Proceedings of the Sixth Asian Logic Conference : Beijing, China, 20-24 May 1996

Proceedings of me

Sixth Asian Logic Conference

This page is intentionally left blank

Proceedings of die

Sixth Asian Logic Conference Beijing, China 20-24 May 1996

wmmwmmmmmmmmmmmm Editors

C T Chong National University of Singapore

QFeng National University of Singapore

DDing Nanjing University

Q Huang Academia Sinica

M Yasugi Kyoto Sang/o University

V f e World Scientific wb [

Singapore* Singapore • New Jersey London* London • Hong Kong

I SINGAPORE UNIVERSITY PRESS / NATIONAL UNIVERSITY OF SINGAPORE

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE and Singapore University Press (Pte) Ltd. Yusof Ishak House National University of Singapore 10 Kent Ridge Cresent Singapore 119260

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PROCEEDINGS OF THE SIXTH ASIAN LOGIC CONFERENCE Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. and Singapore University Press 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3432-5

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

Foreword

The Sixth Asian Logic Conference was held from 20 to 24 of May, 1996 in Beijing, China. There were one hundred and fifty participants from six countries. This volume collects written versions of invited as well as contributed talks presented at the conference. An article by M. Yasugi on the Asian Logic Conference, first appeared in Japanese, is included as it provides a glimpse into the history and development of the series of conferences. We wish to thank all those who help referee the submitted papers, working within a tight deadline. On behalf of the organizers we also wish to thank the Institute of Software, Academia Sinica, China and the Lee Foundation of Singapore for their generous financial support. Editors

31 October 1997

VI

LIST OF I N V I T E D S P E A K E R S

K.

AMBOS-SPIES

S. B.

K.

A.

Genericity and Randomness in Complexity The­ ory

S. Buss

Algebraic Proof Systems

COOPER

Oracles, Enumerations, and Relative Computabil­ ity

D. D I N G

On d-r.e. Degrees

Q.

Principles of Stationary Reflection Sets of Reals, Infinite Games and Core Model The­ ory

FENG

HAUSER

KECHRIS S. L E M P P

W. Li H.

ONO

A. PlLLAY G. E.

SACKS

M. YASUGI

Borel Equivalence Relations and Classifications of Countable Models Decidability and Undecidability in the Enumer­ able Turing Degrees A formalization of Inductive Inference and its Ra­ tionality Recent results on Substructural Logic Differential Fields and Model Theory Infinite Orbits and Vaught's Conjecture A System of Classical logic and natural Reasoning

VII

ORGANIZING COMMITTEE D I N G DECHENG

Nanjing University, Nanjing

G A O HENGSHAN

Graduate School, Academia Sinca, Beijing

HUANG QIEYUAN

Institute of Software, Academia Sinica, Beijing

LI W E I

Beijing Univ. Aeronautics and Astronautics, Bei­ jing

L I XIANG

Guizhou University, Guiyang

Lu

Nanjing Aeronautics and Astronautics Univ., Nanjing

Muo

YIZHONG

Nanjing University, Nanjing

SHAOKUI

SHEN ENSHAO

Shanghai Jiaotong University, Shanghai

SHEN FUXING

Beijing Normal University, Beijing

W A N G SHIQIANG

Beijing Normal University, Beijing

YANG DONGPING

Institute of Software, Academia Sinica, Beijing

ADVISORY COMMITTEE CHONG CHI TAT

National University of Singapore, Singapore

CROSSLEY, J.

Monash University, Australia

N.

T U G U E , TOSIYUKI

Konan Woen's Junior College, Japan

YASUGI, MARIKO

Kyoto Sangyo University, Japan

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Table of C o n t e n t s

ix

Foreword

v

Seminormal Fine Measures on PK(\) Y. A B E

1

Effective Baire Category Concepts K. AMBOS-SPIES AND J. REIMANN

13

Weak Presentations of Computable Partial Orderings M. M. ARSLANOV AND I. S. KALIMULLIN

31

Recursion Theory and Weak Fragments of Peano Arithmetic: A Study of Definable Cuts C. T. CHONG AND Y. YANG

47

Lattice Embedding into d-r.e. Degrees Preserving 0 and 1 D. DING AND L. QIAN

67

On Stationary Reflection Principles Q. F E N G

83

Definable Sets of Real Numbers, Infinite Games and Core Model Theory K. HAUSER

107

The Descriptive Classification of Some Classes of C*-algebras A. S. KECHRIS

121

Decidability and Undecidability in the Enumerable Turing Degrees S. L E M P P

The Theory of Finite Models L. Luo

151

163

A Note on Weak Segments of PFA T. MIYAMOTO

175

On Structual Inference Rules For Gentzen-Style Natural Deduction, Part I K. NAKATOGAWA

199

X

Linear Set Theory with Strict Comprehension M. SHIRAHATA

223

A Solution to a Problem of Marek and Truszcynski K. Su AND H. CHEN

247

Credulous Reasoning about Defaults Y. H. TAN AND L. W.

