## Section: New Results

### Properties of the Formalisms

#### Relational Machines

Participant : Gérard Huet [ correspondent ] .

Gérard Huet and Benoît Razet (who completed this year his Thèse d'Informatique “Machines d'Eilenberg effectives” under the supervision of the former) developed a computational framework in which actions of a non-deterministic machine are executed on on abstract relational machine called an Eilenberg machine, named after the X-machine model of Samuel Eilenberg (1974). Such machines have two components: a control component — a non-deterministic finite-state automaton, whose transitions are labeled with action generators — and a data component — a relational interpretation of the generators over some data domain. Benoît Razet gave a complete definition of the general model, plus variations such as finite machines, for which a termination criterion ensures completeness of bottom-up search. A general machine operates a sequential interpreter called the reactive engine, parameterized by a strategy guiding the resumption structure. Effective relations are curryfied into stream producers, of which two varieties exist, one for partial computability, the other for total computability. Benoît Razet also gave a complete classification of functional implementations of the various compiling schemes of regular expressions into finite automata transition graphs. His work gives the foundation for an ambitious program of high-level relational programming.

#### Formal grammars and type theory

Participants : Pierre Bourreau, Richard Moot, Christian Retoré, Sylvain Salvati [ correspondent ] .

Christian Retoré and Lutz Stassburger (INRIA EPI Parsifal) answered negatively to both an old (1993) conjecture by Retoré and a more recent (1998) by Gugliemi : there exists a Pomset proof net in the sense of Retoré which does not correspond to a sequential proof in the various formulations of Pomset Logic in sequent calculus, but this proof net does correspond to a proof term of the BV-calculus. Hence Pomset Proof net cannot be sequentialized with any standard sequent calculus, and on the other hand, Pomset Logic is not equivalent to BV-calculus.

Sylvain Pogodalla (INRIA EPI Calligramme) and Christian Retoré studied the non commutative proof nets with cuts, trying to extend the proof nets without links for commutative multiplicative linear logic (2003): up to now, they mainly showed that the unique criterion handling non commutative proof nets with cuts, due to Abrusci and Maringelli, is false.

The work of Christian Retore and Sylvain Salvati on embedding of the non-associative Lambek grammars into the ACG framework has been polished and appeared in the Journal of Logic Language and Information [18]

Sylvain Salvati has proved that the problem of generation in the montagovian framework is decidable. A side effect of this work is that it gave rise to a notion of recognizable sets of simply typed lambda-terms. It has several interesting applications outside computational linguistics, such as in formal language theory, in model checking and software verification. Some work is currently being done in collaboration with other members of Méthodes Formelles (LABRI) in order to promote this concept among other formal methods. [29] , [19]

Pierre Bourreau and Sylvain Salvati worked on parsing and generation techniques for abstract categorial grammars, extending Kanazawa's technique to a wider class of -terms, where deletion can be performed.

Sylvain Salvati has started to extend formal language theory to handle free-order languages. He proposed an extension of recognizable languages, context free languages and and multiple context free languages (MCFL) so as to close them under permutation. In the two first cases, he gave some new machine characterizations and complexity results. The last case proves to be much more difficult to understand, since the permutation power of MCFLs is mostly unknown. In order to get a better understanding of this, Sylvain Salvati started to study the MIX problem which has been open for nearly 30 years now. It was believed that this language MIX that contains the words having the same numbers of a's, of b's and c's is not an MCFL, but Sylvain Salvati proved that it actually was using techniques inspired from algebraic topology. This results should resume discussions about the notion of mildly context sensitivity that is supposed to capture natural languages. Concerning these permutation closed languages, Sylvain Salvati emphasized a deep connection with Abstract Categorial Grammars which easily encompass them.