Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource management . . . . . . . . . . . . . . . . . . . . . . 13 3 The syntax-semantics interface: proofs and readings 16 3.1 Term assignment for categorial deductions . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Natural language interpretation: the deductive view . . . . . . . . . . . . . . . . . . . 21 4 Grammatical composition: multimodal systems 26 4.1 Mixedinference:themodesofcomposition........................ 26 4.2 Grammaticalcomposition:unaryoperations ....................... 30 4.2.1 Unary connectives: logic and structure . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Applications: imposing constraints, structural relaxation

The paper is mainly concerned with the extension of proof-nets to additives, for which the best known solution is presented. It proposes two cut-elimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional proof-theory deals with cut-elimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However old-fashioned proof-theory, which is concerned with the ritual question: “is-that-theory-consistent? ” never really cared. The situation changed when subtle algorithmic aspects of cut-elimination became prominent: typically

This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and object-oriented programming and some other applications of LL, like semantics of negation in LP, non-monotonic issues in AI planning, etc. Although the overview covers pretty much the state-of-the-art in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...

Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs. © 2001 Academic Press Key Words: formal proofs; logical flow graphs; cut elimination; bridge groups; Baumslag–Solitar groups; Gersten groups.

We provide a context semantics for Multiplicative-Additive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cut-elimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lvy, who provided a "geometry of optimal h-reduction" (context semantics) for h-calculus and Multiplicative-Exponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a read-back procedure that computes a cut-free proof from the semantics, which is closely related to full abstraction theorems.

It can be very advantageous to borrow key components of a logic for use in another logic. The advantages are both conceptual and practical; due to the existence of software systems supporting mechanized reasoning in a given logic, it may be possible to reuse a system developed for one logic---for example, a theorem-prover---to obtain a new system for another. Translations between logics by appropriate mappings provide a first natural way of reusing tools of one logic in another. This paper generalizes this idea to the case where entire components---for example, the proof theory---of one of the logics involved may be completely missing, so that the appropriate mapping could not even be defined. The idea then is to borrow the missing components (as well as their associated tools if they exist) from a logic that has them in order to create the full-fledged logic and tools that we desire. The relevant structure is transported using maps that only involve a limited aspect of the two logics ...

Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without "lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions.

This paper wants to focus on the generation capabilities of proof nets thanks to their semantic readings as expressed in (de Groote and Reto]'d, 1996). The main features of our proposal consist in the use of proof nets for Lambek calculus, of the Curry-Howard isomorphism (Howard, 1980; Girard et al., 1988), of semantic p]'oo[' nets with semantic expressions 2 la Montague (Montague, 1974; Dowry et al., 1981), and in an algorithm for proof search with a target proof net. Unlike a previous proposal for generation in the Lam- bek calculus framework (Merenciano and Mort'ill, 1997), this point of view avoids the use of the X-term unification to lead the generation process, And the algorithmic undecidability of this latter mechanism (fi'om second order unification) does not occur any more. In this work, we do not consider the choice of lexi- cal items fi'om a given semantic expression the syntactic realization of which we want to generate, but rather tile way we can associate given lexical entries to fit the given semantic expression and generate a syntactically correct expression. For this purpose, we express our problem as a proof search one in (multiplicative) linear logic which is decidable. Moreover, we characterize the semantic recipes of lexical items that provide a polynomial solution for the generation process

Most of the studies in the framework of Lambek calculus have considered the parsing process aud ignored the generation process. This paper wants to rely on the close link between Lambek calculus and linear logic to present a method for the generation process with semantic proof nets. We express the process as a proof search procedure based on a graph calculus and the solutions appear as a matrix computation preserving the decidability properties, and we characterize a polynomial time case.

Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted