Showing papers in "Annals of Pure and Applied Logic in 2005"
TL;DR: This semantics is used to give new proofs of several basic results concerning LP, and the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator.
Abstract: A new semantics is presented for the logic of proofs (LP), (Technical Report MSI 95-29, Cornell University (1995), Bull. Symbolic Logic 7 (2001) 1) based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.
274 citations
TL;DR: A natural, albeit non first-order, axiomatisation is given for the corresponding class of structures and this gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.
Abstract: We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.
133 citations
TL;DR: This paper proves both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms, and defines both algebraic semantics and relational semantics for these logics.
Abstract: In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.
132 citations
TL;DR: A new functional interpretation, based on a novel assignment of formulas, does not care for precise witnesses of existential statements, but only for bounds for them, and new principles are supported, including the FAN theorem, weak Konig's lemma and the lesser limited principle of omniscience.
Abstract: We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Godel's functional "Dialectica" interpretation, the new interpre- tation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including (a version of) the FAN theorem, weak Konig's lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the new interpretation to theories of classical arithmetic and analysis.
90 citations
TL;DR: Properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, are exploited to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy.
Abstract: We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α , we transform a countable directed graph G into a structure G ∗ such that G has a Δ α 0 isomorphic copy if and only if G ∗ has a computable isomorphic copy. A computable structure A is Δ α 0 categorical (relatively Δ α 0 categorical, respectively) if for all computable (countable, respectively) isomorphic copies B of A , there is an isomorphism from A onto B , which is Δ α 0 ( Δ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is a computable, Δ α 0 categorical structure, which is not relatively Δ α 0 categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical. An additional relation R on the domain of a computable structure A is intrinsically Σ α 0 (relatively intrinsically Σ α 0 , respectively) on A if in all computable (countable, respectively) isomorphic copies B of A , the image of R is Σ α 0 ( Σ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is an intrinsically Σ α 0 relation on a computable structure, which not relatively intrinsically Σ α 0 . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable. The Δ α 0 dimension of a structure A is the number of computable isomorphic copies, up to Δ α 0 isomorphisms. We prove that for every computable successor ordinal α and every n ≥ 1 , there is a computable structure with Δ α 0 dimension n . This generalizes the result of Goncharov that there is a structure of computable dimension n for every n ≥ 1 . Finally, we prove that for every computable successor ordinal α , there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that Δ α 0 relative to X is not Δ α 0 . In particular, for every finite n , there is a structure with isomorphic copies in exactly the non- low n Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure A with isomorphic copies in exactly the nonzero Turing degrees.
87 citations
TL;DR: Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic, as well as the logics of dynamic topological systems, just as S4 is the logic of topological spaces.
Abstract: Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and □ can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system be a topological space X together with a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □ (interior), and two temporal modalities, ○ (next) and ∗ (henceforth), both interpreted using the continuous function f. In particular, ○ expresses f’s action on X from one moment to the next, and ∗ expresses the asymptotic behaviour of f.
82 citations
TL;DR: It is proved that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory and also proves a conjecture of Schupp.
Abstract: We prove that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown that the word problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable.
69 citations
TL;DR: It is proved that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index and G/G 00 equipped with the “logic topology” is a compact Lie group.
Abstract: We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G 00 of bounded index and G/G 00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.
59 citations
TL;DR: It is proved that S4 is complete with respect to Boolean combinations of countable unions of convex subsets of the real line, thus strengthening a 1944 result of McKinsey and Tarski.
Abstract: We prove that S4 is complete with respect to Boolean combinations of countable unions of convex subsets of the real line, thus strengthening a 1944 result of McKinsey and Tarski (Ann. of Math. (2) 45 (1944) 141). We also prove that the same result holds for the bimodal system S4+S5+C, which is a strengthening of a 1999 result of Shehtman (J. Appl. Non-Classical Logics 9 (1999) 369).
49 citations
TL;DR: This work presents several deductive systems for common knowledge above epistemic logics –such as K, T, S4 and S5 –with a fixed number of agents and focuses on structural and proof-theoretic properties of these calculi.
Abstract: The notions of common knowledge or common belief play an important role in several areas of computer science (e.g. distributed systems, communication), in philosophy, game theory, artificial intelligence, psychology and many other fields which deal with the interaction within a group of “agents”, agreement or coordinated actions. In the following we will present several deductive systems for common knowledge above epistemic logics –such as K , T , S4 and S5 –with a fixed number of agents. We focus on structural and proof-theoretic properties of these calculi.
