A before any number greater than n is enumerated into the σ6 -stream. Thus, the restriction of A to the numbers not in the σ6 stream is recursive. Now, consider a number a which is enumerated into the σ6 -stream, say at stage sa . By the action of σ6 , for every number a in the σ6 -stream, if a enters A during a stage s greater than or equal to the one during which a was enumerated in the sequence, then the configuration existent when a was enumerated into the stream by σ6 is still available during stage s. Since Θby (BY ) = A and there are infinitely many σ1 -expansionary stages, it must be the case that during the interval from stage t to the next σ1 expansionary stage after t, Y changed below λxy,a (XY, a)[s]. Thus, if a is an element of the σ6 -stream and is enumerated into the stream at stage s, then a enters A no later than the first stage u after stage s such that Y [u] λxy,a (XY, a)[s] = Y λxy,a (XY, a)[s]. It follows that A is recursive relative to Y . 2.2.7. One sequence (X, Y, Λa,x, Λa,y , Λxy,a ) If we were to work only with one sequence (X, Y, Λa,x , Λa,y , Λxy,a ) or equivalently have only one strategy of type σ1 , then our construction would be particularly simple. We would start with the strategies of type σ1 and σ2 and follow them with the strategies of type σ3 , σ4 , σ5 , and σ6 (as well as nonrecursiveness strategies σ0 ). In the simplest case, one of these strategies could have a finite outcome and we could conclude that our requirement is satisfied. In the simplest case, the σ1 -strategy could have a finite outcome and we could conclude that our requirement is satisfied. If not, then either each of the σ6 -strategies would have a finite outcome and we would have satisfied all the necessary requirements Θby (BY ) = A, or one of our strategies would have an infinite outcome and we would conclude that our requirement is satisfied by virtue of A’s being recursive relative to Y . In this last case, how can we conclude that A is not recursive? The σ6 -strategy that generates an infinite stream associates with these numbers an infinite stream of certified configurations. The strategy to ensure


13

Θ = A chooses a number a, preserves its configuration and preserves its certification by preserving B and preserving enough of A to ensure that Y cannot change on any relevant number. If at a later stage t it happens that Θ(a)[t] = 0 then the diagonalization strategy enumerates a into A. 3. The global construction In the previous section, we analyzed the combinations of the strategies associated with a single sequence (X, Y, Λa,x, Λa,y , Λxy,a ). We now combine the strategies for all possible such sequences and thereby present a proof of Theorem 2.1. 3.1. Interactions between σ-strategies In fact, there is very little interaction between the strategies associated with different sequences. For the most part, their constraints apply to different B’s, C’s, D’s, and E’s and so are mutually compatible. The only set that they have in common is A and the only constraints that they put upon A are the finite ones associated with successful diagonalization and the infinite one constraining the enumeration of new elements of A to the conditions of a σ6 -stream. Consider a strategy τ constrained to work within an infinite σ6 -stream. The new constraint on τ is that at stage t, τ can enumerate element a into A if and only if a was enumerated into the σ6 -stream at a stage s < t and the configuration associated with a during the stage s still exists during stage t. If τ is associated with a σ0 -strategy ensuring Θ = A then when τ chooses its number with which to diagonalize, τ chooses that number a from the σ6 -stream. While τ is waiting for Θ(a) = 0, τ preserves enough of A to preserve the σ6 -configuration associated with a. If Θ(a) is seen to be equal to 0 then τ can enumerate a into A consistently with the constraint of σ6 . By inspection of the strategies, this is the only way by which numbers enter A and so we need not make many internal changes within our families of strategies. 3.2. The tree of strategies We fix recursive enumerations (Θi : i ∈ ω) of all recursive functionals relative to the empty set, ((X i , Y i , Λia,x , Λia,y , Λixy,a ) : i ∈ ω) of all sequences

14


as described in σ1 , and, for each i, (Ψi,j : j ∈ ω), (Θi,j a : j ∈ ω), and : j ∈ ω) of all recursive functionals with one set argument. Of course, (Θi,j by i,j i,j i,j the enumerations (Ψ : j ∈ ω), (Θa : j ∈ ω), and (Θby : j ∈ ω) need not depend on i, but the notation will be convenient below. Let ((i, j) : i, j ∈ ω) be a recursive ordering of ω × ω of order type ω. We will assume that for all j, i is less than or equal to the position of (i, j) in this ordering. We define a tree T of sequences of pairs of strategies and outcomes using recursion. We will also order the immediate extensions of each node from left to right. Ordering by first difference, we have a left to right ordering for all incompatible sequences in T . As usual, shorter nodes or nodes to the left will be assigned higher priority than those below or to the right. For η ∈ T , we will speak of the extensions of η as being below η in T . We start with the empty sequence as an element of T . Suppose that η = ((τk , ok ) : k < ) is an element of T . Definition 3.1 Suppose k < and τk is of the form σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a ). Then 1. τk is in effect at η if and only if ok is Π2 , and 2. τk is unresolved at η if and only if for all j, (σ6 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j by ), Π2 ) ∈ η. Definition 3.2 Suppose k < and τk is of the form σ6 (X, Y, Λa,x , Λa,y , Λxy,a , Θby ). Then, τk is in effect at η if and only if ok is Π2 . Strategies below η in T are based on the assumption that there will be infinitely many expansionary stages for the σ1 - and σ6 -strategies in effect at η. If τk is unresolved at η then no strategy in η has determined that A is recursive relative to Y . Let imax be the largest i such that there is a k < such that τk is equal to σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a ). (If there is no such i, let imax = −1.) Case 1. If there is a pair (i∗ , j ∗ ) among the first imax many such pairs such that ∗

∗

∗

∗

∗

A. σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a ) is in effect at η, unresolved at η, and ∗

∗

∗

∗

∗

B. one of σ2 (X i , Y i , Λia,x , Λia,y , Λixy,a ), ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ3 (X i , Y i , Λia,x , Λia,y , Λixy,a , Ψi ,j ),

Generating Sets for the Recursively Enumerable Turing Degrees ∗

∗

∗

∗

∗

∗

∗

σ4 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θia ,j ), ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ5 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θia ,j ), ∗

∗

∗

∗

∗

15

∗

∗

or σ6 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θiby,j ) does not appear in (the first coordinate of an element of) η, then let (i, j) be the least such (i∗ , j ∗ ). We determine the immediate successor of η in T by the first of the following conditions which applies. 1. If σ2 (X i , Y i , Λia,x , Λia,y , Λixy,a ) does not appear in η then η (σ2 (X i , Y i , Λia,x , Λia,y , Λixy,a), Π1 ) ∈ T. 2. If σ3 (X i , Y i , Λia,x , Λia,y , Λixy,a , Ψi,j ) does not appear in η then η (σ3 (X i , Y i , Λia,x , Λia,y , Λixy,a , Ψi,j ), Σ2 ) ∈ T, η (σ3 (X i , Y i , Λia,x , Λia,y , Λixy,a , Ψi,j ), Π2 ) ∈ T, and the Σ2 -extension of η is to the right of the Π2 -extension. 3. If σ4 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j a ) does not appear in η then η (σ4 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j a ), Σ1 ) ∈ T, η (σ4 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j a ), Π1 ) ∈ T, and the Π1 -extension of η is to the right of the Σ1 -extension. 4. If σ5 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j a ) does not appear in η then η (σ5 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j a ), Σ1 ) ∈ T, η (σ5 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j a ), Π1 ) ∈ T, and the Π1 -extension of η is to the right of the Σ1 -extension. 5. If σ6 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j by ) does not appear in η then η (σ6 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j by ), Σ2 ) ∈ T, η (σ6 (X i , Y i , Λia,x , Λia,y , Λixy,a , Θi,j by ), Π2 ) ∈ T, and the Σ2 -extension of η is to the right of the Π2 -extension.

16


Case 2. If there is no such pair (i, j) as above then we set i = imax + 1 and determine the immediate successor of η in T as follows. 1. If σ0 (Θi ) does not appear in η then η (σ0 (Θi ), Σ1 ) ∈ T, η (σ0 (Θi ), Π1 ) ∈ T, and the Σ1 -extension of η is to the left of the Π1 -extension. 2. Otherwise, η (σ1 (X, Y, Λa,x , Λa,y , Λxy,a ), Σ2 ) ∈ T, η (σ1 (X, Y, Λa,x , Λa,y , Λxy,a ), Π2 ) ∈ T, and the Σ2 -extension of η is to the right of the Π2 -extension. 3.2.1. η-configurations Notice that if η ∈ T then there is a unique strategy σ such that σ appears as the first component in the last element of the immediate successors of η. Definition 3.3 Suppose that η is an element of T . 1. Let j j j , Λia,y , Λixy,a ) : j < 1 } {σ1 (X ij , Y ij , Λia,x

be the sequence of σ1 -strategies σ in effect at η. Then an ηconfiguration for a1 is a finite initial segment A and the sets associated with these strategies such that for each j < 1 , there is a ij ij ij , Λa,y , Λxy,a )-configuration for a1 within this initial σ1 (X ij , Y ij , Λa,x segment. 2. We say that an η-configuration is certified if in addition to the above, for every σ6 -strategy τk which is in effect at η, the computation setting Θby (BY, a) = 0 has not changed since the stage during which τk enumerated the configuration for a as certified. For example, if η has only one σ1 -strategy σ1 (X, Y, Λa,x , Λa,y , Λxy,a ) with a Π2 -outcome, then an η-configuration for a1 is the same as a σ1 (X, Y, Λa,x , Λa,y , Λxy,a )-configuration for a1 , as described in Figure 1. With n such strategies, an η-configuration is described by n copies of Figure 1, one for each strategy and involving the sets associated with that strategy, sharing a common value for a1 .


