Higher-order Intuitionistic Logic Research Articles

Kripke semantics for higher-order type theory applied to constraint logic programming languages

We define a Kripke semantics for Intuitionistic Higher-Order Logic with constraints formulated within Church's Theory of Types via the addition of a new constraint base type. We then define an executable fragment, hoHH(C), of the logic: a higher-order logic programming language with typed λ-abstraction, implication and universal quantification in goals and constraints, and give a modified model theory for this fragment. Both formal systems are shown sound and complete for their respective semantics.We also solve the impredicativity problem in λProlog semantics, namely how to give a definition of truth without appealing to induction on subformula structure. In the last section we give a simple semantics-based conservative extension proof that the language hoHH(C) satisfies a uniformity property along the lines of [39].

On fixed-point theorems in synthetic computability

Lawvere's fixed point theorem captures the essence of diagonalization arguments. Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds. We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom, which states that there are countably many countable subsets of $\mathbb{N}$. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory. We show that every countably generated $\omega$-chain complete pointed partial order ($\omega$cppo) is countable, and that countably generated $\omega$cppos are closed under countable products. Therefore, Lawvere's fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated $\omega$cppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains ($\omega$cppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an $\omega$cpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere's fixed point theorem for multi-valued endomaps.

A realizability interpretation of Church's simple theory of types

We give a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda–calculus with pairing and case-construct. The purpose of this interpretation is to provide a foundation for the extraction of verified programs from formal proofs as an alternative to type-theoretic systems. The advantages of our approach are that (a) induction and coinduction are not restricted to the strictly positive case, (b) abstract mathematical structures and results may be imported, (c) the formalization is technically simpler than in other systems, for example, regarding the definition of realizability, which is a simple syntactical substitution, and the treatment of nested and simultaneous (co)inductive definitions.

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (8x9y A(x,y)) ) 9w 8xA(x,wx) and a bound

A Topos-Theoretic Approach to Counterfactual Logic

Topoi are known to be categories with extra properties that make them much alike the category of Sets. In a Topoi it is possible to define adequate notions of membership, elements and subobjects, power “sets”, and finally, every Topoi has an internal logic able to justify any reasoning carried inside it. Most of the cases, this logic is not Classical (Boolean). The general logic for the Topoi is Intuitionistic Higher-order Logic. Topoi have their linguistic counter-part provided by Local Set Theories (LST). There is a deductive apparatus, in the style of Sequent Calculus, able to justify logical consequence inside any LST. Counterfactuals are subtle conditionals largely studied by the philosophical and logic community. Since Lewis, counterfactual have a uniform semantics provided by means of Neighborhood systems on top a possible world style semantics. In this article, taking into account the fundamental theorem on Topoi, we define, by means of the internal logic of Graphs, Lewis counterfactuals and show how to use the LST deductive apparatus to prove properties on counterfactual logic. This article can be also used as an initial step towards the definition of deductive systems for counterfactual logic, taken in an alternative way the ones already existent in the logic literature.

A framework for checking proofs naturally

We propose a logical framework, called Natural Framework (NF), which supports formal reasoning about computation and logic (CAL) on a computer. NF is based on a theory of Judgments and Derivations. NF is designed by observing how working mathematical theories are created and developed. Our observation is that the notions of judgments and derivations are the two fundamental notions used in any mathematical activity. We have therefore developed a theory of judgments and derivations and designed a framework in which the theory provides a uniform and common play ground on which various mathematical theories can be defined as derivation games and can be played, namely, can write and check proofs. NF is equipped with a higher-order intuitionistic logic and derivations (proofs) are described following Gentzen's natural deduction style. NF is part of an interactive computer environment CAL and it is also referred to as NF/CAL. CAL is written in Emacs Lisp and it is run within a special buffer of the Emacs editor. CAL consists of user interface, a general purpose parser and a checker for checking proofs of NF derivation games. NF/CAL system has been successfully used as an education system for teaching CAL for undergraduate students for about 8 years. We will give an overview of the NF/CAL system both from theoretical and practical sides.

Completeness and Cut-elimination in the Intuitionistic Theory of Types

In this paper we define a model theory and give a semantic proof of cut-elimination for ICTT, an intuitionistic formulation of Church's theory of types defined by Miller et al. and the basis for the λProlog programming language. Our approach, extending techniques of Takahashi and Andrews and tableaux machinery of Fitting, Smullyan, Nerode and Shore, is to prove a completeness theorem for the cut-free fragment and show semantically that cut is a derived rule. This allows us to generalize a result of Takahashi and Schutte on extending partial truth valuations in impredicative systems. We extend Andrews' notion of Hintikka sets to intuitionistic higher-order logic in a way that also defines tableau-provability for intuitionistic type theory. In addition to giving a completeness theorem without using cut we then show, using cut, how to establish completeness of more conventional term models. These models give a declarative semantics for the logic underlying the λProlog programming language.