N. VAN DER T O R R E

255

Computational Complexity of Infinite-Valued Lukasiewicz Propositional Logic H. W A G N E R

273

NDK and Natural Reasoning M. YASUGI AND M. NAKATA

285

Default Logic and it's Variants: A Semantical View M. ZHANG

311

Adding Eventual Different Reals Y. ZHANG

329

Asian Logic Conferences M. YASUGI

345

1

Seminormal Fine Measures on VK\ Yoshihiro Abe * Department of Mathematics, Kanagawa University, Kanagawa-ku, Yokohama 221, Japan A b s t r a c t . We first show that K is A-supercompact if seminormal fine mea­ sures on VK\ exist. Second, assuming a supercompact cardinal, we prove the existence of non-normal ideals whose seminormal closures are normal. We also discuss prepicitousness and saturation of the minimal ^-normal ideal. M a t h e m a t i c a l Subject Classification: 03E05, 03E55 K e y w o r d s : Fine measure; Seminormality; ^-normality; nowhere prepicitous; saturation 1. Introduction The VK\ context of closed unbounded sets was investigated in Jech [7], and the structure of normal ideals and filters on VKX have been studied in various aspects. Some weakening of normality was also presented. Seminor­ mal ideals were considered in Johnson [9], [10] in relation to the partition property. Their basic structure is dealt with in [2] and ideals with partial normality are treated. We discuss here large cardinal features of seminormal ideals. But our interest is not in 7\A-combinatorics but in more familiar large cardinal properties such as fine measures, and saturated and prepicitous ideals. In 3, we project a seminormal fine measure to a normal measure, show­ ing the former is in fact a supercompactness measure. We also mention seminormal fine measures in the Rudin-Keisler orderings. In 4, assuming K, is supercompact and A is regular, we build a nonnormal ideal / on VKX such that every seminormal ideal extending i" is normal, answering a question in [2]. In the last section we argue saturation of the minimal ^-normal ideal on VKX and its prepicitousness. Such an ideal was also studied in [2]. Results *This research was partially supported by Grant-in-Aid for Scientific Research (No.06640178), Ministry of Education, Science and Culture of Japan.

2

YOSHIHIRO A B E

in [13] are used. 2. Definition and preliminaries Throughout this paper, K, < X are uncountable cardinals and K is regular. For such a pair (K, A), VKX = {x C A : \x\ < K}. F is a filter on VKX if F is a collection of subsets of VKX such that (1) 0 g F and 7\A G F . (2) I f l c F C 7>KA and X G F , then 7 e F . (3) If X G F and y G F , then X n y G F . 7 is an z'deaZ on PKA if 7 is a collection of subsets of VKX such that (1) 0 G / and P„A g 7. (2) If X C y C PKA and Y G 7, then X G 7. (3) If X G 7 and y G 7, then X U Y G 7. A filter F on 7\A and an ideal 7 on VKX are dual to each other if the following holds: for every X C VKX,

X G F iff PKA - X G 7 .

The dual filter (ideal) of 7 (F) will be denoted by 7* (F*) and each member of 7* (F) is called 7 (F)-measure one. Let 7 + = VVKX - I — {X : X i 7}. X G 7+ is called I-positive. Also let F + = P 7 \ A - F* = {X : 7\A - X g F } . X G F + is called F-positive. For X G 7+, 7|X = { y C PKA : y H X G 7} is an ideal D 7. A filter F on VKX is K-complete if it is closed under intersection of less than K many members. F is /me if for each a < A, {x G 7\A : a G x} G F . For the sake of convenience, throughout this paper, by 'filter', we mean a 'fine ^-complete filter'. Most properties defined for filters are translated into properties for ideals using duality. We often say a measure for an ultrafilter. That is, a filter F is a measure iff for all X G VK\, either X G F or VKX - X G F . The dual ideal of a measure is called a prime ideal. A function / : VKX —► A + 1 is the least unbounded function for a measure U if the following hold. (1) {x G VKX : f(x) >a}

eU for all a < X.