17

The ij th component of an η-configuration for a1 is the initial segment ij ij , Λa,y , of A, B ij , C ij , Dij , and E ij which makes up the σ1 (X ij , Y ij , Λa,x ij Λxy,a )-configuration for a1 . 3.3. The construction We organize our construction by stages s, where s is greater than or equal to 1. Each stage s is divided into at most s many substages t, where t is also greater than or equal to 1. We proceed as follows during stage s. Let η[s, 0] equal the empty sequence. Given η[s, t−1] with t less than or equal to s, let σ be the strategy which appears in the first component of the immediate successors of η[s, t − 1]. We may assume that σ has been assigned a number a1 and an η[s, t − 1]configuration for a1 during an earlier stage and that no component of that configuration has changed since the stage during which it was assigned. (Otherwise, a strategy simply ends the stage since by its hypothesis, it will eventually be assigned such a number a1 by a σ2 - or σ5 -strategy as described below.) We follow the instructions of σ, which depend on its type as described below. At the end of its action, either σ ends stage s and we go to stage s + 1 with t = 0 or σ determines a value for η[s, t]. In the second case, if t is less than s then we continue with substage t + 1 of stage s. We adapt the pure strategies described in the previous section to work within the full construction as follows. 3.3.1. Adding an A-diagonalization strategy σ0 Suppose that σ is a diagonalization strategy σ0 (Θi ) to ensure that Θi = A. If Θi (a1 )[s] is not equal to 0 then we restrain any number from entering any set involved in our η-configuration for a1 . We let η[s, t] be η[s, t − 1] (σ, Π1 ). If Θi (a1 )[s] is equal to 0 and a1 is not an element of A[s], then we enumerate a1 into A and end stage s. If Θi (a1 )[s] is equal to 0 and we have already enumerated a1 into A, then we let η[s, t] be η[s, t − 1] (σ, Σ1 ). 3.3.2. Adding a σ1 (X, Y, Λa,x , Λa,y , Λxy,a ) Suppose that σ is σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a). We alter the pure σ1 strategy (described in the previous section) in the following way.

18


First, we measure σ-expansions in terms of stages when η[s, t − 1] is active in the construction. Second, while waiting for a σ-expansionary stage, we preserve the η[s, t − 1]-configuration for a1 . If s is not σ-expansionary in the above sense then let η[s, t] be η[s, t − 1] (σ, Σ2 ). Otherwise, we enumerate the pair (a1 , a2 ) as in the pure σ1 -strategy, we let η[s, t] be η[s, t − 1] (σ, Π2 ), and we cancel all strategies on nodes to the right of η[s, t]. 3.3.3. Adding a σ2 (X, Y, Λa,x , Λa,y , Λxy,a ) Suppose that σ is σ2 (X i , Y i , Λia,x , Λia,y , Λixy,a). By the definition of T and the previous paragraph, we may assume that the last strategy mentioned in η[s, t − 1] is of the form σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a ) and that s is expansionary for that strategy (i.e., the sequence η[s, t − 1] ends with the pair (σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a ), Π2 )). We let η[s, t] be η[s, t − 1](σ, Π1 ), and we alter the pure σ2 -strategy in the following way. First, we may need to change BC to record a change in X. If during the previous stage s during which σ2 acted, some strategy associated with an extension of η enumerated a number a into A, then let µ be the node in T associated with that strategy. There are two cases to consider. In the first case, there is no strategy σ6 (X i , Y i , Λia,x , Λia,y , Λixy,a, Θby ) above µ with B i = B and C i = C which certified a. In this case, we record the change in X by changing B accordingly. Otherwise, if Y did not change below λixy,a (a) between stage s and the current stage, then X must have changed there. This allows us to record all changes in X by enumerating c into C, where c is the number depicted in Figure 1, and we do so without changing B. Next, let (a1 , a2 ) be the pair just enumerated by σ1 (X i , Y i , Λia,x , i Λa,y , Λixy,a ). We extend the definitions of the functionals Γx,b , Γx,c , and Γbc,x so that we have a σ1 (X i , Y i , Λia,x , Λia,y , Λixy,a )-configuration for a1 and thus an η[s, t]-configuration for a1 . If there is a µ such that η[s, t] ⊆ µ, µ was active during a previous stage, the set of strategies in effect at µ is equal to the set of strategies in effect at η[s, t], and µ does not currently have an η[s, t]-configuration assigned to it, then we assign the configuration for a1 to the leftmost and shortest such µ (that is, the one of highest priority). We cancel all strategies to the right of µ and end stage s.


19

3.3.4. Adding a σ3 (X, Y, Λa,x , Λa,y , Λxy,a , Ψcd , Ψce ) Our only alteration to the pure σ3 -strategy is to make it measure expansionary stages taking into account only those stages during which it is active. For σ = σ3 (X, Y, Λa,x , Λa,y , Λxy,a , Ψcd , Ψce) we let η[s, t] be η[s, t − 1] (σ, Π2 ) if s is σ-expansionary and η[s, t − 1] (σ, Σ2 ) otherwise. 3.3.5. Adding a σ4 (X, Y, Λa,x , Λa,y , Λxy,a , Θa ) or a σ5 (X, Y, Λa,x , Λa,y , Λxy,a , Θa ) We use the pure σ4 - and σ5 -strategies without change. For a σ4 -strategy σ, we let η[s, t] be η[s, t − 1] (σ, Σ1 ) if the diagonalization witness n has been enumerated into D and η[s, t − 1] (σ, Σ2 ) otherwise. (For a σ5 -strategy σ, D is replaced by E.) 3.3.6. Adding a σ6 (X, Y, Λa,x , Λa,y , Λxy,a , Θby ) We do not alter the first phase of the pure σ6 -strategy. We start with a number a for which we have a certified η[s, t − 1]-configuration. We restrain a from entering A and preserve its configuration. We wait for a stage s such that (Θby (BY, a) = 0)[s]. If the current s is not such a stage then, for σ = σ6 (X, Y, Λa,x , Λa,y , Λxy,a , Θby ) we set η[s, t] = η[s, t − 1] (σ, Σ2 ). Otherwise, we set η[s, t] = η[s, t − 1] (σ, Π2 ). If there is an a such that we wait forever for an s such that (Θby (BY, a) = 0)[s], then Θby = A and the requirement is satisfied. Otherwise, according to the pure σ6 -strategy, we should restrict the enumeration of numbers into A to those which appear in the stream it generates. We must describe how the numbers in that stream are distributed to the strategies associated with nodes extending η[s, t] = η[s, t − 1] (σ, Π2 ). For this, we make the same alteration which we made for the σ2 -strategies. We can assume that σ has been assigned a certified η[s, t − 1]configuration for a number a1 . When σ finds a computation setting Θby (BY, a1 ) = 0 and for which there is are Λa,y (A) computations agreeing with Y below θby (BY, a1 ), then we say that these computations certify the η[s, t − 1]-configuration for a1 with respect to σ. Thus, a1 now has a certified η[s, t]-configuration.

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If there is a µ such that η[s, t] ⊆ µ, µ was active during a previous stage, the set of strategies in effect at µ is equal to the set of strategies in effect at η, and µ does not currently have an η[s, t]-configuration assigned to it, then we assign the configuration for a1 to the leftmost and shortest such µ (that is, the one of highest priority). We cancel all strategies η to the right of µ and end stage s. If there is no such µ then we cancel all strategies to the right of η[s, t]. 3.4. Analyzing the construction Let η ∞ be the path through T such that 1. for infinitely stages s and infinitely many substages t, η[s, t] is a subsequence of η ∞ , and 2. for at most finitely many stages s and substages t, η[s, t] is to the left of η ∞ . Following convention, we say that η ∞ is the true path of the construction. Lemma 3.4 The true path η ∞ is an infinite path in T . Proof. Suppose that η is a finite initial segment of η ∞ . We will argue that there is a proper extension of η which is also contained in η ∞ . Note that T is a finite branching tree. Consequently, if there are infinitely many s during which η acts and does not end the stage, then there is a leftmost proper extension which acts infinitely often and the claim is proven. There are four cases in which η peremptorily ends a stage during which it acts. Firstly η might end with a strategy which ends the stage since it is not assigned a number, which can happen at most finitely often in a row by the strategy’s assumption on outcomes of strategies above it. Next, η might end with a strategy for making Θ = A, in which case this strategy can end the stage at most once without being initialized. Otherwise, either η ends with a σ2 -strategy, or it ends with a σ6 -strategy, and, in both cases, η allocates an η-certified configuration to a strategy below it. But, if µ is eligible to be assigned a configuration by η then µ must have been active during an earlier stage of the construction. If η were to end all but finitely many of the stages during which it finds a new certified η-configuration, then there can only be finitely many such µ’s. Eventually, every such µ will have a certified η-configuration assigned to it. But then the next time that


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η finds a new certified η-configuration it will not end the stage and some proper extension of η will be active. Lemma 3.5 For each finite η ⊂ η ∞ , the following conditions hold. 1. We cancel η during the last stage sη during which there is a t such that η[sη , t] is to the left of η. 2. We let η act infinitely often. 3. During every stage greater than or equal to sη , we respect all of the constraints imposed by η during any earlier stage. Proof. Routine.

Lemma 3.6 Our construction satisfies all of the requirements of Section 2.1. Proof. This follows as in the analysis of the individual strategies in Section 2.2. Theorem 2.1 follows directly from Lemma 3.6. References Ambos-Spies, K. (1984). On pairs of recursively enumerable degrees. Trans. Amer. Math. Soc. 283 (2), 507–531. MR MR737882 (85d:03083). Ambos-Spies, K. (1985). Generators of the recursively enumerable degrees. In Recursion theory week (Oberwolfach, 1984), Volume 1141 of Lecture Notes in Math., pp. 1–28. Berlin: Springer. MR 87i:03081. Fejer, P. A. (1982). Branching degrees above low degrees. Trans. Amer. Math. Soc. 273 (1), 157–180. MR 84a:03044. Fejer, P. A. (1983). The density of the nonbranching degrees. Ann. Pure Appl. Logic 24 (2), 113–130. MR MR713296 (85g:03062). Jockusch, Jr., C. G. (1985). Genericity for recursively enumerable sets. In Recursion theory week (Oberwolfach, 1984), Volume 1141 of Lecture Notes in Math., pp. 203–232. Berlin: Springer. MR MR820782 (87f:03117). Lachlan, A. H. (1966). Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. (3) 16, 537–569. MR MR0204282 (34 #4126). Sacks, G. E. (1963). Degrees of unsolvability. Princeton, N.J.: Princeton University Press. MR MR0186554 (32 #4013).

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Slaman, T. A. (1991). The density of infima in the recursively enumerable degrees. Ann. Pure Appl. Logic 52 (1-2), 155–179. International Symposium on Mathematical Logic and its Applications (Nagoya, 1988). MR 92e:03061. Yates, C. E. M. (1966). A minimal pair of recursively enumerable degrees. J. Symbolic Logic 31, 159–168. MR MR0205851 (34 #5677).