The H ASC ASL prologue: Categorical syntax and semantics of the partial [formula omitted]-calculus

We develop the semantic foundations of the specification language H ASC ASL, which combines algebraic specification and functional programming on the basis of Moggi's partial λ -calculus. Generalizing Lambek's classical equivalence between the simply typed λ -calculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc's) and partial λ -theories. Building on these results, we define (set-theoretic) notions of intensional Henkin model and syntactic λ -algebra for Moggi's partial λ -calculus. These models are shown to be equivalent to the originally described categorical models in pccc's via the global element construction. The semantics of H ASC ASL is defined in terms of syntactic λ -algebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higher-order logic of partial functions, which by our results is the logic of pccc's with equality). Finally, we give some applications of the model-theoretic equivalence result to the semantics of H ASC ASL and its relation to first-order C ASL.

A Theory of Local Set Queries

A relational database can be considered as a finite structure for a finite relational signature in first-order logic. This allows finite model theory to be used for investigating database theory. However, database research in the last decade has shifted its emphasis from the relational datamodel to higher-order datamodels including object oriented databases, object-relational databases, and most recently the model of semi-structured data and XML. A common characteristic of these new datamodels of interest is their richness in structure. In order to find a suitable generalisation of database theory that can cope with this structural richness a suitable logical ground is needed. In this paper local set theory, which can be considered as a version of higher-order intuitionistic logic, is investigated for this purpose. The central notions of database schema and instance for a slighly restricted class of higher-order datamodels are re-defined in local languages using a finite set interpretation. Terms of “set type” define simple queries, which provide the analogue of relational calculus queries in the relational model. Then the work on computable queries can be generalised as well. An analogue of the reflective relational machine model called LQ machine is defined and proven to be complete, i.e., able to express exactly the computable queries over local languages. Finally, fixed-point operators are added to simple queries. However, adding fixed-points will not increase expressiveness.

Computable Queries for Object Oriented Databases

A relational database can be considered as a finite structure for a finite relational signature in first-order logic, i.e., there are no function symbols. Interpreting the logic over this signature in such structures allows the expressiveness and complexity of queries to be studied in detail. This is the starting point for finite model theory which has proven to be a viable tool to study relational database theory. In particular, it is known that computable queries expressed as isomorphism-preserving partial recursive functions can be formalized by Reflective Relational Machines. These are extended Turing Machines with an additional relational store, a query tape and the facility to evaluate the query on the tape against the database in the store in a single step.In this paper we start to generalize the theory to post-relational databases. We first consider the case of having set-based complex values and references such that the semantics can still be expressed in finite sets. Following the approach that object oriented databases in general including those, where the underlying type systems does no longer allow the semantics defined by sets, can be expressed as theories in higher-order intuitionistic logic, we use such a logic instead of first-order logic. However, as we are not yet exploiting the full power of such logics, we can interpret the logic in the category FINSET of finite sets, i.e., again in a structure defined by a database.Having done this the definition of computable queries and the model of Reflective Relational Machines carry over easily. We can show that the new model of Reflective Object Machines guarantees completeness, i.e., all computable queries can be expressed by the model.

A General Notion of Realizability

AbstractWe present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We show how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a topos, i.e., a model of impredicative intuitionistic higher-order logic.

A General Notion of Realizability

We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We show how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a topos, i.e., a model of impredicative intuitionistic higher-order logic.

Implementing tactics and tacticals in a higher-order logic programming language

We argue that a logic programming language with a higher-order intuitionistic logic as its foundation can be used both to naturally specify and implement tactic-style theorem provers. The language extends traditional logic programming languages by replacing first-order terms with simply-typed λ-terms, replacing first-order unification with higher-order unification, and allowing implication and universal quantification in queries and the bodies of clauses. Inference rules for a variety of inference systems can be naturally specified in this language. The higher-order features of the language contribute to a concise specification of provisos concerning variable occurrences in formulas and the discharge of assumptions present in many inference systems. Tactics and tacticals, which provide a framework for high-level control over search for proofs, can be directly and naturally implemented in the extended language. This framework serves as a starting point for implementing theorem provers and proof systems that can integrate many diversified operations on formulas and proofs for a variety of logics. We present an extensive set of examples that have been implemented in the higher-order logic programming language λProlog.