SEMINORMAL F I N E M E A S U R E S

(2) If {x G VKX : g(x) < f(x)} some 7 < A.

3

G U, then {x G VKX : g(x) < 7} G U for

Note that the range of the least unbounded function is included in A if c/(A) > K. A filter F on VK A is weakly normal if for any regressive function / : X —> A with X e F+ there is 7 < A such that {x G X : / ( x ) < 7} G F + . Thus, a fine measure is weakly normal iff its least unbounded function assigns x to sup(x). For any x G VKX, x = {y G VKX : x C 2/}, and X C 7\A is unbounded iff X H x ^ 0 for all x G VK\ . Let 7KA = {X C T^A : X is not unbounded }. 7K/\ is the minimal ideal on VKX. Hence FSFK\ = I*x is the minimal filter on VK\, and x G FSFK\ for any x G P^A. X C PKA is c/oserf iff \JD G X for any increasing C -sequence D C X such that |D| < K. C C P^A is said to be a cub iff it is closed and unbounded. X is stationary iff X p| C 7^ 0 for every cub C. Set iV5KA = {X C VKX : X is not stationary}. So NSK\ is the dual ideal of the cub filter on VKX, CFK\, generated by cub sets. X C VKX is strongly closed iff \JD G X for any .D C X with |£>| < ft and we call C C VKX a strong cub iff it is strongly closed and unbounded. SCFK\ is the filter generated by strong cub sets, and its dual ideal is SNSK\ = {X C PKA : X D C = 0 for some strong cub set C C VKX). Such notions were generalized to VKX by Jech [7]. Further study was made in Menas [14] and Carr [3] where the following was proved: hx

C SNSKX

C

NSKX

Definition. For 6 < A, the diagonal union and the diagonal intersection of { X a : a < 6} are defined respectively by: Va<sXa = {x € VKX : x G X a for some a G x } Aa]U = K and Ca G U for all a < K. ([f]u means the transitive collapse of the equivalence class of a function / in the usual ultrapower VVKX/U.) We can not have such a fine measure for some {Ca : a < K+} C SCFKK+ unless K, is ft+-supercompact. T h e o r e m 3.6. There is a family of K+ many strong cub subsets ofVKK+ A such that K, is K+ -supercompact if there is a fine measure U on VKK+ with U D A and [< x n K\X G VKX >]V = K. Proof. For each a < tt+, let ha : a —> K be injective and Ca = ChaLet A = {Ca ' OL < K + } . Suppose that U is a fine measure on VKK+ such that U D A and [< x n K\X G VKX >]U = «. We only have to show U is seminormal by Theorem 3.2. For any (3 < K+ and regressive function / : VKK+ —► f3, define g : V KK +

-> « by

0(a) = h0(f(x))

for all x G P K /^ + .

Since C/3 € U, {x £ VKK+ : p(x) G x D tt} G C7. Hence there is a 7 < K, with £ - 1 ( { 7 } ) ^ U because [< x D /c|a; G 7\A >](/ = K. So, / " H I ^ H T ) } € J7.

□ Definition. For a fine measure U on /„({/) = {X C 7>KA : / _ 1 ( X ) G J7}. VF = /*(£/) for some / : T^A —> 7\A. on some X G U whenever /*(£/) is a fine

VKX and / : VKX —> T^A, let We say U is projected to W if C7 is minimal if / is one to one measure.

7

SEMINORMAL F I N E M E A S U R E S

Theorem 3.2 says that every seminormal measure on VKX can be pro­ jected to some normal measure on VKX. For a given normal measure U on VKX, we easily find a non-normal seminormal measure to which U is projected provided that c/(A) > K. P r o p o s i t i o n 3.7. / / c/(A) > K, then every normal measure on VKX is isomorphic to a non-normal, seminormal measure which does not extend SCFKX. Proof. Let U be a normal measure on VKX and f(x) = x U {sup(x)} for each x G VKX. Since U is normal, there is an X G U such that / is injective on X. So, U = f*(U) and /*(£/) is clearly seminormal by definition. Also SCFKX K there is a nonnormal ideal I such that Sm(I) is normal. The reader should note that any seminormal ideal D SNSK\ is normal if A is singular. For the proof, see [2]. Before proving the theorem, we make some definitions and prove some lemmas. Definition. An ideal 7 is r)-subnormal if there is an 77-normal ideal J D I. A function / is I-small if / _ 1 ( { # } ) G I for any x.

