System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a self-dual, non-commutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, system BV extends the applications of MLL to those where sequential composition is crucial, e.g., concurrency theory. System FBV is an extension of MLL with the rules mix and nullary mix. In this paper, by relying on the fact that system BV is a conservative extension of system FBV, I show that system BV is NP-complete by encoding the 3-Partition problem in FBV. I provide a simple completeness proof of this encoding by resorting to a novel proof theoretical method for reducing the nondeterminism in proof search, which is also of independent interest.

In this paper, we propose new labelled proof systems to analyse the intuitionistic provability in classical and linear logics. An important point is to understand how search in a non-classical logic can be viewed as a perturbation of search in classical logic. Therefore, suitable characterizations of intuitionistic provability and related labelled sequent calculi are defined for linear logic. An alternative approach, based on the notion of proof-net and on the definition of suitable labelled classical proof-nets, allows to directly study the intuitionistic provability by constructing intuitionistic proof-nets for sequents of classical linear logic. Keywords: Proof theory, intuitionistic logic, linear logic, labelled sequent calculus, proofnets, automated deduction. 1. INTRODUCTION Many proof-search methods (sequent calculus, tableaux, resolution, connections) have been naturally developed in classical logic (CL) with a view 1 2 LABELLED DEDUCTION to avoiding the possible redundanc...

In this paper we describe a solution to the problem of proving cut-elimination for FILL, a variant of exponential-free and multiplicative Linear Logic originally introduced by Hyland and de Paiva. In the work of Hyland and de Paiva, a term assignment system is used to describe the intuitionistic character of FILL and a proof of cut-elimination is barely sketched. In the present paper, as well as correcting a small mistake in their work and extending the system to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut-elimination theorem. The formal system is based on a dependency-relation between formulae occurrences within a given proof and seems of independent interest. The procedure for cut-elimination applies to (multiplicative and exponential) Classical Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut-elimination proofs, is...

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We introduce the language CP, a compositional language for place transition petri nets for the purpose of modelling signalling pathways in complex biological systems. We give the operational semantics of the language CP by means of a proof theoretical deductive system which extends multiplicative exponential linear logic with a self-dual non-commutative logical operator. This allows to express parallel and sequential composition of processes at the same syntactic level as in process algebra, and perform logical reasoning on these processes. We demonstrate the use of the language on a model of a signaling pathway for Fc receptor-mediated phagocytosis.

Abstract. We present a sound and complete compositional semantics, structured around certain abstractions of proof nets, for proof-search computation in a linear logic-based language. The model captures the interaction of agents in terms of the actions they engage into and of the dynamic creation of names. The model is adequate for reasoning about a notion of operational equivalence. We will also suggest how a partial order semantics can be derived from the present approach. 1

A Proof of: We present a sound and complete compositional semantics for proof-search computation in a linear logic-based language structured around certain abstractions of proof nets. The model captures the interaction of agents in terms of the actions they engage into and of the dynamic creation of names. The model can also be shown adequate for reasoning about a notion of operational equivalence. We will also suggest how a partial order semantics can be derived from the present approach. 1

In this paper, we propose new labelled proof systems to analyse the intuitionistic provability in classical and linear logics. An important point is to understand how search in a non-classical logic can be viewed as a perturbation of search in classical logic. Therefore, suitable characterizations of intuitionistic provability and related labelled sequent calculi are defined for linear logic. An alternative approach, based on the notion of proof-net and on the definition of suitable labelled classical proof-nets, allows to directly study the intuitionistic provability by constructing intuitionistic proof-nets for sequents of classical linear logic. Keywords: Proof theory, intuitionistic logic, linear logic, labelled sequent calculus, proofnets, automated deduction. 1. INTRODUCTION Many proof-search methods (sequent calculus, tableaux, resolution, connections) have been naturally developed in classical logic (CL) with a view 1 2 LABELLED DEDUCTION to avoiding the possible redundanc...

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Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK ∗ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a selfcontained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK ∗, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria. 1