An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.

This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and object-oriented programming and some other applications of LL, like semantics of negation in LP, non-monotonic issues in AI planning, etc. Although the overview covers pretty much the state-of-the-art in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resource management properties of the ffl connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The first type is obtained by combining a number of unimodal systems into one multimodal logic. The...

1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10

Linear logic is a resource-aware logic that is based on an analysis of the classical proof rules of contraction (copying) and weakening (throwing away). In this paper we study the decision problem for the multiplicative fragment of linear logic without quantifiers or propositions: the constant-only case. We show that this fragment is np-complete. Earlier work by Max Kanovich showed that propositional multiplicative linear logic is np-complete. With Natarajan Shankar, the first author developed a simplified proof for the propositional case. The structure of this simplified proof is utilized here with a new encoding which uses only constants. The end product is the somewhat surprising result that simply evaluating expressions in true, false, and, and or in multiplicative linear logic (\Omega , , 1, and ?) is np-complete. By conservativity results not proven here, the np-hardness of larger fragments of linear logic follows. 1 Introduction When Girard introduced linear logic [7], he bro...

We present a proof-theoretic foundation for automated deduction in linear logic. At first, we systematically study the permutability properties of the inference rules in this logical framework and exploit these to introduce an appropriate notion of forward and backward movement of an inference in a proof. Then we discuss the naturally-arising question of the redundancy reduction and investigate the possibilities of proof normalization which depend on the proof search strategy and the fragment we consider. Thus, we can define the concept of normal proof that might be the basis of works about automatic proof construction and design of logic programming languages based on linear logic. 1 Introduction Linear logic is a powerful and expressive logic with connections to a variety of topics in computer science. We are mainly interested by the significance it may have in different domains as logic programming or program synthesis through theorem proving. As a matter of fact, classical linear ...

We present a general framework for proof search in first-order cut-free sequent calculi and apply it to the specific case of linear logic. In this framework, Herbrand functions are used to encode universal quantification, and unification is used to instantiate existential quantifiers so that the eigenvariable conditions are respected. We present an optimization of this procedure that exploits the permutabilities of the subject logic. We prove the soundness and completeness of several related proof search procedures. This proof search framework is used to show that provability for firstorder MALL is in nexptime, and first-order MLL is in np. Performance comparisons based on Prolog implementations of the procedures are also given. The optimization of the quantifier steps in proof search can be combined effectively with a number of other optimizations that are also based on permutability. 1 Introduction Since proofs contain more information than the theorems they prove, the main challen...

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Linear logic, introduced by J.-Y.Girard, is a refinement of classical logic providing means for controlling the allocation of "resources". It has aroused considerable interest both from proof theorists and computer scientists. In this paper we investigate methods for automated theorem proving in propositional linear logic. Both the "bottom-up" and "top-down" (resolution) proof strategies are analyzed -- various modifications of sequent rules and efficient search strategies are presented along with the experiments performed with the implemented theorem provers.

this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics