A Non-Commutative Extension of MELL

We extend multiplicative exponential linear logic (MELL) by a non-commutative, self-dual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We should then be able to extend the range of applications of MELL, by modelling a broad notion of sequentiality and providing new properties of proofs. We show some proof theoretical results: decomposition and cut elimination. The new operator represents a significant challenge: to get our results we use here for the first time some novel techniques, which constitute a uniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus

Towards a Combinatorial Proof Theory

International audienceThe main part of a classical combinatorial proof is a skew fi-bration, which precisely captures the behavior of weakening and contraction. Relaxing the presence of these two rules leads to certain substruc-tural logics and substructural proof theory. In this paper we investigate what happens if we replace the skew fibration by other kinds of graph homomorphism. This leads us to new logics and proof systems that we call combinatorial

Observed communication semantics for classical processes

Classical Linear Logic (CLL) has long inspired readings of its proofs as communicating processes. Wadler's CP calculus is one of these readings. Wadler gave CP an operational semantics by selecting a subset of the cut-elimination rules of CLL to use as reduction rules. This semantics has an appealing close connection to the logic, but does not resolve the status of the other cut-elimination rules, and does not admit an obvious notion of observational equivalence. We propose a new operational semantics for CP based on the idea of observing communication, and use this semantics to define an intuitively reasonable notion of observational equivalence. To reason about observational equivalence, we use the standard relational denotational semantics of CLL. We show that this denotational semantics is adequate for our operational semantics. This allows us to deduce that, for instance, all the cut-elimination rules of CLL are observational equivalences

Naming Proofs in Classical Propositional Logic

Rapport interne.We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a "real'' sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures, and we get "Boolean'' categories which are not posets

Words as Modules and Modules as Partial Proof-Nets in a Lexicalized Grammar

In this paper we describe a syntactic calculus based on partial proof-nets (or modules) as building blocks of the system. The main idea is to associate with each lexical item one or more modules as syntactic types. These modules are obtained by unfolding the components of formulae that would be associated as types with the lexical items in a typelogical grammar ([Mor 94]). Proof nets are obtained by combining these modules by a uniform set of "plugging" rules. There are other approaches, based on Partiel Proof-Trees as building blocks in a grammar ([Jos 95]), our approach differs from them mainly by the emphasis put on the geometric notion of Proof-Net (cf [Lec 95]). Our main motivation is to obtain a very general and logical model in which it would be possible to embed other calculi like the Lambek grammars on one side and the Lexicalized Tree Adjoining Grammars on the other side. In this paper, we first give an overview of the whole system (called POMSET-logic) : it is based on Multiplicative Linear Logic enriched with the non-commutative "before" ([Ret 93]), and which consequently deals with Partially Ordered Multisets instead of ordinary multisets of formulae. Then, we give more restricted versions which can be used for linguistic purposes. We plan to give illustrative examples (concerning scrambling in German, clitics in French and wh-extraction) in a next article

Typed Hilbert Epsilon Operators and the Semantics of Determiner Phrases

International audienceThe semantics of determiner phrases, be they definite de- scriptions, indefinite descriptions or quantified noun phrases, is often as- sumed to be a fully solved question: common nouns are properties, and determiners are generalised quantifiers that apply to two predicates: the property corresponding to the common noun and the one corresponding to the verb phrase. We first present a criticism of this standard view. Firstly, the semantics of determiners does not follow the syntactical structure of the sentence. Secondly the standard interpretation of the indefinite article cannot ac- count for nominal sentences. Thirdly, the standard view misses the linguis- tic asymmetry between the two properties of a generalised quantifier. In the sequel, we propose a treatment of determiners and quantifiers as Hilbert terms in a richly typed system that we initially developed for lexical semantics, using a many sorted logic for semantical representations. We present this semantical framework called the Montagovian generative lexicon and show how these terms better match the syntactical structure and avoid the aforementioned problems of the standard approach. Hilbert terms rather differ from choice functions in that there is one polymorphic operator and not one operator per formula. They also open an intriguing connection between the logic for meaning assembly, the typed lambda calculus handling compositionality and the many-sorted logic for semantical representations. Furthermore epsilon terms naturally introduce type-judgements and confirm the claim that type judgment are a form of presupposition.Typed Hilbert Epsilon Operators and the Semantics of Determiner Phrases (Invited Lecture

Cyclic Multiplicative Proof Nets of Linear Logic with an Application to Language Parsing

This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e. de-sequentialized proofs, of linear logic (LL). In particular, it presents a simple and intuitive syntax for PNs of the cyclic multiplicative fragment of linear logic (CyMLL). The proposed correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of LL. The main intuition relies on the fact that any DR-switching (i.e. any correction or test graph for a given PN) can be naturally viewed as a seaweed, i.e. a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Dislike the most part of current syntaxes for non-commutative PNs, our syntax allows a sequentialization for the full class of CyMLL PNs, without requiring these latter must be cut-free. Moreover, we give a simple characterization of CyMLL PNs for Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences by means of Lambek PNs

Cyclic multiplicative-additive proof nets of linear logic with an application to language parsing

This paper concerns a logical approach to natural language parsing based on proof nets (PNs), i.e. de-sequentialized proofs, of linear logic (LL). In particular, it presents a syntax for PNs of the cyclic multiplicative and additive fragment of linear logic (CyMALL). Any proof structure (PS), in Girards style, is weighted by boolean monomial weights, moreover, its conclusions Γ (a sequence of formulas occurrences) are endowed with a cyclic order σ, i.e., σ(Γ). Naively, a CyMALL PS π with conclusions σ(Γ) is correct if, for any slice ϕ(π) (obtained by a boolean valuation ϕ of π) there exists an additive resolution (i.e. a multiplicative refinement of ϕ(π)) that is a CyMLL PN with conclusions σ(Γr), where Γr is an additive resolution of Γ (i.e. a choice of an additive subformula for each formula of Γ). In its turn, the correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of LL. The main intuition relies on the fact that any DR-switching (i.e. any correction or test graph for a given PN) can be naturally viewed as a seaweed, i.e. a rootless planar tree inducing a cyclic order on the conclusions of the given PN. Dislike the most part of current syntaxes for non-commutative PNs our syntax allows a sequentialization for the full class of CyMLL PNs, without requiring these latter must be cut-free. Moreover, we give a characterization of CyMALL PNs for the extended (MALL) Lambek Calculus and thus a geometrical (non inductive) way to parse phrases or sentences. In particular additive Lambek PNs allow to parse phrases containing words with syntactical ambiguity (i.e. words with polymorphic type)

Words as Modules: a Lexicalised Grammar in the framework of Linear Logic Proof Nets

We present a syntactic calculus relying on the notion of proof net, ie. a geometric representation for proofs in linear logic. The lexicon associates a module or partial proof net with each word, and this information completely encodes the syntactic behavior of the word. Parsing a sentence consists in combining the modules associated with its words into a complete proof net. In order to handle word order, this model takes place in a non-commutative extension of linear logic called Pomset logic