8

YOSHIHIRO A B E

The following is a VKX-version of well-known fact for ideals on K. L e m m a 4.2. I is not r\-subnormal if there is an I-small regressive f : X —► 7] for some X G I*.

function

L e m m a 4.3. If 6 < X and I is 8+ -normal, and not 8+ -saturated, then there is a J D I which is not 8+ -subnormal. Proof. By our hypothesis there is a disjoint partition of VKX {Xa 8+} Cl+. J is defined by X eJ

iff

X CVKX

and

\{a {0,1} such that AeC

O

( 3 m ) ( V n ) ( $ A ( m , n ) - 1).

For more details on the arithmetical hierarchy of classes we refer to Rogers [Ro67], Chapter 15. T h e o r e m 2.3 For any class C the following are equivalent. (i) C is effectively

meager.

(ii) There is a Ti^-class D and a recursive set A such that C C D , D is c.f.v., and A 0 D . (Hi) There is a T^-class D and a set A such that C C D finite variant A of A, A $ D .

and, for any

For the proof of Theorem 2.3 we need the following observations.

EFFECTIVE BAIRE CATEGORY

P r o p o s i t i o n 2.4 Let f be an effective system of extension functions. is a greatest class which is meager via f, namely M(f) Proof.

= {A:

17

There

(3e)(Vn)(fe(A\n)£A)}

Straightforward.

□

P r o p o s i t i o n 2.5 For any effective system of extension functions f, M ( / ) is a T,®-class. Proof.

Immediate by definition.

□

P r o p o s i t i o n 2.6 For any effective system of extension functions f, R E C 2 M(/). P r o o f . Given / , by an effective finite extension argument inductively define a recursive set A $ M ( / ) as follows. Enumerate A\l(s) in stages, where / : UJ —> LU is a strictly increasing function, and l(s) and the finite extension A\l(s) of A\l(s — 1) are defined at stage s: Given 5 and A\l(s — 1) (where Z(-l) = 0), let A\l(s) = fs(A\l(s - 1)). Then A is recursive and the initial segment A\l(s) ensures that A is not effectively nowhere dense via fs. □ P r o p o s i t i o n 2.7 For any effective system of extension functions f there is an effective system of extension functions f such that M ( / ) C M ( / ) and M ( / ) is closed under finite variants. P r o o f . Given / the function / is obtained by considering all finite variants of the extension functions of the system / in the following sense. For strings x and y such that \x\ > \y\ let var(x,y) be the string obtained from x by replacing it's initial segment of length \y\ by y. Then, for any number e and strings y and z, the extension function f(e,y,z) extends a string x with \x\ > \y\ in the same way as fe extends the string var(x,y) and it maps shorter strings to an extension determined by z. To be more precise, for x with \x\ > \y\, f(e,yjZ){x) = x~v, where v is the unique string satisfying fe(var(x,y)) — var(x,yYv, and f(e,y,z)(x) — x^0^~^z otherwise. □ P r o o f of T h e o r e m 2 . 3 . Since the implication (ii) => (Hi) is obvious, it suffices to show (i) ^ (ii) and (Hi) => (i). For a proof of (i) => (ii) assume that C is effectively meager via / . Then, for / as in Proposition 2.7 and for D = M ( / ) , C C D (by Proposition 2.4), D is a c.f.v. S°- class (by Propositions 2.5 and 2.7) and there is a recursive set A $. D (by Proposition 2.6). . For a proof of the implication (Hi) => (i) let D be a E^-class, say X eB

o

(3m) (Vn) ( $ * ( m , n ) = 1),

18

A M B O S - S P I E S AND R E I M A N N

and let A be a set such that, A £ D for all finite variants of A. By Propo­ sition 2.4, it suffices to define an effective system of extension functions / such that D C M ( / ) . Note that, by assumption on A, for any number m and for any string x there is a string y "3 x such that $ y (m, n) = 0 for some number n. So, by letting fm(x) be the least such string y, / is an effective system of extension functions as required. □ The above characterization of effective meagerness easily implies the previ­ ously mentioned results of Mehlhorn on his concept. Corollary 2.8 (Mehlhorn [Me73]) R E C is not effectively meager.