44 citations
TL;DR: An extension of Heyting arithmetic in finite types called Uniform Heyting Arithmetic ( HA u) is presented that allows for the extraction of optimized programs from constructive and classical proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program.
Abstract: We present an extension of Heyting arithmetic in finite types called Uniform Heyting Arithmetic ( HA u ) that allows for the extraction of optimized programs from constructive and classical proofs. The system HA u has two sorts of first-order quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripke-style Friedman/Dragalin translation which is inspired by work of Coquand and Hofmann and a variant of the refined A-translation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical first-order proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program.
TL;DR: This work provides a combinatorial characterization of P -indestructibility and constructs maximal almost disjoint families which are P - indestructible yet Q -destructible for several pairs of forcing notions ( P, Q ) .
Abstract: Let A ⊆ [ ω ] ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P -indestructible if A is still maximal in any P -generic extension. We investigate P -indestructibility for several classical forcing notions P . In particular, we provide a combinatorial characterization of P -indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P -indestructible yet Q -destructible for several pairs of forcing notions ( P , Q ) . We close with a detailed investigation of iterated Sacks indestructibility.
TL;DR: A weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley, is presented, it is shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statement of logic, e.g. compactness of classical proposition calculus, are equivalent to the (Weak) Fan Theorem in this system.
Abstract: This article presents a weak system of intuitionistic second-order arithmetic, WKV , a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the (Weak) Fan Theorem in this system.
TL;DR: Using a versatile and flexible compression technique, regularity properties of ordinal count functions can be used to prove logical limit laws.
Abstract: This paper is intended to give for a general mathematical audience (including non-logicians) a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below e 0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws.
TL;DR: Every stable theory that is interpretable in a geometric surgical theory is superstable of finite U -rank, which means that every stable theory in the class of geometric surgical theories is superstably interpretable.
Abstract: The class of geometric surgical theories (which includes all o-minimal theories) is examined. The main theorem is that every stable theory that is interpretable in a geometric surgical theory is superstable of finite U -rank.
TL;DR: It is proved that intuitionistic logic can be replaced with any other proper intermediate logic without modifying the resulting semantics, and it is shown that the answer set semantics satisfies an important property, the “extension by definition”, that can be used to construct program translations.
Abstract: We propose an extension of answer sets, that we call safe beliefs, that can be used to study several properties and notions of answer sets and logic programming from a more general point of view. Our definiti on, based on intuitionistic logic and following ideas from D. Pearce [Stable inference as intuitionistic validity, Logic Programming 38 (1999) 79–91], also provides a general approach to define several semantics based on different logics or inference systems. We prove that, in particular, intuitionistic logic can be replaced with any other proper intermediate logic without modifying the resulting semantics. We also show that the answer set semantics satisfies an important property, the “extension by definition”, that can be used to construct program translations. As a result we are able to provide a polynomial translation from propositional theories into the class of disjunctive programs. © 2004 Elsevier B.V. All rights reserved.
TL;DR: In this article, a Kripke model of CZF is presented in which Power Set is false and Subset Collection is replaced by Exponentiation, in which Subset collection fails.
Abstract: CZF is an intuitionistic set theory that does not contain Power Set, substituting instead a weaker version, Subset Collection. In this paper a Kripke model of CZF is presented in which Power Set is false. In addition, another Kripke model is presented of CZF with Subset Collection replaced by Exponentiation, in which Subset Collection fails.
TL;DR: This work captures the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study.
Abstract: We capture the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study. 1
TL;DR: A system of simply typed lambda terms (with fixed point combinators) is introduced and it is shown that a rather comprehensive class of (co-)recursion equations on streams or non-wellfounded trees can be solved in this system.
Abstract: We introduce a system of simply typed lambda terms (with fixed point combinators) and show that a rather comprehensive class of (co-)recursion equations on streams or non-wellfounded trees can be solved in our system. Moreover certain conditions are presented which guarantee that the defined functionals are primitive recursive. As a major example we give a co-recursive treatment of Mints’ continuous cut-elimination operator.
TL;DR: This work proves a full completeness theorem for multiplicative–additive linear logic using a double gluing construction applied to Ehrhard’s *-autonomous category of hypercoherences, and guarantees that every dinatural transformation corresponds to a Girard MALL proof-structure.
Abstract: We prove a full completeness theorem for multiplicative–additive linear logic (i.e. MALL ) using a double gluing construction applied to Ehrhard’s *-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free MALL proof. Our proof consists of three steps. We show: • Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts. • Softness, together with multiplicative full completeness, guarantees that every dinatural transformation corresponds to a Girard MALL proof-structure. • The proof-structure associated with any dinatural transformation is a MALL proof-net, hence a denotation of a proof. This last step involves a detailed study of cycles in additive proof-structures.
The second step is a completely general result, while the third step relies on the concrete structure of a double gluing construction over hypercoherences.
TL;DR: This first paper on constructive concurrent dynamic logic (CCDL) gives a satisfactory treatment of what statements are forced to be true by partial information about the underlying computer, and develops a constructive logic programming tool for specification and modular verification of programs in any imperative concurrent language.
Abstract: This is the first paper on constructive concurrent dynamic logic (CCDL). For the first time, either for concurrent or sequential dynamic logic, we give a satisfactory treatment of what statements are forced to be true by partial information about the underlying computer. Dynamic logic was developed by Pratt [V. Pratt, Semantical considerations on Floyd–Hoare logic, in: 17th Annual IEEE Symp. on Found. Comp. Sci., New York, 1976, pp. 109–121, V. Pratt, Applications of modal logic to programming, Studia Logica 39 (1980) 257–274] for nondeterministic sequential programs, and by Peleg [D. Peleg, Concurrent dynamic logic, Journal of the Association for Computing Machinery 34 (2) (1987), D. Peleg, Communication in concurrent dynamic logic, Journal of Computer and System Sciences 35 (1987)] for concurrent programs, for the purpose of proving properties of programs such as correctness. Here we define what it means for a dynamic logic formula to be forced to be true knowing only partial information about the results of assignments and tests. This informal CCDL semantics is formalized by intuitionistic Kripke frames modeling this partial information, and each such frame is interpreted as an idealized concurrent machine (a concurrent transition system). In CCDL, proofs and deductions are ω -height, ω -branching, well-founded labeled subtrees of ω ω . These are a generalization of the signed tableaux of Nerode [A. Nerode, Some lectures in modal logic, Technical Report, M.S.I. Cornell University, 1989, CIME Logic and Computer Science Montecatini Volume, Springer-Verlag Lecture Notes, 1990, A. Nerode, Some lectures in intuitionistic logic, Technical Report, M.S.I. Cornell University, 1988, Marktoberdorf Logic and Computation NATO Summer School Volume, NATO Science Series, 1990 (in press)] stemming from the prefix tableaux of Fitting [M.C. Fitting, Proof Methods for Modal and Intuitionistic Logic, Reidel, 1983]. We demonstrate the correctness of our tableau proofs, define consistency properties, prove that consistency properties yield models, construct systematic tableaux, prove that systematic tableaux yield a consistency property, and conclude that CCDL is complete. This infinitary semantics and proof procedure will be the primary guide for defining, in a sequel, the correct finitary CCDL (FCCDL) based on induction principles. FCCDL is suitable for implementation in constructive logic software systems such as Constable’s NUPRL or Huet-Coquand’s CONSTRUCTIONS. Our goal is to develop a constructive logic programming tool for specification and modular verification of programs in any imperative concurrent language, and for the extraction of concurrent programs from constructive proofs. Subsequent papers will introduce analogous logics for declarative and functional concurrent languages.
TL;DR: A new proof of the completeness of the modal logic S 4 for all topological spaces as well as for the real line R, the n -dimensional Euclidean space R n and the segment (0, 1) etc is presented.
Abstract: The completeness of the modal logic S 4 for all topological spaces as well as for the real line R , the n -dimensional Euclidean space R n and the segment (0, 1) etc. (with □ interpreted as interior) was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure K for S 4 into a subspace of the Cantor space which in turn encodes (0, 1). This provides an open and continuous map from (0, 1) onto the topological space corresponding to K . The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures.
TL;DR: It is shown that the converse is not true, and any function that is computable in the sense of Markov, i.e., computable with respect to a standard Godel numbering of the computable real numbers, is computability in thesense of Banach and Mazur.
Abstract: We consider two classical computability notions for functions mapping all computable real numbers to computable real numbers. It is clear that any function that is computable in the sense of Markov, i.e., computable with respect to a standard Godel numbering of the computable real numbers, is computable in the sense of Banach and Mazur, i.e., it maps any computable sequence of real numbers to a computable sequence of real numbers. We show that the converse is not true. This solves a long-standing open problem posed by Kushner.
