## Project-Team : calligramme

## Section: New Results

## Proof Nets, Sequent Calculus and Typed Lambda Calculi

### Proof Nets for Multiplicative Constants

In [26] Lutz Straßburger and François Lamarche present a new solution to the problem of including multiplicative constants in proof nets for classical multiplicative linear logic. It is well known that the constant has to be attached somewhere, but that the correct choice of attachment point is very problematic. An example in [26] shows that there is no hope of having a normal form in the standard sense, and that a quotient by a suitable equivalence relation has to be taken. The point of the given approach is that it uses the ordinary technology of proof nets, so there is no need to introduce new special types of links, and ordinary correctness criteria can be used; but some of these ordinary links are used to ``bunch'' several formulas together and thus give a much better range of choices for attaching s. This theory obeys the most stringent possible criterion of success for the problem of the units, in the sense that it is shown that the free *-autonomous category (with units) is obtained.

The full version of this work is [30]. This paper takes pains to make everything as elementary and explicit as possible, with no hand-waving whatsoever—in particular the algebraic calculations are given in full. This make the paper readable by beginners, who have only mastered the basics of the theory of proof nets and of category theory.

### Proof Nets for Classical Logic

In [31] François Lamarche and Lutz Straßburger propose a theory of proof denotations for ordinary, Boolean propositional logic. Two full models are constructed, and they show characteristics of both syntax (proof nets) and denotational semantics. The proof net aspect is due to the fact that the ``essence'' of a proof is captured by the means of axiom links. In the first model, which does not count how many times a formula is used, a full completenes theorem is given, and thus it can be seen as a substitute for ordinary syntax (sequent calculs, -calculus...) for denoting classical proofs. The second model, which does use natural numbers for counting, is not endowed with such a full completeness theorem, and thus is closer to an ordinary denotational semantics. But it shows definite promise from the point of view of complexity: it has a direct relevance to the NP vs. co-NP problem. Both models are deceptively simple; the fact that they have been discovered so late (although Andrew's work of 1976 was a definite precursor) is perhaps more due to widely held ideological positions about the computational nature of cut-elimination than inherent technical difficulty. These models give a very satisfying solution to a well-known problem: it is very easy to find denotational semantics for proofs in intuitionistic logic, as any bi-cartesian closed category will do, and they abound in nature (e.g., the category of sets and functions, or posets and monotone maps). But adapting this naively to classical logic leads to an immediate collapse of a candidate category to a poset (a Boolean algebra, naturally). The present work shows very clearly what standard algebraic property has to be abandoned to avoid the collapse, and it is preservation of the weakening maps. The algebraic treatment is very clean, considerably simpler—but less general in a certain sense—than what is proposed by Fuhrmann and Pym.

### Proof Nets without Links for Lambek Calculus

In a joint work, Christian Retoré (INRIA project Signes) and S. Pogodalla define proof nets without connective links for the Lambek calculus. Proof nets without links were introduced by C. Retoré. In the commutative version, both commutativity and associativity are interpreted as equality and proof structures are defined with graph theoretical notions (series-parallel graphs, perfect matching). To cope with cyclicity (Lambek non commutativity), they also use the notions of path and cyclic order to define proof structures and correctness criterion for the Lambek calculus.