Preservation theorems for Kripke models

There are several ways for defining the notion submodel for Kripke models of intuitionistic first‐order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds A^α and B^α of them, A^α is a subset of B^α and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the structure of kripke models of heyting arithmetic

Since in Heyting Arithmetic (HA) all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic (PA)? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial answers to these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ₁ (PA^- with induction for provably Δ₁ formulas) and that the relation between these classical structures must be that of a Δ₁-elementary submodel. MSC: 03F30, 03F55.     

Submodels of Kripke models

A Kripke model ? is a submodel of another Kripke model ℳ if ? is obtained by restricting the set of nodes of ℳ. In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Łoś-Tarski theorem for the Classical Predicate Calculus. In Appendix A we prove that for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In Appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that ℋT is HA. Here ℋT is the theory of all Kripke models ℳ such that the structures assigned to the nodes of ℳ all satisfy T in the sense of classical model theory. Received: 4 February 1999 / Published online: 25 January 2001     

Constants in Kripke Models for Intuitionistic Logic

We present a technique to extend a Kripke structure (for intuitionistic logic) into an elementary extension satisfying some property (cardinality, saturation, etc.) which can be “axiomatized” by a family of sets of sentences, where, most often, many constant symbols occur. To that end, we prove extended theorems of completeness and compactness. Also, a section of the paper is devoted to the back-and-forth construction of isomorphisms between Kripke structures.     

Z-Quantale及其范畴性质

本文把集系统的概念应用到Quantale理论中,作为Quantale的一般化,引入了Z-quantale的概念,研究了Z-quantale及其范畴的若干性质.主要结果有:证明了Z-quantale范畴是序半群范畴的反射子范畴,凝聚Z-quantale范畴是Z-quantale范畴的余反射子范畴.讨论了Z-quantale范畴中的投射对象,证明了Z-quantale A是E-投射的当且仅当它是稳定Z-连续的.     

Sémantique de type Kripke d'un système logique basé sur un ensemble ordonné fini

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems the use of a set of constants constitutes a fundamental tool. We have introduced in [8] a logic system called without this kind of constants but limited to the case that T is a finite poset. We have proved a completeness result for this system w.r.t. an algebraic semantics. We introduce in this paper a Kripke‐style semantics for a subsystem of for which there existes a deduction theorem. The set of “possible worldsr is enriched by a family of functions indexed by the elements of T and satisfying some conditions. We prove a completeness result for system with respect to this Kripke semantics and define a finite Kripke structure that characterizes the propositional fragment of logic . We introduce a reational semantics (found by E. Orlowska) which has the advantage to allow an interpretation of the propositionnal logic using only binary relations. We treat also the computational complexity of the satisfiability problem of the propositional fragment of logic .     

Cut-Elimination Theorem for the Logic of Constant Domains

The logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that (→) and (?–) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD . In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD , saying that all “cuts” except some special forms can be eliminated from a proof in LD . From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD : fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD . Mathematics Subject Classification : 03B55. 03F05.     

Geometric topological completions with universal final lifts

In this paper necessary and sufficient conditions are given on a concrete category over a category B so that it can be densely embedded (over B) into a geometric topological category E that admits certain universal final lifts. These conditions, as well as the class of universal final lifts, depend upon an a priori given full subcategory Δ of B. For example, E may have, depending upon Δ and B, universal coproducts or quotients or colimits. For appropriate Δ's, if B is cartesian closed then so is E.     

量词模态逻辑的代数语义学（Ⅲ）──关于不含Barcan公式的正规模态系统的情形

本文首先讨论嵌套论域语义的相应代数语义并由Ｈｕｇｈｅｓ和Ｃｒｅｓｓｗｅｌｌ在［５］中建立的关于具有嵌套论域的正规量词模态系统的关系语义完全性定理推出其相应的代数语义完全性定理：然后对于具有任意可变论域语义的正规系统，我们用Ｈｅｎｋｉｎ方法给出其关于狭义Ｋｒｉｐｋｅ语义的关系语义完全性定理，由此通过将关系语义转化为代数语义从而亦推得其代数语义完全性定理。     

Computational adequacy for recursive types in models of intuitionistic set theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.     

Simultaneous representations in categories of algebras

Hedrlin and Pultr proved that every small category is isomorphic to a full subcategory of the category Alg (Δ) of all algebras of type Δ whenever the sum ∑Δ of their arities satisfies ∑Δ>2. This article deals with simultaneous representation in categories of algebras, a generalization of the related question: given a subcategory k′ of a small category k, when does there exist an extension Δ′ of a type Δ with ∑Δ?2 such that k′ is a full subcategory of Alg(Δ′) while the Alg (Δ)-redacts of algebras representing k′ determine a category isomorphic to k? We characterize simultaneous representability by algebras and their reducts completely, and show that it is closely related to Isbell's dominion. A consequence of the main result states that algebraically representable pairs (k′, k) of one-object categories k′ k are exactly those for which k′ coincides with its dominion in k, and provides an alternative characterization of the dominion. Simultaneous representability by partial algebras is not subject to any such restriction.     

Weak Arithmetics and Kripke Models

In the first section of this paper we show that i Π₁ ≡ W⌝⌝lΠ₁ and that a Kripke model which decides bounded formulas forces iΠ₁ if and only if the union of the worlds in any (complete) path in it satisflies IΠ₁. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ₁. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Π_m‐formulas in a linear Kripke model deciding Δ₀‐formulas it is necessary and sufficient that the model be Σ_m‐elementary. This implies that if a linear Kripke model forces PEM_prenex, then it forces PEM. We also show that, for each n ≥ 1, iΦ_n does not prove ℋ︁(IΠ_n's are Burr's fragments of HA.     

Infinitary first-order categorical logic

We present a unified categorical treatment of completeness theorems for several classical and intuitionistic infinitary logics with a proposed axiomatization. This provides new completeness theorems and subsumes previous ones by Gödel, Kripke, Beth, Karp and Joyal. As an application we prove, using large cardinals assumptions, the disjunction and existence properties for infinitary intuitionistic first-order logics.     

A semantic hierarchy for intuitionistic logic

Brouwer’s views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic, and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy.     

On intuitionistic modal and tense logics and their classical companion logics: Topological semantics and bisimulations

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order (or partial-order) with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics.     

Classification of dicoverings

The dicoverings of a “well pointed” d-space are classified as quotients of the universal dicovering space under congruence relations. We prove that the subcategory of d-spaces generated by the subcategory of directed cubes is equal to the category generated by the interval and the directed interval. Similarly, the category of topological spaces generated by simplices may be generated by the interval.     

Induced and higher-dimensional stable independence

We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory to a category as a whole, and examine a number of applications to categories of groups and modules, extending results of [16]. We prove, too, that under the hypotheses of [11], a stable independence notion immediately yields higher-dimensional independence as in [26].     

An Intuitionistic Axiomatisation of Real Closed Fields

We give an intuitionistic axiomatisation of real closed fields which has the constructive reals as a model. The main result is that this axiomatisation together with just the decidability of the order relation gives the classical theory of real closed fields. To establish this we rely on the quantifier elimination theorem for real closed fields due to Tarski, and a conservation theorem of classical logic over intuitionistic logic for geometric theories.     

A Cartan-Eilenberg approach to homotopical algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.     

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg. This project is partially supported by China Postdoctoral Science Foundation (No.s 20070420125 and 200801230). The author also gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong.