There has been considerable work aimed at enhancing the expressiveness of logic programming languages. To this end logics other than classical first order logic have been considered, including intuitionistic, relevant, temporal, modal and linear logic. Girard's linear logic has formed the basis of a number of logic programming languages. These languages are successful in enhancing the expressiveness of (pure) Prolog and have been shown to provide natural solutions to problems involving concurrency, natural language processing, database processing and various resource oriented problems. One of the richer linear logic programming languages is Lygon. In this paper we investigate the implementation of Lygon. Two significant problems that arise are the division of resources between sub-branches of the proof and the selection of the formula to be decomposed. We present solutions to both of these problems.

linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prex unication. We present the tableau calculus used by linTAP, an algorithm for prex unication in linear logic, the linTAP implementation, and some experimental results obtained with linTAP. 1

Recently there has been significant interest in the logic programming community in linear logic, a logic designed with bounded resources in mind. As linear logic is a generalisation of classical logic, a logic programming language based on linear logic subsumes and extends (pure) Prolog. One such language is Lygon, a language based on a certain kind of proof in the linear sequent calculus. However these proofs, whilst providing a logical characterization of the language, still retain some of the non-determinism of the sequent calculus, and hence require further analysis before an implementation can be attempted. In this report we define and discuss a more detailed proof system, which is more deterministic than the original. In particular, this system handles the allocation of resources to different branches of the proof in a lazy manner. The resulting system differs significantly from the original sequent calculus, and so we discuss its properties in some detail. We prove the soundness...

Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrix-based proof search procedure for this fragment and a procedure which translates the machine-found proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found. 1

Linear logic (LL) is the logical foundation of some type-theoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proof-search in different logics. Aiming at using proof nets as a tool for automated deduction in linear logic, we define a connection-based characterization of provability in Multiplicative Linear Logic (MLL). We show that an algorithm for proof net construction can be seen as a proof-search connection method. This central result is illustrated with a specific algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof. From these results we expect to extend to other LL fragments, we analyse what happens with the additive connectives of LL by tackling the additive fragment in a similar way.

Linear Logic is gaining momentum in computer science because it offers a unified framework and a common vocabulary for studying and analyzing different aspects of programming and computation. We focus here on models where computation is identified with proof search in the sequent system of Linear Logic. A proof normalization procedure, called "focusing", has been proposed to make the problem of proof search tractable. Correspondingly,

Proof nets can be seen as a multiple conclusion natural deduction system for Linear Logic (LL) and form a good formalism to analyze some computation mechanisms, for instance in type-theoretic interpretations. This paper presents an algorithm for automated proof nets construction in the non-commutative multiplicative linear logic that is useful for applications including planning, concurrency or sequentiality. The properties of this algorithm can be proved from a recently defined graph-theoretic characterization of non-commutative proof nets. Involving simple construction principles improved in the commutative case, it leads also to a new proof search method for the non-commutative fragment. Moreover because of the relationships between the non-commutative linear logic and the Lambek calculus we can derive from it an alternate method for automatic construction of proof nets in this calculus. 1 Introduction Linear Logic (LL) can offer a framework to study and analyze various notions or ...

In this paper we propose new calculi for the multiplicative fragment of Mixed Linear Logic (MMLL) which is a logic that combines both commutative and noncommutative connectives. These based-on sequent and proof net calculi, that can be seen as a new proof-theoretical formulation of MMLL, are based on the definition of dependency relations. We provide a proof-search procedure for MMLL that is based on proof nets construction with associated sets of dependencies.

We present a matrix characterization of logical validity in the multiplicative fragment of linear logic with exponentials. In the process we elaborate a methodology for proving matrix characterizations correct and complete. Our characterization provides a foundation for matrixbased proof search procedures for as well as for procedures which translate machine-found proofs back into the usual sequent calculus.

Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever---rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how -terms...