Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a clock . Namely let f(x) x ; suppose for convenience x is always of limit length and at least . Theorem 1.9 With f as above: NP . 2 Preliminaries We shall let 1 stand for the rst ordinal not recursive in x. Then L 1 [x] is an admissible set. We refer the reader to [1] for an account of admissible sets and their basic properties. We shall use the following notation for the machine con gurations. Let the cells of the tape be enumerated hC i ji i with the cell C i having value C i ( at time . We assume that the rst n blocks on the tape are enumerated by ....

.... is in the illfounded part of the original On ) then in fact WFP (A) WFP (A) this is because (a) we cannot have 2 WFP (A) as otherwise A would recognize it as ) b) but neither can WFP (A) as L x [x] is inadmissible, and this would contradict the Truncation Lemma ([1]) 7 Hence any instance of illfoundedness in (On) will be detected before the true many steps have been taken. This leaves time to change the contents of C 0 to a zero, and halt here before many steps have been taken. In each case then P n (e a x y) halts in no more than ....

J. Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, 1975.

....s safety which has been used in Section 5 is that in classical ZF only the size of collections matters, not the e ectivity of their construction. It might be interesting to develop a constructive version which is based on ( ZF ; FZF ) safety, and to see its relations with admissible set theory ([Ba75]) and with the intuitionistic CZF and IZF ( TvD88] ....

J. Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer-Verlag, 1975.

....N of KP M Sigma 1 Foundation is a model of KP M Class Foundation; further, WO will hold in p (N) if it held in N. Proof : 1) That KP might hold in the standard part of a non standard model is a result that goes back to M lle Ville. The argument we give is from Barwise s book [K3]. That M j= Delta 0 Separation is readily checked, because Delta 0 formulae are absolute between M and N. M j= Infinity because N is an model. Union is easily checked, and pairing will hold because o (N) is a limit ordinal. Pi 1 Foundation, indeed full Class Foundation, will hold because ....

J. Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, 1975.

....to set theory. We strongly believe that the answer to this question is positive. Our conjecture, in fact, is that a new insight into admissible set theory can be gained using the notion of effective finiteness. Specifically: we conjecture that the separation and collection axioms of KPU (see [Ba75]) are valid in KPU for every formula which is effectively (oe AS ; FAS ) Gammafinite w.r.t. oe AS ; FAS ) was introduced in the previous section, in the third example after Definition 16. Note that proving the converse, i.e. that every Delta 0 formula is effectively (oe AS ; FAS ....

J. Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer-Verlag, 1975.

....is brought out concretely by Theorem A 0 (in section 2) which establishes the non elementary character of bisimilarity via a first order compactness argument. The author suspects that there is more to be mined in compactness, especially in its generalized form involving admissible sets (Barwise [7]) Making this suspicion plausible is one of the aims of the present paper, which proceeds as follows. After reviewing some preliminary definitions and facts in section 1, a couple of basic results are presented in section 2 concerning (respectively) the co inductive characterization of ....

....2 Z a check if a a; f This is a Sigma 0 1 problem easily reducible to bisimilarity. g if not, then a 62 Z a ; otherwise, search the n such that f a (n) a, concluding that a 2 Z a iff n a 0. 2 For orientation, recall that these sets form the hard core of arithmetic see Barwise [7], especially pp. 113 114, where non hyperarithmetic sets are omitted from various models. The present section shows that the notion of a bisimulation requires more types (in contrast to the previous section where types were omitted) 12 By Friedman [20] a can be chosen such that O is ....

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Jon Barwise. Admissible Sets and Structures. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1975.

....theories of truth, of necessity, are complicated. We make some comments on this, but largely pass no judgement here on what that implies. 1. 1 The general setting We shall attempt to make the discussion as self contained as possible; however we shall refer the reader to the relevant sections of [2] or [5] for the basic de nition of the G odel constructible hierarchy L . We shall not use much of admissibility theory, but we shall be generalising the theory of monotone inductive de nitions, and for these the books of [2] and [11] can be consulted. For an in nite cardinal, H is the class ....

....possible; however we shall refer the reader to the relevant sections of [2] or [5] for the basic de nition of the G odel constructible hierarchy L . We shall not use much of admissibility theory, but we shall be generalising the theory of monotone inductive de nitions, and for these the books of [2] and [11] can be consulted. For an in nite cardinal, H is the class of sets of hereditary cardinality less than : that is, those sets x whose transitive closure, TC(x) satis es card(T C(x) HC denotes H 1 , the class of hereditarily countable sets. We identify N with . Let : P( ....

K.J.Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, 1975, Springer Verlag, Berlin, Heidelberg.

....kind of concepts are in general definable in this way, using revision theoretical semantical schemes, and some partial results about the scope of such definitions. We give an explicit approach, mirroring the theory of monotone inductive definitions and the theory of the next admissible set (cf. [2], 15] that yields a reasonably complete account. iii) A rapprochement with Kripke. We propose an approach (realistic variance) to introducing variance into revision sequences that solves many of the puzzles arising in the revision theory of truth of certain intuitively true (or stable . ....

....M ) sets; the class of 1 (S M ) definable sets coincides with the class of S , co S definable subsets of M , that is, again, with the strongly S definable sets. Remark: 2 S N has domain that of L where is the first stable ordinal. For information on the stable ordinals, see for example, [2]. Note that many first order structures M will have jS M j the same. For example S N will have the same domain as SA where A is the least model of analysis, and exactly the same class of sets of integers are revision theoretically definable over each structure. In our terms L is a very large ....

JON BARWISE, Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, 1975.

....definition of the expanded universe is carried out within the language of set theory, but, alternatively, indeterminates could also be added as new symbols in the language. For instance, in [BE88] indeterminates are indeed treated as primitive elements (Urelemente) of a set theory like the one in [Bar75]. But in order to carry out this extension of the language formally, an extension of the axioms of the theory is also required. 4.4 Special Final Coalgebra Theorem The assumption that the universe (greatest fixed point of P) be a final coalgebra of the powerset functor is strong enough to make ....

J. Barwise. Admisible Sets and Structures. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1975.

....a very general definition of a quantifier, allowing practically any class K of structures to be used for defining a new quantifier QK that captures membership in that class. Since then, the study of languages with added quantifiers has been an important line of research of abstract model theory [BF85]. Meanwhile, finite model theory had emerged as an important research area [Fag90] The steadily growing interest of logicians in finite structures was a consequence of the strengthened connections between logic and computer science. Researchers rapidly realized that first order logic (FO) was ....

J. Barwise and S. Feferman, editors. Model theoretic logics. Perspectives in Mathematical Logic (Springer Verlag, Berlin, 1985).

....what kind of concepts are in general de nable in this way, using revision theoretical semantical schemes and some partial results thereto. We give what we believe is an explicit approach, mirroring the theory of monotone inductive de nitions and the theory of the next admissible set (cf. [2], 16] that yields a reasonably complete account. iii) A rapprochement with Kripke. We propose an approach (realistic variance) to introducing variance into revision sequences that solves many of the puzzles arising in the revision theory of truth of certain intuitively true (or stable . ....

Jon Barwise,\Admissible Sets and Structures", Perspectives in Mathematical Logic, Springer Verlag.

....to flesh out the above, and listing some basic facts concerning the nature of this form of computation. For any notions involving constructible sets, the reader may find them in [3] For the basic structure of Turing Machines, see, for example [2] for admissibility theory either [3] or [1]. We use the notation f p (x) g for the notion computation f p (x) has real g on its output tape after steps . This is then also a Delta 1 2 relation of f; x; g and 2 WO (the latter the set of codes of wellorderings) We write OE p (0) x for program p eventually has settled ....

K.J.Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, 1975, Springer Verlag, Berlin, Heidelberg.

....to flesh out the above, and listing some basic facts concerning the nature of this form of computation. For any notions involving constructible sets, the reader may find them in [3] For the basic structure of Turing Machines, see, for example [2] for admissibility theory either [3] or [1]. We should like to warmly thank Joel Hamkins for his introducing this subject to us, and for his helpful suggestions and patient discussions, and to Benedikt Loewe for his comments on an earlier draft. We use the notation f p (x) g for the notion computation f p (x) has real g on ....

K.J.Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, 1975, Springer Verlag, Berlin, Heidelberg.

....Corollary 3.6 AW ( C. Theorem 1.1 is proven at the end of section 2. The other corollaries are deduced in section 3. For any notions of constructible sets, the reader may find them in [3] For the basic structure of Turing Machines, see, for example [2] for admissibility theory either [3] or [1]. We should like to warmly thank Joel Hamkins for introducing us to the subject by asking us some of these questions, giving some patient explanations, and for his suggestions. 2 The Main Proposition The Corollaries and the Theorem 1.1 mentioned are all those to (the proof of) the main ....

K.J.Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, 1975, Springer Verlag, Berlin, Heidelberg.

....approaches: FOL[ATC;Q] will be different from IFPL[Q] when Q is a generalized quantifier treated as a variable. Once spelled out, this is not surprising, but illuminating. Outline of the paper We assume the reader is familiar with the basics of generalized quantifiers and logics as described in [EFT80, MP93, BF85] and with logics capturing complexity classes, as in [Imm87, Imm88, Imm89] In section 2 we introduce our notion of uniformly capturing of relativized complexity classes. In this section we give also a detailed outline of our results. In section 3 we discuss various interpretations of our results ....

J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

....sequence. Assume C is any complexity class which is captured by a logic L over ordered structures. Then both lemmas hold also if we replace P by C and IFP by L. Theorem 4 now follows immediately. 3 Implicit, invariant and Delta definability We recall the following standard definitions, cf. [BF85] Definition8. Let L be a logic. Let K be a class of S structures. i) K is a L projective class if there is a vocabulary R and a sentence OE in L( Rt S) such that A 2 K iff there is an expansion of A to a R[ S structure A such that A j= OE. ii) Delta(L) is the family of ....

J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

....technique can be considered as a description technique where the vocabulary is precisely de ned and sentences are assigned a meaning based on structures and the satisfaction relation in a precise way. These concepts are captured by the notions of institutions, cf. GB92] or abstract logics, cf. Bar85,CK90] Formally, we consider an FDT to be an institution. An institution 3 provides a category Sig of signatures which form the basic vocabulary to describe a system. Formal syntax and semantics are given by functors Sen and Str from the category Sig to sets of sentences and categories of ....

....satisfying the translated sentence can themselves be translated back to structures satisfying the original sentence. The second one (isomorphism) states, that truth of a sentence is invariant under isomorphism of the structures. This condition is taken from the de nition of abstract logics [Bar85,CK90] In the following the category Set denotes the category of all sets as objects and mappings between sets as morphisms. We assume the standard Zermelo Fraenkel axioms (ZFC) for set theory. Taking categories themselves as objects and functors as morphisms, the category Cat is obtained. ....

J. Barwise. Model-theoretic logics. Perspectives in mathematical logic. Springer, 1985.

.... classes such as L (LogSpace) NL (Non deterministic LogSpace) P (Polynomial Time) NP (Non deterministic Polynomial Time) PH (the polynomial hierarchy) Fag74, Imm87, Imm89, Sto87, Ste91] In mathematical logic the theory of abstract model theory and Lindstrom quantifiers is well established [BF85]. In this talk we report our work concerning unification of Descriptive Complexity Theory and Abstract Model Theory. A detailed account has been published in [MP93, MP94] Lindstrom Logics We show in the framework of abstract model theory how to construct, in general, logics capturing complexity ....

J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

.... such as L (LogSpace) NL (Non deterministic LogSpace) P (Polynomial Time) NP (Non deterministic Polynomial Time) PH (the polynomial hierarchy) Fag74, Imm87, Imm89, Sto87, Ste91] In mathematical logic the theory of abstract model theory and generalized quantifiers is well established [BF85]. One purpose of this paper is to show in the framework of abstract model theory how to construct, in general, logics capturing complexity classes. Much of the abstract framework was already presented, in outline, in [MP93] Intuitively, logics over ordered finite structures can be viewed as high ....

....allow substantial revision of this paper. 2 The General Framework We assume the reader is familiar with the basics of complexity theory as presented in [HU80, GJ79] or the excellent surveys [Sto87, Joh90] and with the basics of abstract model theory as presented in [EFT80, CK90] or in [Ebb85] of [BF85]. We start by introducing a notion of relational regular complexity classes, similar in spirit to both Lindstrom s abstract definition of logics and the computable queries of A. Chandra and D. Harel [CH80, CH82] Independently A. Dawar in [Daw94] introduced basically the same concept for ....

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J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

.... It has been motivated by but is distinctly different from the games introduced in [MZ80] It should also be noted that our explicit definition of these games is more straightforward than their derivation in the framework of abstract model theory and generalized quantifiers as described in [BF85] and [MM85] In [dR87] similarly motivated games are introduced for Fixed Point Logic which is, on ordered structures, of the same expressive power as ATC. However, in contrast to our result, the game introduced in [dR87] is only shown to be sound, which suffices for the non definability result ....

J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

....: 28 4.3 Back and Forth for Transitive Closure : 31 4. 4 Ehrenfeucht Fraiss e Games and Definability : 32 5 Conclusions and Further Research 33 1 Introduction The theory of generalized or Lindstrom Quantifiers is by now well established [BF85] Most of the theory developed so far deals with arbitrary first order structures. We propose in this paper a unified treatment dealing with finite structures only. Our treatment was inspired by the fundamental papers by A. Chandra and D. Harel [CH80, CH82] on computable database queries and by ....

....6= PSpace. Of all the other inclusions it is open whether they are proper, cf. Joh90] Next we introduce the notion of L reducibility which is the descriptive analogue of Karp reducibility in complexity theory. To make the analogy work we generalize the usual notion of regularity of logics, cf. BF85] to k regularity. The usual notion then becomes 1 regularity in our terminology. With this framework set we say that a logic L captures a complexity class C if the L definable classes of finite structures are precisely the members of C. Regular relational complexity classes are exactly the ....

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J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

....with it new logics by closing L under some definability notion under which L is not closed such as relativized definability, vectorized definability, implicit definability etc. One way of doing this is using projective ( Sigma 1 1 ) classes. We recall the following standard definitions, cf. [BF85] Definition5. Let K be a class of oe structures. i) K is a L projective class if there is a vocabulary ae and a sentence OE in L(aetoe) such that A 2 K iff there is an expansion A such that A j= OE. ii) Delta(L) is the family of classes K of oe structures such that both K and its ....

....additionally, L is 1 regular and K 0 is closed under substructures INVK0 (L) is a 1 regular logic. For closure under relativization and vectorization (the regular case) one has to be more cautious and add the closure conditions to a modified definition of Delta(L) and INV (L) respectively, cf. [BF85]. 4 Invariant Definability, Delta Closure and Implicit Definability In this section we relate invariant definability over finite structures to other notions of definability. We shall discuss the case for arbitrary finite and infinite structures as well as some open problems in section 4.4. 4.1 ....

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J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

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Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1975.

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J. Barwise. Admissible Sets and Structures. Perspectives in Mathematical Logic, Springer-Verlag, 1975.

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K.J. BARWISE, Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, 1975.

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Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1975.

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J. Barwise and S. Feferman, editors. Model--Theoretic Logics. Perspectives in Mathematical Logic, Springer-Verlag, 1985.

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J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

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J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

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J. Barwise and S. Feferman, editors. Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer Verlag, 1985.

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