There are many situations in logic, theoretical computer science, and category theory where two binary operations---one thought of as a (tensor) "product", the other a "sum"---play a key role. In distributive and -autonomous categories these operations can be regarded as, respectively, the and/or of traditional logic and the times/par of (multiplicative) linear logic. In the latter logic, however, the distributivity of product over sum is conspicuously absent: this paper studies a "linearization" of that distributivity which is present in both case. Furthermore, we show that this weak distributivity is precisely what is needed to model Gentzen's cut rule (in the absence of other structural rules) and can be strengthened in a natural way to generate - autonomous categories. We also point out that this "linear" notion of distributivity is virtually orthogonal to the usual notion as formalized by distributive categories. 0 Introduction There are many situations in logic, theoretical co...

We extend the multiplicative fragment of linear logic with a non-commutative connective (called before), which, roughly speaking, corresponds to sequential composition. This lead us to a calculus where the conclusion of a proof is a Partially Ordered MultiSET of formulae. We firstly examine coherence semantics, where we introduce the before connective, and ordered products of formulae. Secondly we extend the syntax of multiplicative proof nets to these new operations. We then prove strong normalisation, and confluence. Coming back to the denotational semantics that we started with, we establish in an unusual way the soundness of this calculus with respect to the semantics. The converse, i.e. a kind of completeness result, is simply stated: we refer to a report for its lengthy proof. We conclude by mentioning more results, including a sequent calculus which is interpreted by both the semantics and the proof net syntax, although we are not sure that it takes all proof nets into account...

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

Lambek calculus may be viewed as a fragment of linear logic, namely intuitionistic non-commutative multiplicative linear logic. As it is too restrictive to describe numerous usual linguistic phenomena, instead of extending it we extend MLL with a non-commutative connective, thus dealing with partially ordered multisets of formulae. Relying on proof net technique, our study associates words with parts of proofs, modules, and parsing is described as proving by plugging modules. Apart from avoiding spurious ambiguities, our method succeeds in obtaining a logical description of relatively free word order, head-wrapping, clitics, and extraposition (these latest two constructions are unfortunately not included, for lack of space).

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A graph-theoretical look at multiplicative proof nets lead us to two new descriptions of a proof net, both as a graph endowed with a perfect matching. The first one is a rather conventional encoding of the connectives which nevertheless allows us to unify various sequentialisation techniques as the corollaries of a single graph theoretical result. The second one is more exciting: a proof net simply consists in the set of its axioms -- the perfect matching -- plus one single series-parallel graph which encodes the whole syntactical forest of the sequent. We thus identify proof nets which only differ because of the commutativity or associativity of the connectives, or because final par have been performed or not. We thus push further the program of proof net theory which is to get closer to the proof itself, ...

We study some normalisation properties of the deep-inference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain non-commutative self-dual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a self-dual, non-commutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.

It is known that (mix) proof nets admit a coherence semantics, computed as a set of experiments. We prove here the converse: a proof structure is shown to be a proof net whenever its set of experiments is a semantical object --- a clique of the corresponding coherence space. Moreover the interpretation of atomic formulae can be restricted to a given coherent space with four tokens in its web. This is done by transforming cut-links into tensor-links. Dealing directly with non-cut-free proof structure we characterise the deadlock freeness of the proof structure. These results are especially convenient for Abramsky 's proof expressions, and are extended to the pomset calculus.

We consider a class of graphs embedded in R 2 as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. Braided proofnets are proof-nets for multiplicative linear logic with Mix embedded in R 3 . In order to prove the cut-elimination theorem, we consider proof-nets in R 2 as projections of braided proof-nets under regular isotopy. Contents 1 Introduction 2 2 Language 4 2.1 Links and Proof-Structures . . . . . . . . . . . . . . . . . . . . 5 3 A combinatorial characterization. 8 3.1 Subnets of Proof-Nets . . . . . . . . . . . . . . . . . . . . . . 11 Research supported by EC Individual Fellowship Human Capital and Mobility, contract n. 930142. Both authors thank Jacques van de Wiele, f...

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