Showing papers on "Linear logic published in 1996"

Dual Intuitionistic Linear Logic

Abstract: We present a new intuitionistic linear logic, Dual Intuitionistic Linear Logic, designed to reflect the motivation of exponentials as translations of intuitionistic types, and provide it with a term calculus, proving associated standard type-theoretic results. We give a sound and complete categorical semantics for the type-system, and consider the relationship of the new type-theory to the more familiar presentation found for example in [4].

A linear logical framework

Abstract: We present the linear type theory LLF as the formal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination.

Linear logic, monads and the lambda calculus

Abstract: Models of intuitionistic linear logic also provide models of Moggi's computational metalanguage. We use the adjoint presentation of these models and the associated adjoint calculus to show that three translations, due mainly to Moggi, of the lambda calculus into the computational metalanguage (direct, call-by-name and call-by-value) correspond exactly to three translations, due mainly to Girard, of intuitionistic logic into intuitionistic linear logic. We also consider extending these results to languages with recursion.

Coordination programming: mechanisms, models and semantics

Abstract: Part 1 Coordination models: gamma and the chemical reaction model - ten years after, J.-P. Banatre and D. Le Metayer coordination in LO, J.-M. Andreoli truth and action osmosis (the TAO computation model), A. Porto and V.T. Vasconcelos type inference and subtyping for higher-order generative communication, L. Dami. Part 2 Semantics: temporal semantics for gamma, M. Reynolds a programme logic for gamma, S.J. Gay and C.L. Hankin schedules for multiset transformer programmes, M. Chaudron and E. de Jong composed reduction systems, D. Sands an alternative semantics for the parallel operator of the calculas of gamma programmes, P. Ciancarini et al a linear logic view of gamma style computations as proof searches, P. Bruscoli and A. Guglielmi. Part 3 Implementations, application: specifying a reflective and distributed implemenation of LO in higher order gamma, M. Bourgois practical implications of reflection for coordination languages, M. Mourgois Gammalog - a coordination language based on gamma and Godel, P. Ciancarini et al coordination of distributed and parallel programmes in ConCoord, A.A. Holzbacher gamma, chromatic typing and vegetation, H. McEvoy.

Reference counting as a computational interpretation of linear logic

Abstract: We develop formal methods for reasoning about memory usage at a level of abstraction suitable for establishing or refuting claims about the potential applications of linear logic for static analysis. In particular, we demonstrate a precise relationship between type correctness for a language based on linear logic and the correctness of a reference-counting interpretation of the primitives that the language draws from the rules for the `of course' operation. Our semantics is `low-level' enough to express sharing and copying while still being `highlevel' enough to abstract away from details of memory layout. This enables the formulation and proof of a result describing the possible run-time reference counts of values of linear type.

Efficient Resource Management for Linear Logic Proof Search

Abstract: The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses (Prolog) and hereditary Harrop formulas (λProlog) Among these, the problem of efficiently managing the linear context when solving a goal is of crucial importance for the use of these systems in non-trivial applications This paper studies this problem in the case of Lolli [6] (though its results have application to other systems) We first give a proof-theoretic presentation of the operational semantics of this language as a resolution calculus We then present a series of resource management systems designed to eliminate the non-determinism in the distribution of linear formulas that undermines the efficiency of a direct implementation of this system

Bounded contraction and Gentzen-style formulation of Łukasiewicz logics

Abstract: In this paper, we consider multiplicative-additive fragments of affine propositional classical linear logic extended with n-contraction. To be specific, n-contraction (n ⩾ 2) is a version of the contraction rule where (n+ 1) occurrences of a formula may be contracted to n occurrences. We show that expansions of the linear models for (n + 1)- valued Łukasiewicz logic are models for the multiplicative-additive classical linear logic, its affine version and their extensions with n-contraction. We prove the finite axiomatizability for the classes of finite models, as well as for the class of infinite linear models based on the set of rational numbers in the interval [0, 1]. The axiomatizations obtained in a Gentzen-style formulation are equivalent to finite and infinite-valued Łukasiewicz logics.

The Undecidability of Second Order Linear Logic without Exponentials

Abstract: Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics.

Cut-Elimination for Full Intuitionistic Linear Logic

Abstract: We describe in full detail a solution to the problem of proving the cut elimination theorem for FILL, a variant of (multiplicative and exponential-free) Linear Logic introduced by Hyland and de Paiva. Hyland and de Paiva's work used a term assignment system to describe FILL and barely sketched the proof of cut elimination. In this paper, as well as correcting a small mistake in their paper and extending the system to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut elimination theorem. The formal system is based on a notion of dependence between formulae within a given proof and seems of independent interest. The procedure for cut elimination applies to (classical) multiplicative Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut elimination proofs, is a little involved and we have not seen it published anywhere.

