The abuse of structural rules may have damaging complexity effects.

A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is proved for this semantics. 1 Introduction This paper contains two main contributions: ffl the introduction of a new form of game semantics, which we call concurrent games. ffl a proof of full completeness of this semantics for Multiplicative-Additive Linear Logic. We explain the significance of each of these in turn. Concurrent games Traditional forms of game semantics which have appeared in logic and computer science have been sequential in format: a play of the game is formalized as a sequence of moves. The key feature of this sequential format is the existence of a global schedule (or polarization) : in each (finite) position, it is (exactly) one player's turn to move 1 . This seq...

A cornerstone of the theory of proof nets for unit-free multiplicative linear logic (MLL) is the abstract representation of cut-free proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cut-free monomial proof nets can correspond to the same cut-free proof. Thus the problem of finding a satisfactory notion of proof net for unit-free multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory. 1

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.

Vol. 2 (4:3) 2006, pp. 1–44 www.lmcs-online.org

We study a general algebraic framework which underlies a wide range of computational formalisms...

Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative canonicity directly in the cut-free sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multifocused proofs. We validate this definition by proving a bijection to the well-known proof-nets for the unit-free multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments. 1

Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.

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.