Negation-as-Failure in the Base-extension Semantics for Intuitionistic Propositional Logic

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS

Definite Formulae, Negation-as-Failure, and the Base-extension Semantics of Intuitionistic Propositional Logic

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of the latter as provided by uniform proof-search -- the proof-theoretic foundation of logic programming (LP) -- to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS.Comment: submitte

Cut formulae and logic programming

In this paper we present a mechanism to define names for proof-witnesses of formulae and thus to use Gentzen's cut-rule in logic programming. We consider a program to be a set of logical formulae together with a list of such definitions. Occurrences of the defined names guide the proof-search by indicating when an instance of the cut-rule should be attempted. By using the cut-rule there are proofs that can be made dramatically shorter. We explain how this idea of using the cut-rule can be applied to the logic of hereditary Harrop formulae.Junta Nacional de Investigação Científica e Tecnológica (JNICT).União Europeia (UE) - Programa ESPRIT - grant BRA 7232 GENTZEN

Proof-theoretic investigations into integrated logical and functional programming

This thesis is a proof-theoretic investigation of logic programming based on hereditary Harrop logic (as in lambdaProlog). After studying various proof systems for the first-order hereditary Harrop logic, we define the proof-theoretic semantics of a logic LFPL, intended as the basis of logic programming with functions, which extends higher-order hereditary Harrop logic by providing definition mechanisms for functions in such a way that the logical specification of the function rather than the function may be used in proof search. In Chap. 3, we define, for the first-order hereditary Harrop fragment of LJ, the class of uniform linear focused (ULF) proofs (suitable for goal-directed search with backchaining and unification) and show that the ULF-proofs are in 1-1 correspondence with the expanded normal deductions, in Prawitz's sense. We give a system of proof-term annotations for LJ-proofs (where proof-terms uniquely represent proofs). We define a rewriting system on proof-terms (where rules represent a subset of Kleene's permutations in LJ) and show that: its irreducible proof- terms are those representing ULF-proofs; it is weakly normalising. We also show that the composition of Prawitz's mappings between LJ and NJ, restricted to ULF-proofs, is the identity. We take the view of logic programming where: a program P is a set of formulae; a goal G is a formula; and the different means of achieving G w.r.t. P correspond to the expanded normal deductions of G from the assumptions in P (rather than the traditional view, whereby the different means of goal-achievement correspond to the different answer substitutions). LFPL is defined in Chap. 4, by means of a sequent calculus. As in LeFun, it extends logic programming with functions and provides mechanisms for defining names for functions, maintaining proof search as the computation mechanism (contrary to languages such as ALF, Babel, Curry and Escher, based on equational logic, where the computation mechanism is some form of rewriting). LFPL also allows definitions for declaring logical properties of functions, called definitions of dependent type. Such definitions are of the form: (f,x) =def(A, w) : EX:RF, where f is a name for A and x is a name for w, a proof-term witnessing that the formula [A/x]F holds (i.e. A meets the specification Ex:rF). When searching for proofs, it may suffice to use the formula [A/x]F rather than A itself. We present an interpretation of LFPL into NNlambdanorm, a natural deduction system for hereditary Harrop logic with lambda-terms. The means of goal-achievement in LFPL are interpreted in NNlambdanorm essentially by cut-elimination, followed by an interpretation of cut-free sequent calculus proofs as normal deductions. We show that the use of definitions of dependent type may speed up proof search because the equivalent proofs using no such definitions may be much longer and because normalisation may be done lazily, since not all parts of the proof need to be exhibited. We sketch two methods for implementing LFPL, based on goal-directed proof search, differing in the mechanism for selecting definitions of dependent type on which to backchain. We discuss techniques for handling the redundancy arising from the equivalence of each proof using such a definition to one using no such definitions

Proof search in constructive logics

We present an overview of some sequent calculi organised not for "theorem-proving" but for proof search, where the proofs themselves (and the avoidance of known proofs on backtracking) are objects of interest. The main calculus discussed is that of Herbelin [1994] for intuitionistic logic, which extends methods used in hereditary Harrop logic programming; we give a brief discussion of similar calculi for other logics. We also point out to some related work on permutations in intuitionistic Gentzen sequent calculi that clarifies the relationship between such calculi and natural deduction.Centro de Matemática da Universidade do Minho (CMAT).União Europeia (UE) - Programa ESPRIT - BRA 7232 Gentzen

Type-driven natural language analysis

The purpose of this thesis is in showing how recent developments in logic programming can be exploited to encode in a computational environment the features of certain linguistic theories. We are in this way able to make available for the purpose of natural language processing sophisticated capabilities of linguistic analysis directly justified by well developed grammatical frameworks. More specifically, we exploit hypothetical reasoning, recently proposed as one of the possible directions to widen logic programming, to account for the syntax of filler-gap dependencies along the lines of linguistic theories such as Generalized Phrase Structure Grammar and Categorial Grammar. Moreover, we make use, for the purpose of semantic analysis of the same kind of phenomena, of another recently proposed extension, interestingly related to the previous one, namely the idea of replacing first-order terms with the more expressive λ-terms of λ-Calculus

Proof search issues in some non-classical logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed.This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin’s cutfree LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1–1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called ‘permutation-free’ calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a ‘permutation-free’ calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to besound and complete with respect to a proof-theoretic semantics and (weak) cutelimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter