A proof procedure for the logic of hereditary Harrop formulas (original) (raw)

A Proof-theoretic Analysis of Goal-directed Provability

Journal of Logic and Computation, 1994

Uniform proofs have been presented as a basis for logic programming, and it is known that by restricting the class of formulae it is possible to guarantee that uniform proofs are complete with respect to provability in intuitionistic logic. In this paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform proofs arise naturally as a normal form for proofs in rst-order intuitionistic sequent calculus. Next we show that the class of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae from uniform proofs in two di erent ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proofs are guaranteed to be complete. Finally we brie y discuss some possibilities for a slightly more general approach using intermediate and in nitary logics.

Goal-Directed Proof Search in Multiple-Conclusions Intuitionistic Logic

2000

A key property in the definition of logic programming languages is the completeness of goal-directed proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (single-conclusioned) sequent calculus LJ, but has subsequently been adapted to multiple-conclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goal-directed proofs for a multiple-conclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both single-conclusioned and multiple-conclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multiple-conclusioned system differs from that for the single-conclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contraction-free fragments of the logic.

Uniform proofs as a foundation for logic programming

Annals of Pure and Applied Logic, 1991

A proof-t heoret ic characterization of logical languages that form suit able bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed.

A connection based proof method for intuitionistic logic

Lecture Notes in Computer Science, 1995

We present a proof method for intuitionistic logic based on Wallen's matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof search the computed first-order and intuitionistic substitutions are used to simultaneously construct a sequent proof which is more human oriented than the matrix proof. This allows to use our method within interactive proof environments. Furthermore we can consider local substitutions instead of global ones and treat substitutions occurring in different branches of the sequent proof independently. This reduces the number of extra copies of formulae to be considered.

Bottom-up abstract interpretation of logic programs

Theoretical Computer Science, 1994

λ-Prolog is a logic programming language that extends Prolog by incorporating notions of higher-order functions, λ-terms, higher-order unification, polymorphic types, and mechanisms for building modules and secure abstract data types. These new features are provided in a principled fashion by extending the classical first-order theory of Horn clauses to the intuitionistic higher-order theory of hereditary Harrop formulas. The justification for considering this extension a satisfactory logic programming language is provided through the proof-theoretic notion of a uniform proof. The correspondence between each extension to Prolog and the new features in the stronger logical theory is discussed. Also discussed are various aspects of an experimental implementation of λ-Prolog.

On Goal-Directed Proofs in Multiple-Conclusioned Intuitionistic Logic

A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (single-conclusioned) sequent calculus LJ, but has subsequently been adapted to multiple-conclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goal-directed proofs for a multipleconclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both single-conclusioned and multiple-conclusioned systems (although the latter are less well known than the former). In this paper we show that the language obtained for the multiple-conclusioned system differs from that for the single-conclusioned case, and discuss the consequences of this result.

THE JUDGEMENT CALCULUS FOR INTUITIONISTIC LINEAR LOGIC: PROOF THEORY AND SEMANTICS

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1992

In this paper we propose a new set of rules for a judgement calculus, i.e. a typed lambda calculus, based on Intuitionistic Linear Logic; these rules ease the problem of definiag a suitable mathematical semantics. A proof of the canonical form theorem for this new system is given: it assures, beside the consistency of the calculus, the termination of the evaluation process of every well-typed element. The defmition of the mathematical semantics and a completeness theorem, that turns out to be a representation theorem, follow. This semantics is the basis to obtain a semantics for the evaluation process of every 1991 MSC: 03B20, O3B40

Structural Properties of Logic Programs

1990

Miller has shown that disjunctions are not necessary in a large fragment of hereditary Harrop formulae, a class of formulae which properly includes Horn clauses. In this paper we extend this result to include existential quanti cations, so that for each program ...

A Strong Logic Programming View for Static Embedded Implications

Lecture Notes in Computer Science, 1999

A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (programs and goals) of the underlying logic L, provided that: firstly, any program P (viewed as a L-theory) has a canonical model MP which is initial in the category of all its L-models; secondly, the L-satisfaction of a goal G in MP is equivalent to the L-derivability of G from P , and finally, there exists an effective (computable) proof-subcalculus of the L-calculus which works out for derivation of goals from programs. In this sense, Horn clauses constitute a strong (first-order) logic programming language. Following the methodology suggested in [15] for designing logic programming languages, an extension of Horn clauses should be made by extending its underlying first-order logic to a richer logic which supports a strong axiomatization of the extended logic programming language. A well-known approach for extending Horn clauses with embedded implications is the static scope programming language presented in [8]. In this paper we show that such language can be seen as a strong FO ⊃ logic programming language, where FO ⊃ is a very natural extension of first-order logic with intuitionistic implication. That is, we present a new characterization of the language in [8] which shows that Horn clauses extended with embedded implications, viewed as FO ⊃-theories, preserves all the attractive mathematical and computational properties that Horn clauses satisfy as first-order-theories.