James Lipton

Logic Programming in Tau Categories

Many features of current logic programming languages are not captured by conventional semantics. Their fundamentally non-ground character, and the uniform way in which such languages have been extended to typed domains subject to constraints, suggest that a categorical treatment of constraint domains, of programming syntax and of semantics may be closer in spirit to declarative programming than conventional set theoretic semantics.

A new framework for declarative programming

We propose a new framework for the syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over τ-categories: finite product categories with canonical structure.Constraint information is directly built-in to the notion of signature via categorical syntax. Many-sorted equational are a special case of the formalism which combines features of uniform logic programming languages (moduels and hypothetical implication) with those of constraint logic programming. Using the cannoical structure supplied by τ-categories, we define a diagrammatic generalization of formulas, goals, programs and resolution proofs up to equality (rather than just up to isomorphism).We extend the Kowalski-van Emden fixed point interpretation, a cornerstone of declarative semantics, to an operational, non-ground, categorical semantics based on indexing over sorts and programs.We also introduce a topos-theoretic declarative semantics and show soundness and completeness of resolution proofs and of a sequent calculus over the categorical signature. We conclude with a discussion of semantic perspectives on uniform logic programming.

Some Intuitions Behind Realizability Semantics for Constructive Logic: Tableaux and Läuchli Countermodels

Abstract We use formal semantic analysis based on new constructions to study abstract realizability, introduced by Lauchli in 1970, and expose its algebraic content. We claim realizability so conceived generates semantics-based intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Well-known semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the completeness proofs for these semantics do not generate confidence in the sufficiency of the Heyting Calculus, since we have no reason to believe that every intuitively constructive truth is valid in the formal semantics. Lauchli has proved completeness for a realizability semantics with formal concepts analogous to constructions. We argue in some detail that, in spite of a certain inherent inexactness of the analogy, every intuitively constructive truth is valid in Lauchli semantics, and therefore the Heyting Calculus is powerful enough to prove all constructive truths. Our argument is based on the postulate that a uniformly constructible object must be communicable in spite of imprecision in our language, and that the permutations in Lauchlis semantics represent conceivable imprecision in a language, while allowing a certain amount of freedom in choosing the particular structure of the language. We give a detailed generalization of Lauchlis proof of completeness for the propositional part of the Heyting Calculus, in order to make explicit constructive and algebraic content. In our treatment, we establish several new results about Lauchli models. We show how to extend the sconing and gluing constructions familiar from Kripke and Frame semantics and Topos theory, to Lauchli models, and use them to give an algebraic approach to countermodel construction. In particular, the Lauchli arguments are given without the restriction to the integers, Z, as a group of permutations, which makes much of the coding scheme used in Lauchlis original paper transparent. We also make use of a new propositions-as-types syntax for the Heyting calculus, with limited nondeterminism, in which validity of formulae can be decided without loop-detection.

Indexed Categories and Bottom-Up Semantics of Logic Programs

We propose a categorical framework which formalizes and extends the syntax, operational semantics and declarative model theory of a broad range of logic programming languages. A program is interpreted in an indexed category in such a way that the base category contains all the possible states which can occur during the execution of the program (such as global constraints or type information), while each fiber encodes the logic at each state. We define appropriate notions of categorical resolution and models, and we prove the related correctness and completeness properties.

Higher-order logic programming languages with constraints: a semantics

A Kripke Semantics is defined for a higher-order logic programming language with constraints, based on Churchs Theory of Types and a generic constraint formalism. Our syntactic formal system, hoHH(C) (higher-order hereditary Harrop formulas with constraints), which extends λPrologs logic, is shown sound and complete. A Kripke semantics for equational reasoning in the simply typed lambda-calculus (Kripke Lambda Models) was introduced by Mitchell and Moggi in 1990. Our model theory extends this semantics to include full impredicative higher-order intuitionistic logic, as well as the executable hoHH fragment with typed lambda-abstraction, implication and universal quantification in goals and constraints. This provides a Kripke semantics for the full higher-order hereditarily Harrop logic of λProlog as a special case (with the constraint system chosen to be β,η-conversion).

Encapsulating Data in Logic Programming via Categorial Constraints

We define a framework for writing executable declarative specifications which incorporate categorical constraints on data, Horn Clauses and datatype specification over finite-product categories. We construct a generic extension of a base syntactic category of constraints in which arrows correspond to resolution proofs subject to the specified data constraints.

A Constructive Semantic Approach to Cut Elimination in Type Theories with Axioms

We give a fully constructive semantic proof of cut elimination for intuitionistic type theory with axioms. The problem here, as with the original Takeuti conjecture, is that the impredicativity of the formal system involved makes it impossible to define a semantics along conventional lines, in the absence, a priori, of cut, or to prove completeness by induction on subformula structure. In addition, unlike semantic proofs by Tait, Takahashi, and Andrews of variants of the Takeuti conjecture, our arguments are constructive. Our techniques offer also an easier approach than Girards strong normalization techniques to the problem of extending the cut-elimination result in the presence of axioms. We need only to relativize the Heyting algebras involved in a straightforward way.

Logic Programming in Tabular Allegories

We develop a compilation scheme and categorical abstract machine for execution of logic programs based on allegories, the categorical version of the calculus of relations. Operational and denotational semantics are developed using the same formalism, and query execution is performed using algebraic reasoning. Our work serves two purposes: achieving a formal model of a logic programming compiler and efficient runtime; building the base for incorporating features typical of functional programming in a declarative way, while maintaining 100% compatibility with existing Prolog programs.

Intuitive Counterexamples for Constructive Fallacies

Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are not constructively valid. We argue that the intuitive impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Lauchli realizability models. We introduce a new semantics for constructive reasoning, called relational realizability, which strengthens further the intuitive impact of Lauchli realizability. But, none of these model theories provides countermodels with the compelling impact of classical truth-table countermodels for classically unprovable formulae.

Kripke Semantics for Higher-Order Type Theory Applied to Constraint Logic Programming Languages ☆

Abstract We define a Kripke semantics for Intuitionistic Higher-Order Logic with constraints formulated within Churchs 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] .