## Section: New Results

### Properties of the Formalisms

#### Formal grammars and type theory

Participants : Pierre Bourreau, Sylvain Salvati [correspondent] .

Pierre Bourreau and Sylvain Salvati have put in correspondence a syntactic criterion (negatively non-duplicating) on types and the -terms (almost-affine) that inhabit them. This result has been proved using a game theoretic approach. Furthermore, the interest in this correspondence lies in the fact that negatively non-duplicating types have at most one inhabitant.

Pierre Bourreau and Sylvain Salvati completed the extension of Kanazawa's technique for parsing almost linear ACGs to parsing almost affine ACGs.

Sylvain Salvati studied a newly defined class of automata,
*higher-order pushdown automata with collapse*, and proved they
where computing the (possibly infinite) tree generated by a
higher-order programming scheme using Krivine machines. This result
main interest consists in showing that Krivine machine may well be a
good way of studying the properties of higher-order programming
schemes.

Sylvain Salvati worked further on the notion of recognizability in the simply typed -calculus. He has given a definition in terms of congruences of finite index in order to study a possible extension of Eilenberg variety Theorem to recognizable sets of -terms. Surprisingly a difficulty of this line of research is to prove that the congruential definition of recognizability is equivalent to the ones that use standard models or intersection types.

#### Properties of mildly context sensitive formalisms

Participant : Sylvain Salvati [correspondent] .

Sylvain Salvati obtained, in a collaboration with Makoto Kanazawa, a precise account of the copying power of well-nested Multiple Context-Free Language (MCFL). This result is, to the best of our knowledge, providing the simplest way of separating well-nested MCFL from MCFL.

Sylvain Salvati has pursued his research on MCFL and tried to understand the iteration (or pumping) properties of this class of languages. It turns out that he now conjectures that there are languages in the class of MCFL that are not iterable. He gave a candidate of a language that he believes not to be iterable.

Still on the difference between MCFL and well-nested MCFL, Sylvain
Salvati worked on the problem of whether MIX is a well-nested MCFL
or not. As a starting point, he tried to prove that it was not a
well-nested MCFL of rank 2 (*i.e.* a Tree Adjoining
Language). He was able to prove that MIX is not in a certain
subclass well-nested MCFL of rank 2 and he is now working on
extending this result so as to prove that MIX is actually not a Tree
Adjoining Language.