□

Recall that a class C is recursively presentable (r.p.) if there is a recursive set U C u x UJ such that C = {Ue : e > 0}. A class of recursive sets is called bounded if it is contained in an r.p. class. Corollary 2.9 (Mehlhorn [Me73]) Every bounded class of recursive sets is effectively meager. Proof. By Theorem 2.3 it suffices to show that every r.p. class C is a £§class. But given a recursive set U such that C = {Ue : e > 0}, X eC

& (3e) (Vn) (U(e,n) =

X(n))

whence C is a S^-class.

□

By Corollary 2.9, the standard complexity classes are effectively meager. An interesting example of an effectively meager class containing nonrecursive sets is the following. Corollary 2.10 (Mehlhorn [Me73]) The class C = {A : PA = NPA} effectively meager.

is

P r o o f (Sketch). Let {Me : e > 0} and {Ne : e > 0} be recursive enumerations of the deterministic respectively nondeterministic polynomial time bounded oracle Turing machines. Then, for any oracle set A, the set KA = { ( e , n , 2 m ) : iVe accepts n in less than m steps} is NP A -complete (see Baker et al. [BGS75]). It follows that C = {A:

KA ePA}

= {A:

(3n) (Vm) (KA(m) A

=

Mn(m))}

B

whence C is a S^-class. Moreover, for A =* B, P = P and N P A = N P B whence C is c.f.v. Finally, Baker et al. [BGS75] have shown that there is a set A £ C. Hence effective meagerness of C follows from Theorem 2.3. □ Mayordomo [May94] pointed out some limitations of the effective cate­ gory concept^For any class C, a set A is C-bi-immune if neither A nor its complement A contain any infinite set X G C as a subset.

E F F E C T I V E BAIRE CATEGORY

19

T h e o r e m 2.11 (Mayordomo [May94]) The class B I ( P ) fl R E C of the re­ cursive sets which are not P-bi-immune is not effectively meager though B I ( P ) is meager in the classical sense. This theorem can be interpreted as follows: The effective category concept formalizes the effective finite extension method. A finite extension con­ struction of a P-bi-immune set, however, cannot be effective, i.e. the finite extension method yields only nonrecursive P-bi-immune sets. On the other hand, however, Balcazar and Schoning [BS85] have constructed recursive P-bi-immune sets by a slow diagonalization or wait and see argument. The extension of Mehlhorn's category concept introduced in the next section is designed to formalize this more general diagonalization technique. For a detailed discussion of this topic we refer to Ambos-Spies [Am96].

3

The Extended Effective Category Concept

The extension of Mehlhorn's effective category concept which we will con­ sider here is based on partial extension functions. For such a function / we say that a class C is nowhere dense via / if (VA€C)(3°°n)(/(Arn)i) and

(?x)(f(x)l=> B / ( l ) nC = 0). Note that we can extend / to a total extension function / such that C is nowhere dense via / by letting f(x) = f(y) for the least y II x such that f(y) I if such a string y exists and by letting f(x) = x otherwise. Hence, in the classical case, partial extension functions lead to the same category concept. In case of effective extensions, however, we obtain a stronger concept as we shall show below. Definition 3.1 (a) An effective partial extension function f is a par­ tial recursive function f : 2 2l otherwise

Now, to show that C C M ( / ) , fix A £ M ( / ) . It suffices to define a strategy g for Player I such that R(g, h) = A. For the definition of g distinguish the

25

E F F E C T I V E BAIRE CATEGORY

following two cases: CO

Case 1: (3k) (3 n) (hw(A\n) = k). Let k0 be the least such k and fix lQ such that hw(A\n) > k0 for n > IQ. Then /*° is dense along A for a l l / > 0 whence, by A & M ( / ) , A meets all these extension functions. So for x C A we may fix n = n(x) minimal such that / ™ x ( N ) / o ) ( ^ M z ) ) H A. Note that x C A\n(x), hw(A\n(x)) = k0 e and h (A\n(x)) = f^x^xiJo)(A\n(x)). Now define g as follows: For x C A let g(x) = Afn(x); otherwise let #(x) = x~0. To show that R(g,h) = A, by induction on m show that, for R = R(g,h), R\rm C A. Since Player II cannot cut the first move of Player I, R\r0

=

he(g(\))

and, by definition of p, he(9(X)) = he(A\n(X))

= /£°(^n(A)) C A

and ro > /o- F ° r the inductive step let x = R\rm and assume that x C A. Then 0(i*|V m ) - A\n(x), hw(A\n(x)) = k0 and, by rm > r0 > l0, hw(y) > kQ for all strings y with R\rm C i / C