TL;DR: It is shown that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed points in semirings, and eliminations of chain and deletion rules depend on their inequational properties (and the idempotence of addition).
Abstract: We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed points in semirings, and eliminations of chain and deletion rules depend on their inequational properties (and the idempotence of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed points.
TL;DR: It is proved that a quantifier elimination result for existentially closed n -truncated Hasse fields is proved and the fields are characterized as reducts of existenceentially closed Hasse Fields.
Abstract: We give geometric axioms for existentially closed Hasse fields. We prove a quantifier elimination result for existentially closed n -truncated Hasse fields and characterize them as reducts of existentially closed Hasse fields.
TL;DR: An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions and covers second-order universal quantification and also the extension of the logic with fixed points of non-Strictly negative operators, which appears to be a new result.
Abstract: Termination for classical natural deduction is difficult in the presence of commuting/permutative conversions for disjunction. An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions. It covers second-order universal quantification and also the extension of the logic with fixed points of non-strictly positive operators, which appears to be a new result. Finally, the relation to Parigot’s strictly positive inductive definition of his set of reducibility candidates and to his notion of generalized reducibility candidates is explained.
TL;DR: The properties of focalization and reversion of linear proofs are at the heart of the analysis: it is shown that the tq-protocol of normalization for the classical systems LK pol η and LK Pol η , ρ perfectly fits normalization of polarized proof-nets.
Abstract: We give the precise correspondence between polarized linear logic and polarized classical logic. The properties of focalization and reversion of linear proofs are at the heart of our analysis: we show that the tq-protocol of normalization for the classical systems LK pol η and LK pol η , ρ perfectly fits normalization of polarized proof-nets. Some more semantical considerations allow us to recover LC as a refinement of multiplicative LK pol η .
TL;DR: A certain class of pretoposes are constructed, consisting of what one might call “predicative realizability toposes”, that can act as categorical models of certain predicative type theories, including Martin-Lof Type Theory.
Abstract: Using the theory of exact completions, I construct a certain class of pretoposes, consisting of what one might call “predicative realizability toposes”, that can act as categorical models of certain predicative type theories, including Martin-Lof Type Theory.
TL;DR: This paper proves exponential separations between depth d-LK and depth (d + 1 )-K for every d 2 1 N utilizing the order induction principle, and deduces width lower bounds for resolution refutations of the Ramsey principle.
Abstract: This paper proves exponential separations between depth d-LK and depth (d + 1 )-LK for every d 2 1 N utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d-LK and depth (d+1)-LK ford2N. We investigate the relationship between the sequence-size, tree-size and height of depth d-LK-derivations for d 2 1 N, and describe transformations between them. We deflne a general method to lift principles requiring exponential tree-size (d + 1 )-LK-refutations for d 2 N to principles requiring exponential sequence-size d-LK-refutations, which will be described for the Ramsey principle and d = 0. From this we also deduce width lower bounds for resolution refutations of the Ramsey principle. Constant-depth propositional proof systems have been extensively studied because of their connection with the complexity of constant-depth circuits and fragments of bounded arithmetic (c.f. [2, 10, 14, 15, 17]). Kraj¶‡•cek [10] deflned an alternative notion of constant-depth proofs: a formula is deflned to have §-depth d ifi if is depth d+1 and the bottommost level of connectives = = = = i i;’
TL;DR: A categorical model of polymorphism, based on game semantics, which contains a large collection of generic types and solves a recursive equation involving this relative product to obtain a universe in a suitably “absolute” sense.
Abstract: Genericity is the idea that the same program can work at many different data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type ∀ X . A [ X ] , are equal at any given instance A [ T ] , then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but finding a semantically motivated model satisfying this principle remained an open problem. In the present paper, we construct a categorical model of polymorphism, based on game semantics, which contains a large collection of generic types. This model builds on two novel constructions: •A direct interpretation of variable types as games, with a natural notion of substitution of games. This allows moves in games A [ T ] to be decomposed into the generic part from A , and the part pertaining to the instance T . This leads to a simple and natural notion of generic strategy. •A “relative polymorphic product” Π i ( A , B ) which expresses quantification over the type variable X i in the variable type A with respect to a “universe” which is explicitly given as an additional parameter B . We then solve a recursive equation involving this relative product to obtain a universe in a suitably “absolute” sense. Full Completeness for ML types (universal closures of quantifier-free types) is proved for this model.