Yves Lafont

Soft linear logic and polynomial time

We present a subsystem of second-order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice versa.

Towards an algebraic theory of Boolean circuits

Abstract Boolean circuits are used to represent programs on finite data. Reversible Boolean circuits and quantum Boolean circuits have been introduced to modelize some physical aspects of computation. Those notions are essential in complexity theory, but we claim that a deep mathematical theory is needed to make progress in this area. For that purpose, the recent developments of knot theory is a major source of inspiration. Following the ideas of Burroni, we consider logical gates as generators for some algebraic structure with two compositions, and we are interested in the relations satisfied by those generators. For that purpose, we introduce canonical forms and rewriting systems . Up to now, we have mainly studied the basic case and the linear case , but we hope that our methods can be used to get presentations by generators and relations for the (reversible) classical case and for the (unitary) quantum case .

Interaction Combinators

It is shown that a very simple system ofinteraction combinators, with only three symbols and six rules, is a universal model of distributed computation, in a sense that will be made precise. This paper is the continuation of the authors work oninteraction nets, inspired by Girards proof nets forlinear logic, but no preliminary knowledge of these topics is required for its reading.

The Finite Model Property for Various Fragments of Linear Logic

To show that a formula A is not provable in propositional classical logic, it suuces to exhibit a nite boolean model which does not satisfy A. A similar property holds in the intuitionistic case, with Kripke models instead of boolean models (see for instance TvD88]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a nite model property. In particular, they are decidable: there is a semi-algorithm for provability (proof search) and a semi-algorithm for non provability (model search). For that reason, a logic which is undecidable, such as rst order logic, cannot satisfy a nite model property. The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces Gir87, Gir95], but it is undecidable LMSS92]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete LMSS92], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For aane logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable Kop95a]. Here, we show that the nite phase semantics is complete for MALL and for LLW, but not for MELL. In particular, this gives a new proof of the decidability of LLW. The noncommutative case is mentioned, but not handled in detail. 1. Syntax of linear logic Roman capitals A, B stand for formulas. The connectives of propositional linear logic are: the multiplicatives A (?A) ? = !A ? : One writes A ? B for A ? & B. Greek capitals ?, stand for sequents, which are multisets of formulas, so that exchange is implicit. Identity and cut are written as follows:

A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier)

Abstract Recently, Craig Squier introduced the notion of finite derivation type to show that some finitely presentable monoid has no presentation by means of a finite complete rewriting system. A similar result was already obtained by the same author using homology, but the new method is more direct and more powerful. Here, we present Squiers argument with a bit of categorical machinery, making proofs shorter and easier. In addition we prove that if a monoid has finite derivation type, then its third homology group is of finite type.

Equational Reasoning with Two-Dimensional Diagrams

The significance of the 2-dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street. Independently, Burroni has introduced a general notion of n-dimensional presentation and he has shown that the equational logic of terms is a special case of 2-dimensional calculus.

The undecidability of second order multiplicative linear logic

The multiplicative fragment of second order propositional linear logic is shown to be undecidable.

The Undecidability of Second Order Linear Logic without Exponentials

Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics.

Diagram Rewriting for Orthogonal Matrices: A Study of Critical Peaks

Orthogonal diagramsrepresent decompositions of isometries of ?ninto symmetries and rotations. Some convergent (that is noetherian and confluent) rewrite system for this structure was introduced by the first author. One of the rules is similar to Yang-Baxter equation. It involves a map h: ]0, ?[3?]0, ?[3. In order to obtain the algebraic properties of h, we study the confluence of critical peaks (or critical pairs) for our rewrite system. For that purpose, we introduce parametric diagramsdescribing the calculation of angles of rotations generated by rewriting. In particular, one of those properties is related to the tetrahedron equation(also called Zamolodchikov equation).