CODING INTO H(ω2), TOGETHER (OR NOT) WITH FORCING AXIOMS. A SURVEY David Asper´ o Instituci´ o Catalana de Recerca i Estudis Avan¸cats (ICREA) and Departament de L` ogica, Hist` oria i Filosofia de la Ciència Universitat de Barcelona Baldiri Reixac, s/n, 08028 Barcelona, Spain E-mail: [email protected] URL: http://www.icrea.es/pag.asp?id=David.Aspero

This paper is mainly a survey of recent results concerning the possibility of building forcing extensions in which there is a simple definition, over the structure hH(ω2 ), ∈i and without parameters, of a prescribed member of H(ω2 ) or of a well-order of H(ω2 ). Some of these results are in conjunction with strong forcing axioms like P F A++ or M M , some are not. I also observe (Corollary 4.4) that the existence of certain objects of size ℵ1 follows outright from the existence of large cardinals. This observation is motivated by an (unsuccessful) attempt to extend a P F A++ result to one mentioning M M ++ .

1. Main starting questions and some pieces of notation The work presented here deals mostly with the problem of finding optimal definitions of well-orders of the reals and other objects. More precisely, it addresses the following two questions.1 Question 1: Suppose A is a subset of ω1 . Suppose we are given the task of going over to a nice set-forcing extension2 in which A admits a simple definition Φ(x), without parameters, over the structure hH(ω2 ), ∈i or over 1 As

quoted from [As2]. if nice is to be interpreted as preserving stationary subsets of ω1 in the ground model and A is a stationary and co-stationary subset of ω1 , then A will remain stationary and co-stationary in the extension. 2 So,

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some natural (definable) extension of this structure, like hH(ω2 ), ∈, N Sω1 i. What is the lowest degree of logical complexity that can be attributed to a definition Φ(x) for which we can perform the above task? Question 2: What is the lowest degree of logical complexity of formulas for which there is a formula Φ(x, y) (again without parameters) with that complexity and with the property that we can go over to a set-forcing extension in which the set of real numbers admits a well–order defined by Φ(x, y) (again over the structure hH(ω2 ), ∈i or over some natural extension of it)? For some background on these problems the reader is referred to [As2]. Logical complexity – for formulas of a language extending the language of set theory – will be measured in this paper by the familiar Levy hierarchy S n 0, a formula is Σn (respectively, Πn ) if it is of the form (∃x)ϕ for a Πn−1 formula ϕ (respectively, if it is of the form (∀x)ϕ for a Σn−1 formula ϕ). Note that, in any model M of ZF without the Power Set Axiom, if P is a definable class in M and ϕ(x0 , . . . xk ) is a formula in the language of the structure hM, ∈, P i, then S there is some formula ψ(x0 , . . . xk ) ∈ n |T C(P)|, every countable N 4 H(θ) containing P and every p ∈ P ∩ N there is some ˇ q ∈ P extending p such that q is (N, P)–generic, i.e. such that q P τ ∈ N whenever τ ∈ N is a P–name for an ordinal. Also, P is semiproper in case for every θ, N and p as above there is a condition q extending p such that 3 In other words, if all its quantifiers occur in a subformula of the form (∀x)(x ∈ y → ϕ) or (∃x)(x ∈ y ∧ ϕ). 4 That is, hM, ∈, P i |= (∀x , . . . x ) (ϕ(x , . . . x ) ↔ ψ(x , . . . x )). 0 0 0 k k k

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ˇ for every name τ ∈ N q is (N, P)–semigeneric, i.e. such that q P τ ∈ N for an ordinal in ω1V . Every proper partial order is semiproper and, if P is semiproper, then every stationary subset of ω1 remains stationary after forcing with P. L(R) is the ⊆–minimal transitive inner model of ZF containing all reals and all ordinals. Some arguments in Section 4, and in the proofs of Theorems 2.8 and 2.9 in Section 2, involve Pmax forcing. Pmax is a poset belonging to L(R) and definable in L(R) (without parameters). If x† exists for every real x,5 then Pmax is a homogeneous forcing and is σ–closed (in V and in L(R)). In particular, forcing with it over L(R) does not add new reals. The standard reference for Pmax forcing is [W]. It will be convenient to fix a notion of incompatibility, for pairs of formulas, which is absolute with respect to sufficiently arbitrary models of set theory. We will say that two L–formulas Φ0 (x) and Φ1 (x) are ZF C– provably incompatible if ZF C proves that for every uncountable regular cardinal κ and every x ∈ H(κ+ ), hH(κ+ ), ∈, N Sκ i |= ¬(Φ0 (x) ∧ Φ1 (x)). Also, for an L–formula in two free variables Φ(x, y), we will say that Φ(x, y) is ZF C–provably antisymmetric if ZF C proves that for every uncountable regular cardinal κ and every x, y ∈ H(κ+ ), x 6= y, hH(κ+ ), ∈, N Sκ i |= ¬(Φ(x, y) ∧ Φ(y, x)).6 Acknowledgment. Many of the results in this paper were presented in two talks that I gave in Singapore in July 2005. These talks were part of the IMS workshop “Computational Prospects of Infinity”. I thank the members of the Organizing Committee for inviting me. 2. Results not mentioning forcing axioms The following theorems are proved in [As4]. Theorem 2.1 ([As4]) There are Σ3 L–formulas Φ0 (x) and Φ1 (x) and Π3 L–formulas Ψ0 (x) and Ψ1 (x) with the following two properties. (1) (Φ0 (x), Φ1 (x)) and (Ψ0 (x), Ψ1 (x)) are two pairs of ZF C–provably incompatible formulas over the structure hH(ω2 ), ∈, N Sω1 i. 5 Which follows from all large cardinal assumptions used in the arguments alluded to here. 6 The notions of provably incompatible pairs of formulas over hH(ω ), ∈, N S ω1 i and 2 of provably incompatible formulas in the language of set theory (over hH(ω 2 ), ∈i) are of course defined in the natural way. And the same goes for the corresponding notions of provably antisymmetric formulas.

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(2) Given any A ⊆ ω1 there is a proper poset forcing that (a) A is defined, over hH(ω2 ), ∈, N Sω1 i, by Φ0 (x) and by Ψ0 (x), and (b) ω1 \A is defined, over hH(ω2 ), ∈, N Sω1 i, by Φ1 (x) and by Ψ1 (x). Theorem 2.2 ([As4]) There is a Σ3 L–formula Φ(x, y) and a Π3 L– formula Ψ(x, y) with the following two properties. (1) Φ(x, y) and Ψ(x, y) are ZF C–provably antisymmetric formulas over the structure hH(ω2 ), ∈, N Sω1 i. (2) If there is an inaccessible cardinal, then there is a proper poset P forcing the existence of a well–order ≤ of H(ω2 ) of order type ω2 such that ≤ is defined, over hH(ω2 ), ∈, N Sω1 i, both by Φ(x, y) and by Ψ(x, y). In fact, Theorem 2.1 can be easily derived7 from the following result. Theorem 2.3 ([As4]) There is a Σ2 L–formula Φ(x) such that for every uncountable regular cardinal κ and every A ⊆ κ there is a poset P with the following properties. (1) P is κ–distributive,8 proper, and preserves κ. Also, if 2µ = µ+ whenever µ is an infinite cardinal with µ+ < κ, then P preserves all stationary subsets of κ. Finally, if 2 ω1 , then it satisfies (1) from Theorem 2.3. (b) P forces that {(x, y) ∈ H(κ+ ) × H(κ+ ) : hH(κ+ ), ∈, N Sκ i |= Φ(x, y)} is equal to {(x, y) ∈ H(κ+ ) × H(κ+ ) : hH(κ+ ), ∈, N Sκ i |= Ψ(x, y)} and is a well–order of H(κ+ ) of order type κ+ . Note that none of Theorems 2.3 and 2.4 can hold for κ = ω: By [Mar-St], Projective Determinacy holds if there are infinitely many Woodin cardinals. In particular, under this large cardinal assumption there can be no well– order of the reals definable over hH(ω1 ), ∈i (even allowing parameters). And it can be seen that if δ < κ are such that δ is a limit of infinitely many Woodin cardinals and κ is a measurable cardinal, then given any poset P of size less than δ, any P–generic filter G over V and any real r ∈ V [G]\V , r is not definable over hH(ω1 )V [G] , ∈i by any formula with a real number in V as parameter.9 Given a regular cardinal κ ≥ ω1 , the proofs of Theorems 2.3 and 2.4 involve the manipulation, by forcing, of certain weak club–guessing properties for club–sequences defined on stationary subsets of κ, in such a way that the Σ2 theory of hH(κ+ ), ∈, N Sκ i with ordinals in κ as parameters codes any prescribed subset of κ. Given an ordinal γ, Lim(γ) denotes the set of nonzero limit ordinals in γ. A club–sequence will be a sequence of the form α = hαδ : δ ∈ Lim(γ)i – for some ordinal γ – such that each αδ is a subset of δ. The set S of δ ∈ Lim(γ) such that αδ is a club of δ is called the domain of α. It will also be denoted by dom(α). We may say that α is defined on S. If α is a club–sequence and γ is such that sup(dom(α)) = γ, then we may say that α is a club–sequence on γ. If τ is an ordinal such that the order type of αδ 9 This follows from a result of Woodin to the effect that the theory of L(R) with real numbers as parameters cannot be changed by forcing with P whenever P is a poset with |P| < δ and δ < κ are as above (see [L], Theorem 3.1.12 for a proof).

