Section: New Results

Complexity in Counter Systems and Substructural Logics

Participant : Sylvain Schmitz.

The ties between propositional substructural logics (like linear logic, relevance logic, affine logic, etc.) on the one hand and extensions of vector addition systems on the other hand have long been known, as they lie for instance at the heart of undecidability proof of provability in linear logic. In a series a papers we recently revisited these connections with an eye on complexity issues. This allowed us to prove tight complexity bounds on provability in affine and contractive fragments of linear logic [20] , in affine $(!,\oplus )$-Horn linear logic [16] , and in implicational relevance logic [21] (an open problem for more than 25 years, with consequences on type inhabitation in the $\lambda I$-calculus). Our work also yields a new Tower lower bound on reachability in branching vector addition systems [20] , which entails the same lower bound for logics on XML trees [4] , for which decidability is still open.

Although the connection with data logics might not seem obvious at first, the models of counter systems considered in these papers are tightly connected with logics for XML processing [5] , [4] . Further investigations in the relationships between data logics, substructural logics, and counter systems are the main thrust behind the just accepted ANR PRODAQ project (see Section  8.1.1 ).