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 ...

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.

In this paper, we investigate automated proof construction in classical linear logic (CLL) by giving logical foundations for the design of proof search strategies. We propose common theoretical foundations for top-down, bottom-up and mixed proof search procedures with a systematic formalization of strategies construction using the notions of immediate or chaining composition or decomposition, deduced from permutability properties and inference movements in a proof. Thus, we have logical bases for the design of proof strategies in CLL fragments and then we can propose sketches for their design.

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, there is a normalization procedure mapping formulae of Linear Logic into a syntactic fragment of that logic, called LinLog, and in which the focusing normalization for proofs can be most conveniently expressed. In this paper, we propose to push this compilation/normalization process further, by applying abstract interpretation and partial evaluation techniques to (focused) proofs in LinLog. These techniques provide information concerning the evolution of the computational resources (formulae) during the execution (proof construction). The practical outcome that we expect from this theoretical effort is the definition of a general tool for statically analyzing and reasoning about the runtime behavior of programs in frameworks where computations can be accounted for in terms of proof search in Linear Logic.

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.

. In this paper, we consider the multiplicative fragment of linear logic (MLL) from an automated deduction point of view. Before to use this new logic to make logic programming or to program with proofs, a better comprehension of the proof construction process in this framework is necessary. We propose a new algorithm to construct automatically a proof net for a given sequent in MLL and its proofs of termination, correctness and completeness. It can be seen as an implementation oriented way to consider automated deduction in linear logic. 1 Introduction Computer scientists can consider logic with two points of view: it can be an external reference to which they report during their activity, for example, to make correctness proofs of programs or it can be integrated as an internal tool for programming. In this work, we consider the second one. Intuitionism [16] has given at first a logical tool to approach programming in this connection. Centered around a constructive vision of truth, ...

We discuss here the proof-theoretic foundations for theorem proving and logic programming in linear logic, mainly studying how to define canonical proofs (that are complete) for efficient proof search in fragments of linear logic. We analyze the conception of such proof forms, for frameworks based on proof-construction as computation, emphasizing the relationship between the logical fragment and its proof search strategies. This point is essential for the definition and implementation of logic programming languages within linear logic.

This paper introduces a general backward proof search strategy for multiplicative additive linear logic. This strategy, which is based on Isabelle's basic tactics and tacticals, has been implemented and appears to be rather eOEcient. Its eOEciency derives from several heuristics that we introduce in the paper. We prove that these heuristics preserve completeness.