Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1

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...

We introduce a new category of finite, fair games, and winning strategies, and use it to provide a semantics for the multiplicative fragment of Linear Logic (mll) in which formulae are interpreted as games, and proofs as winning strategies. This interpretation provides a categorical model of mll which satisfies the property that every (history-free, uniformly) winning strategy is the denotation of a unique cut-free proof net. Abramsky and Jagadeesan first proved a result of this kind and they refer to this property as full completeness. Our result differs from theirs in one important aspect: the mix-rule, which is not part of Girard's Linear Logic, is invalidated in our model. We achieve this sharper characterization by considering fair games. A finite, fair game is specified by the following data: ffl moves which Player can play, ffl moves which Opponent can play, and ffl a collection of finite sequences of maximal (or terminal) positions of the game which are deemed to be fair. N...

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 ...

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.

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.

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.

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. 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 give a class of proof nets for Intuitionistic Linear Logic with the connectives (; !, prove a correctness criterion for them and show that a games semantics can be directly derived from these nets, along with a full completeness theorem. It is well-known that games semantics is intimately connected to linear logic, but there is an important example of games semantics where the connection is far from clear, namely Hyland and Ong's [9] for the simply typed lambda-calculus and PCF. Although in this semantics the construction of the function space (intuitionistic implication) depends quite explicitly on the standard decomposition X)Y = !X (Y , it is not clear at all how one would be able to describe the semantics of these two linear operators independently. In particular if one naively follows the spirit of the constructions given in that paper, it seems one would get that the natural morphism !X ! !!X (comultiplication in the comonad) is an isomorphism. It follows from the theory of...