Decidability problems for the prenex fragment of intuitionistic logic

Abstract: We develop a constraint-based technique which allows one to prove decidability and complexity results for sequent calculi. Specifically, we study decidability problems for the prenex fragment of intuitionistic logic. We introduce an analogue of Skolemization for intuitionistic logic with equality, prove PSPACE-completeness of two fragments of intuitionistic logic with and without equality and some other results. In the proofs, we use a combination of techniques of constraint satisfaction, loop-free sequent systems of intuitionistic logic and properties of simultaneous rigid E-unification.

! and ? – Storage as tensorial strength

Abstract: We continue our study of the negation-free structure of multiplicative linear logic, as represented by the structure of weakly distributive categories, to consider the ‘exponentials’! and ? in the weakly distributive context. In addition to the usual triple and cotriple structure that one would expect on each of the two operators, there must be some connection between them to replace the de Morgan relationship found in the linear logic context. This turns out to be the notion of tensorial strength. We analyze coherence for this situation, using a modification of the usual nets due to Danos, which is a form suitable for linear logic with exponentials but without negation.

A compilation-chart method for linear categorial deduction

Abstract: Recent work in categorial grammar has seen proposals for a wide range of systems, differing in their 'resource sensitivity' and hence, implicitly, their underlying notion of 'linguistic structure'. A common framework for parsing such systems is emerging, whereby some method of linear logic theorem proving is used in combination with a system of labelling that ensures that only deductions appropriate to the relevant categorial formalism are allowed. This paper presents a deduction method for implicational linear logic that brings with it the benefit that chart parsing provides for CFG parsing, namely avoiding the need to recompute intermediate results when searching exhaustively for all possible analyses. The method involves compiling possibly higher-order linear formulae to indexed first-order formulae, over which deduction is made using just a single inference rule.

Enhancing Coordination and Modularity Mechanisms for a Language with Objects-as-Multisets

Abstract: We introduce COOLL, a programming language combining the concepts of multiple tuple spaces and objects-as-multisets. COOLL extends Linear Objects (LO), an object oriented language based on the proof theory of Linear Logic (LL), with new mechanisms aiming at improving the ability of expressing coordination and modularity. Our goal is to explore the new mechanisms in the theoretical framework offered by Linear Logic.

Proofs as Computations in Linear Logic.

Abstract: The notions of uniform proof and of resolution represent the foundations of the proof-theoretic characterization of logic programming. The class of Logic Programming Languages nicely captures these concepts for a wide spectrum of logical systems. In the logic programming setting, however, the structure of the formulas, e.g. Horn clauses and hereditary Harrop formulas, plays a crucial role in discriminating between programming and theorem proving. In the paper, and in the framework of the proofs as computations interpretation of linear logic, we present an extension of hereditary Harrop formulas and a corresponding logical system which are the foundations of the logic programming language . The starting point of this study is Forum (Miller, Theoret. Comput. Sci. 165 (1) (1996) 201–232), a presentation of higher-order linear logic in terms of uniform proofs. A subset of its formulas have been isolated and proved to be well-suited to encode descriptions of various programming paradigms.

Directed Virtual Reductions

Abstract: This note defines a new graphical local calculus, directed virtual reductions. It is designed to compute Girard's execution formula EX, an invariant of closed functional evaluation obtained from the “geometry of interaction” interpretation of λ-calculus [5].

A linear conservative extension of zermelo-fraenkel set theory

Abstract: In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF−i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF−. This implies that LZF is a conservative extension of ZF− and therefore the former is consistent relative to the latter.

A Linear Logic Calculus of Objects

Abstract: This paper presents a linear logic programming language, called O−• , that gives a complete account of an object-oriented calculus with inheritance and override. This language is best understood as a logical counterpart the object and record extensions of functional programming that have recently been proposed in the literature. From these proposals, O−• inherits the representation of objects as composite data structures, with attribute and method fields, as well as their interpretation as first-class values. O−• also gives a direct logical modeling of the self-application semantics of method invocation that justifies the view of objects as elements of recursive types. As such, the design of O−• appears interesting, in perspective, as a basis for developing flexible and powerful type systems for logical object-based languages.

An Abstract Machine for Reasoning about Situations, Actions, and Causality

Abstract: Over the last years several new approaches for modeling situations, actions, and causality within a deductive framework were proposed. These new approaches treat the facts about a situation as resources, which are consumed and produced by actions. In this paper we extend one of these approaches, viz.\ an equational logic approach, by reifying actions to become resources as well. Using the concept of a membrane we show how abstractions and hierarchical planning can be modeled in such an equational logic. Moreover, we rigorously prove that the extended equational logic program can be mapped onto the so-called chemical abstract machine. As this machine is a model for parallel processes this may lead to a parallel computational model for reasoning about situations, actions, and causality.