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is τ for each δ ∈ dom(α), then we say that the height of α is τ . ht(α) will denote the height of α (if it exists). As in [A-Sh] (for ladder systems, that is, for club–sequences of height ω), if α is a club–sequence and δ ∈ dom(α), then αδ will denote α(δ) (and similarly with other Greek letters). We will say that α is a coherent club–sequence if there is a club–sequence β with dom(α) ⊆ dom(β) and β dom(α) = α dom(α),10 and such that β is coherent in the usual sense, that is, such that for every δ ∈ dom(β) and every limit point γ of βδ , γ ∈ dom(β) and βδ ∩ γ = βγ . The concepts in this paragraph are defined in [A-Sh] for ladder systems.11 Let α be a club–sequence on an ordinal γ of uncountable cofinality. We say that α is guessing in case for every club C ⊆ γ there is some δ ∈ C ∩ dom(α) such that αδ \C is bounded in δ. Furthermore, we say that α is strongly guessing if for every club C ⊆ γ there is a club D ⊆ γ such that αδ \C is bounded in δ for every δ ∈ D ∩ dom(α).12 α is avoidable if there is a club C ⊆ γ such that αδ ∩ C is bounded in δ for each δ ∈ dom(α) ∩ C. Given two club–sequences α and β on the same ordinal γ, γ of uncountable cofinality, β is disjoint from α if βδ ∩ αδ = ∅ for every δ ∈ dom(α) ∩ dom(β). Given a strongly guessing club–sequence α on an ordinal γ and a set X ⊆ γ including dom(α), α is said to be maximal for X in case every ladder system defined on X and disjoint from α is avoidable. The proofs of Theorems 2.3 and 2.4 make use of a certain weak club– guessing property for club–sequences,13 which is best defined after introducing the following strong version of intersection of two sets of ordinals: Given two sets of ordinals, X and Y , X∩∗ Y is defined as the set of δ ∈ X∩Y such that δ is not a limit point of X. Now, given a club–sequence α on an ordinal γ of uncountable cofinality, we will say that α is type-guessing in case for every club C ⊆ γ there is some δ ∈ C ∩ dom(α) with ot(αδ ∩∗ C) as high as possible, that is, with ot(αδ ∩∗ C) = ot(αδ ). We will say that α is strongly type-guessing in case for every club C ⊆ γ there is a club D ⊆ γ such that ot(αδ ∩∗ C) = ot(αδ ) for every δ ∈ D ∩ dom(α). Also, for a set X of ordinals and an ordinal δ, the Cantor–Bendixson rank of δ with respect to X, rnkX (δ), is defined by specifying that rnkX (δ) = 0 if and only if δ is not a limit point of X, that rnkX (δ) ≥ 1 10 Where, given a club–sequence α and a set X, the restriction of α to X, to be denoted by α X, is that club–sequence which is equal to α on X and is ∅ elsewhere. 11 They will occur in Definition 2.3. 12 Note that a strongly guessing club–sequence is guessing if and only if its domain is stationary. 13 Defined in [As4].

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if and only if δ is a limit point of X and, for each ordinal η ≥ 1, that rnkX (δ) > η if and only if δ is a limit ordinal and there is a sequence (δξ )ξ r(α)[s] and at t, α acted to confirm d(α)[t] (case 2b(i) below). In this case we let α_ ∞ be accessible. [We believe totality as the “length of confirmation” has increased. We are not worried about any of the markers entering C because we already protected computations by removing dangerous markers. We now let weaker nodes act.] (2) Otherwise. There are two cases. (a) Ψi (C)(d(α)[s]) ↑. In this case there is nothing we can do on behalf of the requirement. (i) If γ(m) > r(α_ f )[s] then we let α_ f be accessible. (ii) If γ(m) 6 r(α_ f )[s] then α imposes finite restraint by enumerating γ(m) into C. (b) Ψi (C)(d(α)[s]) ↓. In this case we want to make progress. (i) If γ(m) > d(α)[s] then we confirm d(α): we enumerate all markers greater than d(α)[s] into A and halt the stage. Note that d(α)[s] itself (if it is a marker), is not enumerated. As described later, we now pick new large markers; we raise the value of d(α) and redefine d(α)[s + 1] to be the least marker among the new ones just chosen; we set it to be mobile. [At the next stage at which α is accessible, we follow case 1.] (ii) However, if γ(m) 6 d(α)[s] then there is no point in confirming d(α)[s] as smaller numbers will have to enter a set at this stage. We let α_ ∞ be accessible and do nothing else.

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At the end of the stage we initialise nodes that lie to the right of the accessible nodes (and all nodes of length > s), and reassign a large value (greater than s) to any marker γ(k) (k 6 s) which is not defined (γ(k) was just enumerated into a set or k = s). [Note we do not initialise extensions of the final accessible node.] Let γ(k) be the least marker enumerated at stage s. If γ(k) = d(α)[s] for a Q-node α, define d(α)[s + 1] to point to a new marker. On the other hand, if γ(k) < d(α)[s], then d(α) will no longer point to a marker value and hence has become stationary until α acts to confirm it. 3.3. Verification Claim 3.1. Suppose that α is accessible at stage s, and that m enters 00 at s. Then r(α) 6 γ(m) [s]. Proof. This is by induction on α. For the root, we know that r(hi)[s] = 0 for all s. Suppose that α is accessible at s. If α is a P -node then r(α) = r(β) [s] where β is α’s only child. If α is a Q- or N -node, then r(α) = r(α_ ∞) [s]. If γ(m) < r(α_ f ) [s] then α_ f cannot be accessible at s: instead of going to α_ f , α enumerates γ(m) into C and halts the stage. Corollary 3.2. If a node β is initialised at some stage r, then after r, β never enumerates a number smaller than r into any set. Proof. There are three kinds of enumerations: enumeration of γ(m) (where m enters 00 ); enumeration of a follower into B (by a P -node); and enumeration of markers into A by a Q-node that is confirming its test point. By claim 3.1, if at stage s > r, m enters 00 and β enumerates γ(m), then (as β is accessible) γ(m) > r(β)[s] > r. If β is a P -node, then after r it picks new followers, all greater than r. And if β is a Q-node that is confirming d(β), then by definition, d(β) > r(β) [s] (it is initially assigned to a larger value and can then only increase), and only markers greater than d(β) are enumerated. Lemma 3.3. For every m, the marker γ(m) is eventually fixed. Proof. Suppose that after some stage s0 , the markers γ(0), . . . , γ(m) do not change. Also suppose that after s0 , m + 1 does not enter 00 , so the only way γ(m + 1) can enter a set after stage s0 is when some Q-node confirms a smaller d(α)[s] (necessarily d(α)[s] > γ(m)), in which case γ(m + 1) will be enumerated into A.

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Let α be such a node. After confirming at stage s, a new value is set for γ(m+1), which is also the new value of d(α). At s, d(α) is set to be mobile; and it can never be reverted to be stationary (only γ(m) enumeration can do that). So after s, we always have d(α) > γ(m + 1) (even if they increase together). Therefore, after stage s0 , there is at most one stage s at which α’s action causes γ(m + 1) to be enumerated into A. Consider a Q-node of length > s0 , γ(m). Any test point it ever appoints is at least γ(m+1) and as in the previous paragraph, if it does appoint it to be γ(m + 1) it remains mobile until it is lifted to be greater than γ(m + 1). So such a node never enumerates any γ(m + 1) into A. This shows that after s0 , only finitely many nodes enumerate a γ(m + 1) into A. [An alternative construction could have been: do not allow nodes of length k to enumerate γ(k) (i.e., always set d(α) > γ(k)). Then you would need this current lemma to show finite injury on the true path.] Corollary 3.4. 00 6T A ⊕ C. Proof. Whenever m goes into 00 , γ(m) enters either A or C, and is redefined with a large value. Lemma 3.5. Suppose that α is a node on the tree that is accessible infinitely often and that is eventually never initialised. Then there is an immediate successor of α on the tree that is accessible infinitely often and is eventually never initialised. Since the root is always accessible and is never initialised, inductively applying the lemma shows the existence of an infinite true path. Proof. Let r0 = r(α)[ω] be the last stage at which α is initialised. Of course, everything depends on the requirement to which α is assigned. α is assigned to Pe . After stage r0 , α is assigned a follower x; that follower is never cancelled. After r0 , α acts at most once; after that, whenever α is accessible, so is its child. Also, the child is never initialised without α also being initialised. α is assigned to Ne . Suppose that there is a last α-expansionary stage r > r0 . After r, α_ f is never initialised. After r, α only enumerates a marker γ(m) into C if γ(m) 6 r, which is to say, finitely often. At other stages, whenever α is accessible, so is α_ f . If there are infinitely many expansionary stages, then α_ ∞ is accessible infinitely often; it is not initialised after r0 .

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α is assigned to Qi . If there are infinitely many stages at which α confirms its test point, then α_ ∞ is accessible infinitely often. Suppose that there is a last stage r1 > r0 at which α confirms d(α). Suppose that at the end of r1 , we set d(α) = γ(k). Let r2 > r1 be a stage after which the value of γ(k) is fixed. Then after r2 , the value of d(α) is also fixed (it may or may not have become stationary before r2 ). After r2 , case (2b(ii)) only holds if γ(m) 6 d(α), so finitely many times. So there is a stage r3 > r2 after which α_ ∞ is never accessible (so α_ f is not initialised after stage r3 ). Later, case (2a(ii)) applies only if γ(m) 6 r3 ; again, finitely many times. So after some stage r4 > r3 , whenever α is accessible, so is α_ f . Corollary 3.6. Every P -requirement is met, so B is not computable. Proof. Standard.

Lemma 3.7. Every Qi requirement is met. Hence C is array computable. Proof. Let α on the true path work for Qi . Suppose that Ψi (C) is total. There are infinitely many stages at which α confirms its test point. For otherwise, as argued in the proof above, there would be a final fixed value for d(α), and a stage after which case (2b(ii)) doesn’t hold; but this contradicts Ψi (C) converging permanently on that final value. Definition 3.8. We say that a number x < ω is confirmed at stage s if Ψi (C)(x) ↓ [s] with use smaller than the least marker greater than x. The point, of course, is that a confirmed computation can only be injured by a number not greater than the input. Suppose that α is last initialised at stage r0 . At the next stage r1 at which α is accessible, α sets d(α) = γ(k0 ) for some k0 . Let r2 > r1 be the least stage after which γ(k0 ) is always fixed. At the end of r2 we have x0 := d(α)[r2 + 1] 6 γ(k0 )[r2 + 1]. We in fact must have equality, because at a later stage α confirms its test point. We prove three claims that suggest a way to approximate Ψi (C). Claim 3.9. Let s > r2 be a stage at which α acts (to confirm its test point). Suppose that x ∈ [x0 , d(α)[s]] is confirmed at the end of s. Then it is confirmed at any later stage at which α acts.


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Proof. Suppose that at the end of stage s, γ(m − 1) < x 6 γ(m). We have γ(m) 6 d(α) [s + 1]. By assumption, ψi (C)(x) < γ(m) [s + 1]. Let t > s be the next stage at which α acts. If γ(m−1) is not enumerated between s and t, then the computation is preserved and so it is of course still confirmed. Otherwise, let γ(n) be the smallest marker enumerated between t and s, say at stage u. Then at the end of u we fix d(α) to be stationary and redefine a greater γ(n). At t, d(α) is confirmed; since there is no marker in (x, d(α)[t]], x is also confirmed. Claim 3.10. Let s > r2 be a stage at which α acts and let n ∈ [x0 , d(α)[s]] be a marker. Then n is confirmed at the end of s. Proof. If d(α)[s] = n, then n is confirmed at stage s. Otherwise, let t < s be the least stage at the end of which d(α) > n. Then d(α)[t] = n (n is already a marker at t) and α acts at t and confirms n. Claim 3.11. Let s > r2 be a stage at which α acts. Let x ∈ [x0 , d(α)[s]]. Suppose that Ψi (C)(x)[s] is injured at a later stage u by some y > x. Then at the end of the next stage t at which α acts, x is confirmed. Proof. Let γ(m − 1)[s] < x 6 γ(m) [s]. We must have γ(m) 6 d(α) [s]; otherwise, γ(m)[s] is redefined at s to be larger than ψi (C)(x) and then x is confirmed. Then γ(m) is confirmed at the end of s and so y = γ(m)[s]. Also at the end of s we have d(α) > y. It follows that at the end of u we have γ(m − 1) < x < d(α) < γ(m) (since d(α) is stationary). We thus see that at t, the confirmation of d(α) is also a confirmation of x. The following is an approximation for g = Ψi (C): at a stage s > r2 at which α confirms d(α), guess g(x) = Ψi (C)(x) for all x 6 d(α)[s]. Then the claims ensure that a guessed computation can be injured by some y > x at most once after s. It follows that this approximation is id + 1-c.e. Lemma 3.12. Every Ne requirement is met. As B >T 0, it follows that A and B form a minimal pair. Proof. Let α on the true path work for Ne . Suppose that Φe (A) = Φe (B) = g are total and equal. We first argue that `(e)[s] → ∞ (and so α_ ∞ is on the true path). This is because for all x, eventually there is a permanent Φe (A)(x) = Φe (B)(x) with total use u; for Q-nodes β such that β _ ∞ ⊆ α we know that d(β)[s] → ∞; and there are are finitely many such β (admissibility rocks).

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Assume that after stage r1 , α is never initialised and no P -node β ⊂ α enumerates a follower into B. The familiar argument of Lachlan’s that shows that g is computable follows through, provided that we can show that: (1) If s > r1 is an α-expansionary stage, then at s perhaps numbers enter A or numbers enter B but not both. (2) If s > r1 is an α-expansionary stage and x < `(e)[s], then numbers below ue (x)[s] do not enter either A or B until the next αexpansionary stage. The first is immediate; it holds at every stage of the construction. We verify the second point. Now suppose that s > r1 is α-expansionary and let x < `(e)[s]. • Nodes that lie to the right of α_ ∞ are initialised at stage s; they never enumerate anything smaller than s (which is in turn greater than ue (x)[s]) into any set. • Nodes that lie to the left of α are not accessible after r1 . • P -nodes β ⊂ α do not enumerate numbers into B after stage r1 . • Nodes extending α_ ∞ are not accessible between s and the next α-expansionary stage. Thus the only possible culprits are Q-nodes β ⊂ α that confirm d(β). If β _ f ⊆ α then β does not confirm d(β) after r1 ; if it does, then at the next stage at which it is accessible, α will be initialised. However, if β confirms d(β) at t > s, then it only enumerates numbers greater than d(β)[t] > d(β)[s]. But x is believable at s, which means ue (x)[s] 6 d(β)[s]. References [1] K. Ambos-Spies, C. Jockusch Jr., R.A. Shore, and R.I. Soare, An algebraic decomposition of recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Trans. Amer. Math. Soc., Vol. 281 (1984), 109-128. [2] M. Bickford and C. Mills, Lowness properties of r.e. sets, typewritten unpublished manuscript. [3] P. Cholak, R. Coles, R. Downey, and E. Herrmann, Automorphisms of the lattice of Π10 classes: Perfect thin classes and anr degrees, Trans. Amer. Math. Soc. Vol. 353 (2001), 4899-4924. [4] P. Cholak, M. Groszek, and T. Slaman, An almost deep degree, J. Symbolic Logic, Vol. 66(2) (2001), 881-901. [5] P. Cholak, R. Downey, and M. Stob, Automorphisms of the lattice of recursively enumerable sets: promptly simple sets, Trans. American Math. Society, 332 (1992), 555-570.


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[6] R. Downey and N. Greenberg, Domination and definiability II: The ω ω case, in preparation. [7] R. Downey, N. Greenberg and R. Weber, Domination and definability I: The ω case, in preparation. [8] R. Downey and D. Hirschfeldt, Algorithmic Randomness and Complexity, SpringerVerlag, to appear. [9] R. Downey, D. Hirschfeldt, A. Nies, and F. Stephan, Trivial reals, extended abstract in Computability and Complexity in Analysis Malaga, (Electronic Notes in Theoretical Computer Science, and proceedings, edited by Brattka, Schr¨ oder, Weihrauch, FernUniversit¨ at, 294-6/2002, 37-55),July, 2002. Final version appears in Proceedings of the 7th and 8th Asian Logic Conferences, (R. Downey, Ding Decheng, Tung Shi Ping, Qiu Yu Hui, Mariko Yasuugi, and Guohua Wu (eds)), World Scientific, Singapore (2003) 103-131. [10] R. Downey, D. Hirschfeldt, A. Nies, and S. Terwijn, Calibrating randomness, to appear, Bulletin Symbolic Logic. [11] R. Downey, C. Jockusch, and M. Stob, Array nonrecursive sets and multiple permitting arguments, in Recursion Theory Week (Ambos-Spies, Muller, Sacks, eds.) Lecture Notes in Mathematics 1432, Springer-Verlag, Heidelberg, 1990, 141-174. [12] R. Downey, C. Jockusch, and M. Stob, Array nonrecursive degrees and genericity, in Computability, Enumerability, Unsolvability (Cooper, Slaman, Wainer, eds.), London Mathematical Society Lecture Notes Series 224, Cambridge University Press (1996), 93-105. [13] L. Harrington and R. Soare, Post’s Program and incomplete recursively enumerable sets, Proc. Natl. Acad. of Sci. USA 88 (1991), 10242-10246. [14] S. Ishmukhametov, Weak recursive degrees and a problem of Spector, in Recursion Theory and Complexity, (ed. M. Arslanov and S. Lempp), de Gruyter, (Berlin, 1999), 81-88. [15] M. Kummer, Kolmogorov complexity and instance complexity of recursively enumerable sets, SIAM Journal of Computing, Vol. 25 (1996), 1123-1143. [16] A. Lachlan, A recursively enumerable degree which will not split over all lesser ones, Ann. Math. Logic, Vol. 9, (1975), 307-365. [17] W. Maass, Characterization of the recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets, Trans. Amer. Math. Soc., Vol. 279 (1983), 311-336. [18] Ng Keng Meng, F. Stephan and G. Wu, The degrees of weakly computable reals, in preparation. [19] A. Nies, Lowness properties and randomness, Advances in Mathematics Vol. 197 (2005), 274-305. [20] R.W. Robinson, Jump restricted interpolation in the recursively enumerable degrees, Annals of Math., Vol 93 (1971), 586-596. [21] B. Schaeffer, Dynamic notions of genericity and array noncomputability, Ann. Pure Appl. Logic, Vol. 95(1-3) (1998), 37-69. [22] R.I. Soare, Automorphisms of the lattice of recursively enumerable sets Part II: Low sets, Annals of of Math. Logic, Vol. 22 (1982), 69-107. [23] R.I. Soare, Recursively enumerable sets and degrees (Springer, Berlin, 1987). [24] S. Terwijn and D. Zambella, Algorithmic randomness and lowness, Journal of Symbolic Logic, vol. 66 (2001), 1199-1205.


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LOWNESS FOR COMPUTABLE MACHINES

Rod Downey School of Mathematics, Statistics and Computer Science, Victoria University P.O. Box 600, Wellington, New Zealand E-mail: [email protected] Noam Greenberg School of Mathematics, Statistics and Computer Science, Victoria University P.O. Box 600, Wellington, New Zealand E-mail: [email protected] Nenad Mihailović Mathematical Institute, University of Heidelberg Im Neuenheimer Feld 294, 69120 Heidelberg, Germany E-mail: [email protected] André Nies Department of Computer Science, University of Auckland Private Bag 92019, Auckland, New Zealand E-mail: [email protected] Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by Kjos-Hanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lowness for computable machines, which by results of Downey and Griffiths [3] is an analog of lowness for K. We show that the reals The first, second and fourth authors are partially supported by the New Zealand Marsden Fund for basic research. This work was carried out while Mihailović was visiting Victoria University and was also partially supported by the Marsden Fund and by a “Doktorandenstipendium” from the DAAD (German Academic Exchange Service). 79

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that are low for computable machines are exactly the computably traceable ones, and so this notion coincides with that of lowness for Schnorr randomness and for Schnorr tests.

1. Introduction A central set of results in the theory of algorithmic randomness were established by Nies and his co-authors. They prove the coincidence of a number of natural “anti-randomness” classes associated with prefix-free Kolmogorov complexity. Recall that A is called low for K if for all x, K A (x) ≥ K(x)−O(1),a A is called K-trivial if for all n, K(A n) ≤ K(n)+O(1), and A is called low for Martin-L¨ of randomness if the collection of reals MartinL¨ of random relative to A is the same as the collection of Martin-L¨ of random reals. We have the following. Theorem 1.1. (Nies, Hirschfeldt, [12, 13]) For every real A, the following are equivalent. (i) A is low for K. (ii) A is K-trivial. (iii) A is low for Martin-L¨ of randomness. The situation for other notions of randomness is less clear. In this paper we look at the situation for Schnorr randomness. Recall that a real A is said to be Schnorr random iff for all Schnorr tests {Un : n ∈ N}, A 6∈ ∩n Un , where a Schnorr test is a Martin L¨ of test such that µ(Un ) = 2−n for all n. −n (Of course 2 is a convenience. As Schnorr [16] observed, any uniformly computable sequence of reals with effective limit 0 would do.) The reader might note that there are two possible lowness notions associated with Schnorr randomness. A real A is low for Schnorr randomness if no Schnorr random real becomes non-Schnorr-random relative to A. But since there is no universal Schnorr test, we can also define the stronger (and more technical) notion of lowness for tests; a real A is low for Schnorr tests if for every A-Schnorr test {UnA : n ∈ N}, there is a Schnorr test {Vn : n ∈ N} such that ∩n UnA ⊆ ∩n Vn . a In this paper K will denote prefix-free Kolmogorov complexity and we will refer to members A = a0 a1 . . . of Cantor space as reals, with A n being the first n bits of A. We assume that the reader is familiar with the theory of algorithmic randomness. For details we refer to the monographs of Li and Vit´ anyi [10], of Downey and Hirschfeldt [5], and of Nies [14].

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Terwijn and Zambella [19] proved that there are reals that are low for Schnorr tests. In fact, they classified the collection of reals which are low for Schnorr tests. For any n, we let Dn denote the nth canonical finite set. Definition 1.2. (Terwijn and Zambella [19]) We say that a real A is computably traceable if there is a computable function h(x) such that for all functions g ≤T A, there is a computable collection of canonical finite sets Dr(x) with |Dr(x) | ≤ h(x) and such that g(x) ∈ Dr(x) . We remark that (as noticed by Terwijn and Zambella) if A is computably traceable then for the witnessing function h we can choose any computable, non-decreasing and unbounded function. Terwijn and Zambella proved the following attractive result. Theorem 1.3. (Terwijn and Zambella [19]) A is low for Schnorr tests iff A is computably traceable. We remark that while all K-trivials are ∆02 by a result of Chaitin [1], the computably traceable reals are all hyperimmune-free, and there are 2ℵ0 many of them. Subsequently, Kjos-Hanssen, Stephan, and Nies [7] proved that A is low for Schnorr randomness iff A is low for Schnorr tests. The reader might wonder about analogs of the other results for K. The other members of the coincidence involve K-triviality and lowness for K. What about the Schnorr situation? We want some analog of the characterization of Martin-L¨ of randomness in terms of prefix-free complexity. (R is Martin-L¨ of random iff for all n, K(R n) ≥ n − O(1).) Such a characterization was discovered by Downey and Griffiths [3]. They define a prefix-free Turing machine M to be computable if the domain of M has computable P measure, that is, {σ : M (σ)↓} 2−|σ| is a computable real. They then establish the following: Theorem 1.4. (Downey and Griffiths [3]) R is Schnorr random iff for all computable machines M , for all n, KM (R n) ≥ n − O(1).b

that since the range of M need not be all of 2 0. At any stage s, we have σj−1,s ≺ σj,s and λ(P | σj ) ≥ 2−2j−1 .

(3.1)

We let m0 = 0 and hence σ0 = ∅, so (3.1) is also true for j = 0. We construct a Kraft-Chaitin set L. Each Rj may enumerate into L in order to ensure K(Z k) ≤+ βk. The idea behind the construction is as follows. We are playing the following Rj strategy. We define a length kj where we intend to compress Z, and let mj be the use bound of Ψe , ge (kj ). We define σj of length mj in a σ way that, if x = Ψe j is defined then we compress it down to βkj , by putting an appropriate request into L. The opponent’s answer could be to remove σj from P. But in that case, the measure he spent for this removal exceeds what we spent for our request, so we can account ours against his. Of course, usually σj is much longer than x. So we will only compress x when the measure of oracle strings computing it is large, and use Lemma 3.2. For each j, m ∈ N and each stage t, let Gj,m,t = {σ : |σ| = m & λ(Pt | σ) ≥ 2−2j }.

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Informally, let us call a string σ of length mj good for Rj at stage t if σ σj−1,t and σ ∈ Gj,mj ,t . These are the only oracle strings Rj looks at. The reason to allow the conditional measure to drop from 2−2j+1 at σj−1 down to 2−2j is that we want a sufficiently large measure of them. Lemma 3.3. There is an effective sequence (uj ) of natural numbers such that the following holds. Whenever ρ is a string such that λ(Pt ∩ [ρ]) ≥ 2−(2j−1)−|ρ| , then for each m > |ρ|, λ(Gj,m,t ∩ [ρ]) ≥ 2−uj −|ρ| . Proof. For each measurable C, one obtains a martingale by letting MC (σ) = λ(C | σ) = λ(C ∩ [σ])2|σ| . Now let C = 2ω − P. Let d = 2j − 1. By hypothesis MC (ρ) ≤ 1 − 2−d, so we may apply the so-called Kolmogorov inequality, which bounds the measure of strings σ ρ where M can reach 1 − 2−(d+1) : λ({σ : |σ| = m & MC (σ) ≥ 1 − 2−(d+1) } | ρ) ≤

1 − 2−d . 1 − 2−(d+1)

Now it suffices to determine uj such that 1 − 2−uj ≥ (1 − 2−d )/(1 − 2−(d+1) ), since then λ(Gj,m,t | ρ) ≤ 2−uj . This proves the lemma. Rj compresses x when the measure of good strings computing it is large (4. below). If each good string σ computing x later becomes very bad, in the sense that the conditional measure λ(P | σ) dropped down to half, then we can carry out the accounting argument mentioned above and therefore choose a new x (5. below). The construction at stage s > 0 consists in letting the requirements Rj , for j = 0 . . . s, carry out one step of their strategy. Each time σj is newly defined, all the strategies Rl , l > j, are initialized. Suppose j > 0, that ρ = σj−1 is defined already, (3.1) holds for j − 1 and mj−1 = |ρ|. Let init(j) = j + 2 + the number of times Rj has been initialized. 1. Let kj be so large that βkj ≥ max(init(j), 2mj−1 ), and, where rj = uj + mj−1 + kj , α(rj + 2j + 1) ≤ βkj − init(j).

(3.2)

2. While ge (kj ) is undefined, let mj = mj−1 +1, let σj,s be the leftmost extension of σj−1 of length mj such that λ(Ps | σj,s ) ≥ 2−2j , and stay at 2. Else let mj = ge (kj ) and go to 3.

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3. Let Gs = Gj,mj ,s ∩ [ρ]. While there exists, let σj,s be the leftmost string σ ∈ Gs such that Ψσe kj ↑, and stay at 3. Else, for each string y of length kj , let Sy = {σ ρ : |σ| = mj & Ψσe,s = y}, and go to 4. 4. Let x be the leftmost string of length kj such that λ(Gs ∩Sx ) ≥ 2−rj where rj = uj + mj−1 + kj as above. Put a request hdβkj e, xi into L. Note that x exists since λGs ≥ 2−uj −mj−1 by Lemma 3.3, and (Gs ∩ Sy )|y|=kj is a partition of Gs into at most 2kj sets (because we passed 3). Of course, Gs ∩Sx may shrink later, in which case we have to try a new x. Eventually we will find the right one. 5. From now on, let σj,s be the leftmost string σ in Sx that satisfies (3.1), namely, λ(Ps | σ) ≥ 2−2j−1 . If there is no such σ, then we need to pick a new x, so go back to 4. (We had picked the wrong x. As indicated earlier, we will have to verify that this change is allowed, i.e., the contribution to L does no become too large. The reason is that the opponent had to add at least a measure 2−α(rj +2j+1) of new descriptions to the universal prefix free machine in order to make Gs ∩ Sx shrink sufficiently, and we will account the cost of our request against the measure of his descriptions.) Verification. Claim 3.1. For each j, σj = lims σj,s exists. This is trivial for j = 0. Suppose j > 0 and inductively the claim holds for j − 1. Once σj−1 is stable at stage s0 , we define a final value kj in 1. If mj = ge (kj ) is undefined then |σj | = |σj−1 | + 1 and σj can change at most once after s0 . Otherwise σj can change at most 2mj times till we reach 4. As remarked above, we always can choose some x in 4. If we cannot find σ in 5. any longer, then λ(Gs ∩ Sx ) < 2−rj , so we will discard this x and not pick it in 4. any more. Since there is an x such that λ(Gt ∩ Sx ) ≥ 2−rj for all t ≥ s0 , eventually we will stay at 5. Since all strings σj we try here have length mj , eventually σj stabilizes. This proves the claim. S S Note that |σj | > |σj−1 |, so A = j σj defines a set. Let As = j σj,s . It is easy to verify that for each x, the number of changes of As (x) is computably bounded in x, since the values mj,s are nondecreasing in s, and in 3. and 5., we only move to the right. Thus A ≤wtt ∅0 .

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Claim 3.2. L is Kraft-Chaitin set. By the definition of init(j), it suffices to verify that for each value v = init(j), the weight of the contributions of Rj to L is at most 2−v+1 . When a request hr, xi is enumerated by Rj at stage s, we distinguish two cases. Case 1. The strategy stays at 5. after s or Rj is initialized. Then this was the last contribution, and it weighs at most 2−v since we chose kj in a way that βkj ≥ v. Case 2. Otherwise, that is, the strategy gets back to 4. at a stage t > s. Then, for each σ ∈ Gs ∩ Sx , λ(Pt | σ) < 2−2j−1 (while, by the definition of Gs , we had λ(Ps | σ) ≥ 2−2j ). Now consider the clopen class C = Ps ∩ [Gs ∩ Sx ] − Pt . Since λ(Gs ∩ Sx ) ≥ 2−rj , X λC ≥ 2−|σ| (λ(Ps |σ) − λ(Pt |σ)) σ∈Gs ∩Sx

≥

X

2−|σ| (2−2j − 2−2j−1 ) =

σ∈Gs ∩Sx

X

2−|σ| 2−2j−1 ≥ 2−rj −2j−1 .

σ∈Gs ∩Sx

Clearly C ⊆ Ps and C ∩ Pt = ∅, so by Lemma 3.2 and (3.2), Ωt − Ωs ≥ 2−α(rj +2j+1) ≥ 2−βkj +v , hence the total contributions in Case 2 weigh at most 2−v Ω. Together, the contribution is at most 2−v+1 . Claim 3.3. If Z ≤wtt A then K(Z) ≤ α. It suffices to show that each requirement Rj is met. For then, if Z ≤wtt A either, the reduction has small use (see remarks at the beginning of the proof) or it is included in the list (Ψe ). In the latter case, meeting Rhe,bi for each b ensures K(Z) ≤ α. By the first claim, σj and kj reach a final value at some stage s0 . If ΨA e kj ↑ then Rj is met. Otherwise, the strategy forRj gets to 4. after s0 , and enumerates a request into L which ensures K(Z kj ) ≤+ βkj . This concludes the proof of Theorem 3.1. 4. Concluding Remarks It remains an open problem whether there exists a Turing lower cone of non-integral effective dimension (see [12]). This case appears to be much

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harder. It is, for instance, not even known whether there exists a set of non-integral dimension which does not compute a Martin-L¨ of random set. The best known result in this direction is that there exists a computable, non-decreasing, unbounded function f and a set A such that K(A n) ≥ f (n) and A does not compute a Martin-L¨ of random set. This has been independently proved by Reimann and Slaman [16] and KjosHanssen, Merkle, and Stephan [5]. References 1. H. G. Eggleston. The fractional dimension of a set defined by decimal properties. Quart. J. Math., Oxford Ser., 20:31–36, 1949. 2. P. G´ acs. Every sequence is reducible to a random one. Inform. and Control, 70(2-3):186–192, 1986. 3. D. R. Hirschfeldt and J. S. Miller. personal communication, 2005. 4. S. M. Kautz. Degrees of Random sequences. PhD thesis, Cornell University, 1991. 5. B. Kjos-Hanssen, W. Merkle, and F. Stephan. Kolmogorov complexity and the Recursion Theorem. Manuscript, submitted for publication, 2005. 6. A. Kuˇcera. Measure, Π01 -classes and complete extensions of PA. In Recursion theory week (Oberwolfach, 1984), volume 1141 of Lecture Notes in Math., pages 245–259. Springer, Berlin, 1985. 7. M. Li and P. Vit´ anyi. An introduction to Kolmogorov complexity and its applications. Graduate Texts in Computer Science. Springer-Verlag, New York, 1997. 8. J. H. Lutz. Dimension in complexity classes. In Proceedings of the Fifteenth Annual IEEE Conference on Computational Complexity, pp. 158–169. IEEE Computer Society, 2000. 9. J. H. Lutz. Gales and the constructive dimension of individual sequences. In Automata, languages and programming (Geneva, 2000), pp. 902–913. Springer, Berlin, 2000. 10. J. H. Lutz. The dimensions of individual strings and sequences. Inform. and Comput., 187(1):49–79, 2003. 11. E. Mayordomo. A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inform. Process. Lett., 84(1):1–3, 2002. 12. J. S. Miller and A. Nies. Randomness and computability: Open questions. Submitted for publication, 2005. 13. A. Nies, Computability and Randomness, to appear. 14. P. Odifreddi. Classical recursion theory, volume 125 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1989. 15. J. Reimann. Computability and fractal dimension. Doctoral dissertation, Universit¨ at Heidelberg, 2004. 16. J. Reimann and T. A. Slaman. Randomness, entropy, and reducibility. Manuscript, in preparation, 2005.

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17. B. Y. Ryabko. Noise-free coding of combinatorial sources, Hausdorff dimension and Kolmogorov complexity. Problemy Peredachi Informatsii, 22(3):16– 26, 1986. 18. G. E. Sacks. Degrees of unsolvability. Princeton University Press, Princeton, N.J., 1963. 19. R. I. Soare. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. 20. L. Staiger. Kolmogorov complexity and Hausdorff dimension. Inform. and Comput., 103(2):159–194, 1993. 21. L. Staiger. Constructive dimension equals Kolmogorov complexity. Information Process. Lett., 93:149–153, 2005. 22. F. Stephan. Hausdorff dimension and weak truth-trable reducibility. Submitted for publication, 2005. 23. K. Tadaki. A generalization of Chaitin’s halting probability Ω and halting self-similar sets. Hokkaido Math. J., 31(1):219–253, 2002. 24. S. A. Terwijn. Complexity and randomness. Course Notes, 2003. 25. A. K. Zvonkin and L. A. Levin. The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms. Uspehi Mat. Nauk, 25(6(156)):85–127, 1970.

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A MINIMAL rK-DEGREE

Alexander Raichev Department of Computer Science, University of Auckland Private Bag 92019, Auckland 1001, New Zealand E-mail: [email protected] Frank Stephan School of Computing and Department of Mathematics National University of Singapore, 3 Science Drive 2, Singapore 117543 E-mail: [email protected] We construct a minimal rK-degree, continuum many, in fact. We also show that every minimal sequence, that is, a sequence with minimal rK-degree, must have very low descriptional complexity, that every minimal sequence is rK-reducible to a random sequence and that there is a random sequence with no minimal sequence rK-reducible to it.

1. Introduction This article continues the study of relative randomness via rK-reducibility initiated in [3] and pursued in [9]. One of the most popular definitions of absolute algorithmic randomness states that an infinite binary sequence R is random if it is incompressible, that is, if ∃d ∀n . K(R[n]) ≥ n − d, where K(σ) is the prefix-free descriptional complexity of the string σ. Under this same paradigm of incompressibility, one can define relative algorithmic randomness as follows. An infinite binary sequence A is less random than an infinite binary sequence B if A is completely compressible given B, that is, if ∃d ∀n . K(A[n]|B[n]) < d, 261

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where K(σ|τ ) is the conditional prefix-free descriptional complexity of σ given τ . In this case, we write A ≤rK B for short and say “A is rK-reducible to B”.a The rK-reducibility, which is fairly easily seen to be reflexive and transitive, enjoys the following properties, all of which we will use throughout. Theorem 1.1 (Downey, Hirschfeldt and LaForte [3]). For infinite binary sequences A and B, A ≤rK B is equivalent to both of • ∃d ∀n . C(A[n] | B[n]) < d, • There exists a partial computable function ϕ such that ∃d ∀n ∃i < d . ϕ(i, B[n]) = A[n], and implies all three of • ∃d ∀n . K(A[n]) ≤ K(B[n]) + d, • ∃d ∀n . C(A[n]) ≤ C(B[n]) + d, • A ≤T B. Notice that the fifth bullet says that ≤rK really is a reducibility in the sense of computability theory as there is a way to compute A from B. Moreover, a sequence is computable iff it is rK-reducible to any other sequence, that is, the computable sequences are those of least relative randomness, as they should be. In what follows we answer a basic question: is there a sequence of minimal relative randomness, that is, a sequence with only the computable sequences strictly less random ( 0 the construction runs as follows, where each check is performed only when the markers involved have indices of length ≤ s; also, the computations involved are only up to stage s. • If there exist σ, i < 2 and e ≤ |σ| such that for all x ≤ |σ|, Φe (x) = mσi (x), then Cut(mσ , mσ(1−i) ). • If there exist σ, δ, and e ≤ |σ| such that Φemσ0 and Φemσ1 are compatible for all arguments up to |σ|, but Φemσ0δ and Φemσ1 are incompatible at some argument up to |σ|, then Cut(mσ0 , mσ0δ ) and Cut(mσ1 , mσ1 ). • If there exist σ, τ , υ and e ≤ |σ| such that σ ⊂ τ ⊂ υ and |mτ | ≤ Φe (|mσ |) < |mυ |, then Cut(mτ , mυ ). It is not difficult to check that each marker eventually settles and that, in the limit, T satisfies properties (1)-(3). A suitable path of T . Let A be a path of T of hyperimmune-free Turing degree.b Such a path exists by the Hyperimmune-Free Basis Theorem [5] since [T ] is a nonempty Π01 class (since T is a Π01 tree). We show that A has minimal rK-degree. By (1), A is incomputable. Let B ≤rK A be an incomputable set. We need to show that A ≤rK B. To this end, observe that B ≤T A and, in fact, B ≤tt A since A has hyperimmune-free Turing degree; see page 589 in Odifreddi’s book [8]. Let Φ be a computable functional (total on all oracles) that witnesses the truth-table reduction. We come now to the heart of the argument: building an rK-reduction from B to A. Let ? be the magic string of (2) for Φ and A. Given B[n] for n sufficiently large, run through the computable approximation (that thins) to T until a stage t is reached such that Tt (the stage t approximation of T ) has at most two extensions of ? of length n with extensions in Tt that map to B[n] under Φ. The key here is that such a stage is guaranteed to exist by Lemma 2.2 below. To find these extensions effectively from B[n] we use the fact that Φ is total on all oracles and has a computable use function. Output the (at most) two strings of length n found; one will be A[n]. Except for finitely many short lengths, this procedure describes an rK-reduction from B to A. Extending it to all lengths gives the final reduction. b That

is, for every total function f ≤T A, there exists a computable function g such that for all x, g(x) ≥ f (x). Put more concisely, every total function computable from A is majorized by a computable one.

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Lemma 2.2. Let ? be the magic string of (2) for Φ and A. For almost all lengths n and almost all stages t, Tt has at most two extensions of ? of length n with extensions in Tt that map to B[n] under Φ. Proof . Let ϕ be the computable use function for the tt-reduction Φ. Let f be the function defined for m ≥ | ? | by f (m) equals the first stage s such that for all strings ν ⊃ A[m] b(1 − A(m)) of length ϕ(s) on Ts , there exists x ≤ s such that Φν (x) ↓ 6= ΦA (x). (Notice that all ν extend ?.) For m < | ? |, define f (m) to be 0, say. It is unimportant. Note that f is total, for if not, then for all s there exists a string νs ⊃ A[m] b(1 − A(m)) of length ϕ(s) on Ts such that for all x ≤ s, Φνs (x) = ΦA (x). (Remember that Φ is total on all oracles.) Then the sequence Y defined by Y (n) = lim inf s νs (n) is a path of T different from A such that ΦY = ΦA . But this is a contradiction, because (2b) holds for X = A since B is incomputable. Also, f is A-computable by definition. Thus, since A has hyperimmune-free Turing degree, there is a computable function g majorizing f . Now, fix n bigger than the length of ?, the length that (3) takes effect for g and the length of the first splitting node of A on T . Let τ be the last splitting node of T on A[n] and let σ ⊂ τ be any other splitting node of T extending ?. Then by (3) we have for s = f (|σ|) that s ≤ g(|σ|) < |τ | ≤ n. So by stage s, every ν ∈ Ts extending A[|σ|] b(1 − A(|σ|)) = σ b(1 − A(|σ|)) will have some number x ≤ s with Φν (x) ↓ 6= ΦA (x) = B(x), so that ν cannot map to B[n] under Φ. Since σ was an arbitrary splitting node of T below the last splitting node of A[n], we see that only the strings extending the last splitting node of A[n] can map to B[n] under Φ. So the result holds. In fact, by a generalized Hyperimmune-Free Basis Theorem below, the tree of the proof of Theorem 2.1 has continuum many paths of hyperimmunefree Turing degree. Thus, since every rK-degree is countable, there are continuum many minimal rK-degrees. Theorem 2.3. Every nonempty Π01 class with no computable members has 2ℵ0 paths of hyperimmune-free Turing degree. Proof . By basic facts from the theory of Π01 classes, we can assume without loss of generality that our Π01 class is the set of paths through a binary tree T0 that is infinite, computable and has no computable paths. We modify

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slightly the proof of the Hyperimmune-Free Basis Theorem in [5] by way of an extra parameter sequence X. For each sequence X we construct (uniformly relative to X ⊕ ∅00 ) computable subtrees S1 ⊃ T1 ⊇ S2 ⊃ T2 ⊇ . . . of T0 such that their only common path Y has hyperimmune-free Turing degree. We then show that the map X 7→ Y is one-to-one. To this end, fix X and, starting from T0 , let Se and Te be defined recursively as follows. Let Ue,x be the computable tree {τ | Φτe,|τ | (x) ↑ }. (1) If for all x, Te ∩ Ue,x is finite, then let Se = Te . Otherwise, choose x least such that Ue,x is infinite and let Se = Te ∩ Ue,x . (2) Since Se is an infinite tree with no computable paths, it has at least two paths. Let σ be the length-lexicographic least node of Se such that σ0 and σ1 have paths in Se through them. (3) Let Te+1 = {τ ∈ Se | τ ⊆ σ bX(e) ∨ τ ⊃ σ bX(e)}. T By induction each [Te ] and [Se ] is nonempty, so that e [Te ] ∩ [Se ] is nonempty, being the intersection of a decreasing sequence of closed nonempty sets in the compact space {0, 1}∞. Choose (the unique) sequence T Y ∈ e [Te ] ∩ [Se ]. It will have hyperimmune-free Turing degree, for fix a natural e and consider the function ΦYe . If for every x, Te ∩ Ue,x is finite, then there is a total and computable function g majorizing ΦYe where g is chosen as g(x) = max{Φτe,|τ | (x) | τ ∈ Te ∧ |τ | = lx } and lx is the least value such that Φτe,|τ | (x) is defined for each τ ∈ Te of length lx . If there exists some x such that Te ∩ Ue,x is infinite, then Φτe,|τ | (x) is undefined for infinitely many τ ∈ Te and Se is the set of all these τ . Since all prefixes of Y are in Se , this means ΦYe (x) is undefined, so that ΦYe is not total. Also, the map X 7→ Y is one-to-one, for if two sequences X1 and X2 differ and e is the first place at which this happens, then the corresponding trees Se (X1 ) and Se (X2 ) are the same, but the intersection of Te+1 (X1 ) and Te+1 (X2 ) is finite since one contains the nodes above σ0 and the other the ones above σ1. Thus Y (X1 )(|σ|) 6= Y (X2 )(|σ|). 3. Three notes From now on let us call a sequence with minimal rK-degree a ‘minimal sequence’. As one might expect, minimal sequences have low initial segment complexity. Indeed, so low that they are close to being computable in the

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sense of Chaitin’s characterization (see [1]): a sequence X is computable iff ∃d ∀n . C(X[n]) ≤ C(n) + d. Proposition 3.1. If A is a minimal sequence, then for any computable unbounded increasing function g : N → N, ∃d ∀n . C(A[n]) ≤ C(n) + g(n) + d

and

∃d ∀n . K(A[n]) ≤ K(n) + g(n) + d. In particular, A cannot be random. For the proof, we need the notion of dilutions. Definition 3.2. For X ∈ {0, 1}∞ and f : N → N strictly increasing, the f -dilution of X is the sequence defined by ( X(m) if n = f (m) for some (unique) m Xf (n) = 0 else. Notice that for any sequence X and any strictly increasing computable function f , Xf ≤rK X and Xf ≡T X. Now we prove Proposition 3.1. Proof . Fix A and g as in the hypothesis. The idea is that since A is a minimal sequence, it is rK-reducible to every one of its computable dilutions. Picking a dilution appropriate to g will give the desired complexity bound. We prove the bound for K. The argument for C is identical. Define the function f : N → N recursively by f (0) = 0; f (x) = the least n such that n > f (x − 1) and g(n) ≥ 4x. Since g is unbounded and increasing, f is well-defined. Also, by construction f is computable, total and strictly increasing. Furthermore, for any given n, if x is the greatest number such that f (x) ≤ n, then g(n) ≥ 4x. Since A is minimal, A ≤rK Af via some [ϕ, e]. Now fix n and choose x greatest such that f (x) ≤ n. Observe that inserting zeros into A[x] in the appropriate computable places produces Af [n]. So to describe A[n], besides a few computable partial functions given ahead of time, one only needs the correct i < e such that ϕ(i, Af [n]) = A[n], the value n and A[x]. This information can be coded, up to a uniform constant, by a string of length K(n) + 2K(A[x]). The factor of 2 comes from concatenating strings in a prefix-free way. So, up to a uniform additive constant, for all n, K(A[n]) ≤ K(n) + 2K(A[x]) ≤ K(n) + 4x ≤ K(n) + g(n),

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as desired. Now fixing g as, say, g(n) = blg(n + 1)c, we see that A cannot be random. Using dilutions again, we also get the following. Proposition 3.3. Every minimal sequence is rK-reducible to a random sequence. Proof . Fix a minimal sequence A and choose a random sequence R ≥wtt A with the use being majorized by f (n) = 2n. This is possible since every sequence has such a random; see [4, 6] and also [7] for a more recent proof using martingales. Then R ≥rK Af ≥rK A, by the minimality of A, as desired. Do all sequences have randoms rK-above them? That question is still open and seemingly difficult. We end with one last note, a contrast to Proposition 3.3. Proposition 3.4. There is a random sequence with no minimal sequence rK-reducible to it. Proof . Let R be a random sequence of hyperimmune-free Turing degree. Such a sequence exists by the Hyperimmune-Free Basis Theorem applied to the complement of any member of a universal Martin-L¨ of test. Then R has no minimal sequence reducible to it. To see this, assume (toward a contradiction) there is some minimal sequence A such that A ≤rK R. Since R has hyperimmune-free Turing degree, so does A and A ≤tt R. Since A is incomputable and truth-table reducible to a random, A is Turing equivalent to some random S (see [2]). Since A has hyperimmune-free Turing degree, S ≤tt A via some computable function with computable use function f . Thus, disregarding floor functions and uniform constants for ease of reading, we have that for all n, n

≤ ≤ ≤ ≤ ≤

a contradiction.

K(S[n]) K(A[f (n)]) + K(n) K(f (n)) + lg n + K(n) 2K(n) + lg n 4 lg n,

(since S is random) (using the tt-reduction) (by Proposition 3.1) (since f is computable)

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Remark 3.5. Do maximal rK-degrees exist? That basic question is still open. Acknowledgements We would like to thank the Institute for Mathematical Sciences at the National University of Singapore for organizing the outstanding Computational Prospects of Infinity workshop which made this work possible. Also, thanks to Steffen Lempp and Sasha Rubin for their helpful comments. References 1. Gregory J. Chaitin. Information-theoretic characterizations of recursive infinite strings. Theoretical Computer Science, 2(1):45–48, 1976. 2. Osvald Demuth. Remarks on the structure of tt-degrees based on constructive measure theory. Commentationes Mathematicae Universitatis Carolinae, 29(2):233–247, 1988. 3. Rod G. Downey, Denis R. Hirschfeldt and Geoff LaForte. Randomness and reducibility. Journal of Computer and System Sciences, 68(1):96–114, 2004. 4. Péter G´ acs. Every sequence is reducible to a random one. Information and Control, 70(2-3):186–192, 1986. 5. Carl G. Jockusch, Jr. and Robert I. Soare. Π01 classes and degrees of theories. Transactions of the American Mathematical Society, 173:33–56, 1972. 6. Anton´ın Kuˇcera. Measure, Π01 -classes and complete extensions of PA. In Recursion Theory Week (Oberwolfach, 1984), volume 1141 of Lecture Notes in Mathematics, pages 245–259. Springer, Berlin, 1985. 7. Wolfgang Merkle and Nenad Mihailović. On the construction of effective random sets. The Journal of Symbolic Logic, 69(3):862–878, 2004. 8. Piergiorgio Odifreddi. Classical recursion theory, volume 125 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1989. 9. Alexander Raichev. Relative randomness and real closed fields. The Journal of Symbolic Logic, 70(1):319–330, 2005. 10. Robert I. Soare. Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. 11. Frank Stephan. On the structures inside truth-table degrees. The Journal of Symbolic Logic, 66(2):731–770, 2001.


DIAMONDS ON Pκ λ Masahiro Shioya Institute of Mathematics, University of Tsukuba Tsukuba, 305-8571, Japan E-mail: [email protected] Dedicated to Professor Katsuya Eda on the occasion of his 60th birthday

Assuming 2ω = 2 κ. We also construct a model in which 2ω < 2ω1 holds and Pω2 λ carries a diamond for every λ > ω2 .

1. Introduction Let κ be a regular uncountable cardinal. The combinatorial principle ♦κ was introduced by Jensen [9] to construct Suslin trees in the constructible universe L. In [8] Jech extended the notion to the context of Pκ λ: Suppose that λ is a cardinal ≥ κ and S is a stationary subset of Pκ λ. A diamond on S is a map of the form g : S → Pκ λ such that if A ⊂ λ, then {x ∈ S : g(x) = A ∩ x} is stationary. It is easy to see that ♦κ holds iff Pκ κ carries a diamond. In this paper we are concerned with diamonds on Pκ λ in the case κ < λ. Building on work of Shelah [14], Donder and Matet [3] proved Theorem 1. Pκ λ carries a diamond for every λ > 2 ω1 . Corollary 3. The club filter on Pω1 λ is not 2λ -saturated for any λ > ω1 . Partially supported by JSPS Grant-in-Aid for Scientific Research No. 16540094. The author wishes to thank Professor Shelah for sending a draft of his unpublished book in 2000. He also acknowledges the referee’s suggestions for improving the presentation. 271

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What about diamonds on Pκ λ in the case ω1 < κ < λ? In [4] Foreman and Magidor proved that the club filter on Pκ λ can be 2ω1 -saturated. In particular it is consistent that there is no diamond on Pκ λ. Their model is the forcing extension of L by Add(ω1 , λ++ )L if cf L λ ≥ κ, and by Add(ω1 , λ+++ )L otherwise. Here Add(ω1 , ρ) denotes the poset for adjoining ρ Cohen subsets of ω1 . Through an attempt to understand the proof of Theorem 2, we get Theorem 4. Assume 2ω = 2 κ. We prove Theorem 4 in §3. Our proof invokes Theorem 1 and presumably differs from Shelah’s proof of Theorem 2. Unfortunately we could follow his argument only in the case λ = ω2 . In §6 we present Shelah’s proof (as we understand it) in somewhat greater generality. Subsequently K¨ onig and Todorˇcević [11] found their own proof of Theorem 2, which works uniformly for every λ > ω1 . Is it consistent that Pκ λ carries a diamond in the case 2ω < 2 κ after forcing with Col(κ, ν) × Add(ω1 , ρ). Here Col(κ, ν) denotes the Levy collapse of ν to the successor of κ. In the model of Theorem 5, Pκ κ+ can be partitioned into (κ+ ) κ. Let f : [λ]

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