We prove the undecidability of Core XPath 1.0 (CXP) [G. Gottlob and C. Koch, in Proc. of 17th Ann. IEEE Symp. on Logic in Computer Science, LICS ’02 (Copenhagen, July 2002). IEEE CS Press (2002) 189–202.] extended with an Inflationary Fixed Point (IFP) operator. More specifically, we prove that the satisfiability problem of this language is undecidable. In fact, the fragment of CXP+IFP containing only the self and descendant axes is already undecidable.

We describe a sequent calculus μLJ with primitives for inductive and coinductive datatypes and equip it with reduction rules allowing a sound translation of Gödel’s system T. We introduce the notion of a μ-closed category, relying on a uniform interpretation of open μLJ formulas as strong functors. We show that any μ-closed category is a sound model for μLJ. We then turn to the construction of a concrete μ-closed category based on Hyland-Ong game semantics. The model relies on three main ingredients: the construction of a general class of strong functors called open functors acting on the category of games and strategies, the solution of recursive arena equations by exploiting cycles in arenas, and the adaptation of the winning conditions of parity games to build initial algebras and terminal coalgebras for many open functors. We also prove a weak completeness result for this model, yielding a normalisation proof for μLJ.

A standard bridge between automata theory and logic is provided by the notion of characteristic formula. This paper investigates this problem for the class of event-recording automata (ERA), a subclass of timed automata in which clocks are associated with actions and that enjoys very good closure properties. We first study the problem of expressing characteristic formulae for ERA in Event-Recording Logic (ERL ), a logic introduced by Sorea to express event-based timed specifications. We prove that the construction proposed by Sorea for ERA without invariants is incorrect. More generally, we prove that timed bisimilarity cannot in general be expressed in ERL for the class of ERA , and study under which conditions on ERA it can be. Then, we introduce the logic WTμ , a new logic for event-based timed specifications closer to the timed logic ℒν that was introduced by Laroussinie, Larsen and Weise. We prove that it is strictly more expressive than ERL , and that its model-checking problem against ERA is EXPTIME -complete. Finally, we provide characteristic formulae constructions in WTμ for characterizing the general class of ERA up to timed (bi)similarity and study the complexity issues.

We consider μ-calculus formulas in a normal form: after a prefix of fixed-point quantifiers follows a quantifier-free expression. We are interested in the problem of evaluating (model checking) such formulas in a powerset lattice. We assume that the quantifier-free part of the expression can be any monotone function given by a black-box – we may only ask for its value for given arguments. As a first result we prove that when the lattice is fixed, the problem becomes polynomial (the assumption about the quantifier-free part strengthens this result). As a second result we show that any algorithm solving the problem has to ask at least about n2 (namely Ω(n2/log n)) queries to the function, even when the expression consists of one μ and one ν (the assumption about the quantifier-free part weakens this result).

We study iteration and recursion operators in the denotational semantics of typed λ-calculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhard’s finiteness spaces.