Showing papers in "Journal of Logic and Computation in 1992"

Abstract Interpretation Frameworks

Abstract: Interpretation Frameworks Patrick Cousot LIENS, Ecole Normale Superieure 45, rue d’Ulm 75230 Paris cedex 05 (France) cousot@dmi.ens.fr Radhia Cousot LIX, Ecole Polytechnique 91128 Palaiseau cedex (France) radhia@polytechnique.fr

Logic Programming with Focusing Proofs in Linear Logic

Abstract: The deep symmetry of linear logic [18] makes it suitable for providing abstract models of computation, free from implementation details which are, by nature, oriented and nonsymmetrical. I propose here one such model, in the area of logic programming, where the basic computational principle is Computation = Proof search Proofs considered here are those of the Gentzen style sequent calculus for linear logic. However, proofs in this system may be redundant, in that two proofs can be syntactically different although identical up to some irrelevant reordering or simplification of the applications of the inference rules. This leads to an untractable proof search where the search procedure is forced to make costly choices which turn out to be irrelevant. To overcome this problem, a subclass of proofs, called the 'focusing' proofs, which is both complete (any derivable formula in linear logic has a focusing proof) and tractable (many irrelevant choices in the search are eliminated when aimed at focusing proofs) is identified. The main constraint underlying the specification of focusing proofs has been to preserve the symmetry of linear logic, which is its most salient feature. In particular, dual connectives have dual properties with respect to focusing proofs. Then, a programming language, called LinLog, consisting of a fragment of linear logic, in which focusing proofs have a more compact form, is presented. Linlog deals with formulae which have a syntax similar to that of the definite clauses and goals of Horn logic, but the crucial difference here is that it allows clauses with multiple atoms in the head, connected by the 'par' (multiplicative disjunction). It is then shown that the syntactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.

Using the Universal Modality: Gains and Questions·

Abstract: The paper investigates a simple and natural enrichment of the usual modal language !£ = !£

Modal Theorem Proving: An Equational Viewpoint

Abstract: We propose a new method for automated theorem proving in first order modal logic. Essentially, the method consists in a translation of modal logic into a specially designed typed first order logic cal led Path Logic, such that classical modal systems (first order Q, T, 04, S4, S5) can be characterized by sets of equations. The question of modal theorem proving then amounts to classical theorem proving in some equational theories. Different methods can be investigated and in this paper we cons ider Resolution. We may use Resolution with Paramodulation, or a combination of Resolution and Rewriting techniques. In both cases, known results provide "free of charge" a framework immediately applicable to Path Logic, with completeness theorems. Considering efficiency, the Rewriting method seems better and we present here in details its application to Path Logic. In particular we show how it is possible to define a special kind of skolemisation and design a unification algorithm which insures that two clauses will always have a finite set of resolvents.

An Alternative Order for the Failures Model

Abstract: The failures-divergences model for CSP is usually presented with the refinement order being the one used for fixed points in the semantics of recursion. The requirement that this order be complete means that the model needs a compactness axiom closely related to an assumption of finite nondeterminism. We show that a second and stronger order exists which does not need compactness to make it complete, but which gives exactly the same least fixed points as the refinement order. The new order allows us to prove some new results about the semantics, and to justify versions of recursion induction. In pursuit of this last topic we develop various topologies over the model.

Proof Nets for Lambek Calculus

Abstract: The proof nets of linear logic are adapted to the non-commutative Lambek calculus. A different criterion for soundness of proof nets is given, which gives rise to new algorithms for proof search. The order sensitiveness of the Lambek calculus is reflected by the planarity condition on proof nets; this gives rise to a new non-provability check: balance.

A Module System for a Programming Language Based on the LF Logical Framework

Abstract: : We describe a module system for Elf, a logic programming language based on the LF logical framework. The static part of module calculus addresses name-space management and structured presentation of deductive systems. The dynamic part addresses search-space management and modularization of logic programs.

Bounded Fixed-Point Iteration

Abstract: In the context of abstract interpretation for languages without higher-order features we study the number of times a functional need to be unfolded in order to give the least fixed point. For the cases of total or monotone functions we obtain an exponential bound and in the case of strict and additive (or distributive) functions we obtain a quadratic bound. These bounds are shown to be tight in that sufficiently long chains of functions can be shown to exist. Specializing the case of strict and additive functions to functionals of a form that would correspond to iterative programs we show that a linear bound is tight. This is related to several analyses studied in the literature (including strictness analysis).

Assumption-free semantics for ordered logic programs: on the relationship between well-founded and stable partial models

Abstract: Ordered logic programming is an extension of logic programming that includes, besides classical inference mechanisms, object-oriented abstractions and amenities for nonmonotonic reasoning. Ordered logic programs are partially-ordered sets of 'traditional' logic programs where negation may also occur in the rule heads. The central issue of this paper is the definition of a new unifying semantics for ordered logic programs, called assumption-free semantics, capable of capturing different interesting semantics such as the well-founded and stable (partial model) semantics. It turns out that every ordered logic program possesses exactly one minimal assumption-free partial model which we call the well-founded partial model and one or more maximal assumption-free partial models called stable partial models. This stable model semantics can be viewed as taking the best of the previous approaches for ordered logic programs while keeping their (common) underlying intuition. We discuss the relationship between stable and well-founded partial models, the main result being that the intersection of all stable partial models is exactly the well-founded partial model in all cases but a special type of ordered logic programs. We also define the notion of prooftree and discuss the soundness and completeness results. The new concepts for ordered logic programs are proper generalizations of the corresponding concepts for classical logic programs, thus giving a new unifying definition for the traditional notions of well-founded and stable (partial) models. Hence, all results obtained for ordered programs (e.g. the relationship between stable and well-founded partial models) can be mapped to the more restricted class of traditional logic programs.

A Constructive Presentation for the Modal Connective of Necessity (

Abstract: This work provides a constructive presentation of modal logics in natural deduction style. The modal connective • is presented in a constructive form, which can be considered as an operational semantics for it. Modal connectives have been recognized as intentional connectives for a long time, but modal logicians have insisted in using extensional techniques to deal with them. In this paper, the modal connective is presented as a higherorder connective defined on top of the object level logical connectives. The simplest version of our system of modal logic with classical negation coincides with the classical modal logic K. The most important modal logics have an elegant presentation in this system. T, S4, S5, D, D4, D5 are presented, without any side effect condition on the structure of the deductions. Most presentations of intuitionistic modal logics fail in giving an intuitionistic interpretation to the modal connective. In general, such interpretations are based on some alien element (for instance, the accessibility relation), which are by no means intuitionistic. In the system described here it is not only possible to present an intuitionistic interpretation of the modal connective D, which takes into account only deducibility issues, but also to give a constructive natural deduction presentation for the D.

Relational Reversal of Abstract Interpretation

Abstract: Many semantic analyses of functional languages have been developed using the Cousots' abstract interpretation framework. Some operate on abstract values representing the past history of the computation, and are therefore called forwards analyses. Others propagate abstract contexts representing the future of the computation, and are called backwards analyses. Each form of analysis brings its own insights, and has the potential to influence the other form. For example, it may be very easy to see how to analyse a particular programming language construct in one direction, but not in the direction needed for a particular analysis. Potentially, one might be able to draw on the given analysis to aid in the design of a corresponding reversal. This is the topic of this paper. We show how to reverse any given analysis (forwards or backwards), obtaining a relational reversal which is equivalent to the original. This allows the accuracy of two analyses originally defined in opposite directions to be compared directly. Furthermore, we demonstrate that the relational reversal may be safely approximated by a (more-efficient but slighly less accurate) locally-relational analysis. That is, relational and non-relational reversals may be combined.