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CLASSICAL NONASSOCIATIVE LAMBEK CALCULUS
"... We introduce nonassociative linear logic, which may be seen as the classical version of the nonassociative Lambek calculus. We define its sequent calculus, its theory of proof nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polyno ..."
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We introduce nonassociative linear logic, which may be seen as the classical version of the nonassociative Lambek calculus. We define its sequent calculus, its theory of proof nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.
Resource logics and minimalist grammars
 Proceedings ESSLLI’99 workshop (Special issue Language and Computation
, 2002
"... This ESSLLI workshop is devoted to connecting the linguistic use of resource logics and categorial grammar to minimalist grammars and related generative grammars. Minimalist grammars are relatively recent, and although they stem from a long tradition of work in transformational grammar, they are lar ..."
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This ESSLLI workshop is devoted to connecting the linguistic use of resource logics and categorial grammar to minimalist grammars and related generative grammars. Minimalist grammars are relatively recent, and although they stem from a long tradition of work in transformational grammar, they are largely informal apart from a few research papers. The study of resource logics, on the other hand, is formal and stems naturally from a long logical tradition. So although there appear to be promising connections between these traditions, there is at this point a rather thin intersection between them. The papers in this workshop are consequently rather diverse, some addressing general similarities between the two traditions, and others concentrating on a thorough study of a particular point. Nevertheless they succeed in convincing us of the continuing interest of studying and developing the relationship between the minimalist program and resource logics. This introduction reviews some of the basic issues and prior literature. 1 The interest of a convergence What would be the interest of a convergence between resource logical investigations of
Intuitionistic Multiplicative Proof Nets as Models of Directed Acyclic Graph Descriptions
 in &quot;8th International Conference on Logic for Programming, Artificial Intelligence and Reasoning  LPAR 2001, Havana
, 2001
"... Given an intuitionistic proof net of linear logic, we abstract an order between its atomic formulas. From this order, we represent intuitionistic multiplicative proof nets in the more compact form of models of directed acyclic graph descriptions. If we restrict the logical framework to the implic ..."
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Given an intuitionistic proof net of linear logic, we abstract an order between its atomic formulas. From this order, we represent intuitionistic multiplicative proof nets in the more compact form of models of directed acyclic graph descriptions. If we restrict the logical framework to the implicative fragment of intuitionistic linear logic, we show that proof nets reduce to models of tree descriptions.
Handsome NonCommutative ProofNets: perfect matchings, seriesparallel orders and Hamiltonian circuits
 INRIA
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Relational Interpretation and Geometrical Form
"... We describe the possibility of a systematic correspondence between combined algebraic and relational interpretation of categorial logic, and the form of proof structures and paths in proof nets, illustrating with reference to medial extraction, in situ binding and discontinuity. Type logic for ling ..."
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We describe the possibility of a systematic correspondence between combined algebraic and relational interpretation of categorial logic, and the form of proof structures and paths in proof nets, illustrating with reference to medial extraction, in situ binding and discontinuity. Type logic for linguistic description (e.g. Moortgat 1988, 1997; Morrill 1994; Carpenter 1997) is based on what we may refer to as a Lambekvan Benthem correspondence: (logical) formulas as (linguistic) categories. Lexical signs are classified by category formulas, and the language model projected by a lexicon is determined by the consequence relation induced on category formulas by their interpretation. In this logical model of language, (logical) proofs correspond to (linguistic) derivations, but such syntax serves just to calculate what is generated, not to define it. Although syntax plays no definitional role linguistically, from a computational linguistic point of view we are interested in the processing ...
Constructing Different Phonological Bracketings From a ProofNet
, 1996
"... . We state and prove Roorda's interpolation theorem in the framework of proofnet theory. This allows us to transform any proofnet in some other proofnet that matches some given (phonological or prosodic) bracketing. 1 Introduction Almost a decade ago, Girard invented linear logic together wi ..."
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. We state and prove Roorda's interpolation theorem in the framework of proofnet theory. This allows us to transform any proofnet in some other proofnet that matches some given (phonological or prosodic) bracketing. 1 Introduction Almost a decade ago, Girard invented linear logic together with the notion of proofnet [7]. Girard's proofnets have been subsequently adapted to the Lambek calculus by Roorda [16] and, since then, many authors have advocated the notion of proofnet as the right parsing structure in the framework of categorial grammars [11, 13, 14, 16]. Nevertheless, if one wants to take this proposal seriously, one must be able to perform, on the proofnets, all the computations that one usually performs on Gentzen's sequential derivations. From a theoretical standpoint, the above possible objection is actually not a problem. Indeed, by Girard's sequentialisation theorem, one may always associate to any proofnet some corresponding sequential derivation. Therefore, any p...
Learning Lambek grammars from proof frames
"... Abstract. In addition to their limpid interface with semantics, the original categorial grammars introduced by Lambek 55 years ago enjoys another important property: learnability. After a short reminder on grammatical inference à la Gold, we provide an algorithm that learns rigid Lambek grammars wit ..."
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Abstract. In addition to their limpid interface with semantics, the original categorial grammars introduced by Lambek 55 years ago enjoys another important property: learnability. After a short reminder on grammatical inference à la Gold, we provide an algorithm that learns rigid Lambek grammars with product from proof frames that are name free proof nets a generalisation of functor argument structures to those grammars — that are already known to be unlearnable from strings, as shown by Foret and Le Nir. This result strictly encompasses our previous positive results on learning Lambek grammars without product The result can be extended to kvalued versions of these grammars using kunification although, as expected, algorithmic complexity becomes qui high. Our algorithm combines a proof net version of the principal type scheme algorithm of lambda calculus together with the unification algorithm for syntactic categories, as first explored by Buszkowski and Penn. We thereafter we provide a simple proof of the convergence of this algorithm inspired from the one by Kanazawa. Proof frames may seem complex structures to learn from, but they look like dependency structure that can be found in annotated corpora, and, as we show at the end of the paper, when the product is not used, proof frames exactly correspond to natural deduction frames that extend the functor argument structures that are commonly used for learning basic categorial grammars. We are sad to dedicate the present paper to Philippe Darondeau, with whom we started to study such questions in Rennes at the beginning of the millennium, and who passed away prematurely. We are glad to dedicate the present paper to Jim Lambek for his 90 birthday: he shows that research is an endless learning. 1
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"... Abstract We present here a stochastic point of vue on Lambek Categorial Grammars whichhave shown their utility in Natural Language Processing. The key point comes from the fact that the parsing in the Lambek Calculus is based on an essentiallynondeterministic algorithm. We claim that a stochasting ..."
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Abstract We present here a stochastic point of vue on Lambek Categorial Grammars whichhave shown their utility in Natural Language Processing. The key point comes from the fact that the parsing in the Lambek Calculus is based on an essentiallynondeterministic algorithm. We claim that a stochasting treatment may be a good way to resolve that issue. 1 Introduction Statistical methods have turned out to be quite successful in natural languageprocessing. During the recent years, several models of stochastic grammars have been proposed, including models based on lexicalised contextfree grammars [3], tree adjoining grammars [16], dependency grammars [2,5], or categorial AB grammars [13].In this exploratory paper, we propose a new model of stochastic grammar, whose originality derives from being based on Lambek categorial grammars[8]. This model presents interesting properties: ffl Probabilities are attached to syntactic dependencies and not to derivationrules. Moreover, they are expressed at the level of the lexicon. As a consequence, our model is fully lexicalised.ffl A treatment of lexical ambiguities is provided. This treatment is basedon unresolved dependencies. Consequently, long distance dependencies are