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International audience; Guardedness restrictions are one of the principal means to obtain decidable logics — operators such as negation are restricted so that the free variables are contained in an atom. While guardedness has been applied fruitfully in the setting of first-order logic, the ability to add fixpoints while retaining decidability has been very limited. Here we show that one of the main restrictions imposed in the past can be lifted, getting a richer decidable logic by allowing fixpoints in which the parameters of the fixpoint can be unguarded. Using automata, we show that the resulting logics have a decidable satisfiability problem, and provide a fine study of the complexity of satisfiability. We show that similar methods apply to decide questions concerning the elimination of fixpoints within formulas of the logic.

Given a formula phi(x, X) positive in X, the bounded ness problem asks whether the fix point induced by phi is reached within some uniform bound independent of the structure (i.e. Whether the fix point is spurious, and can in fact be captured by a finite unfolding of the formula). In this paper, we study the bounded ness problem when phi is in the guarded fragment or guarded negation fragment of first-order logic, or the fix point extensions of these logics. It is known that guarded logics have many desirable computational and model theoretic properties, including in some cases decidable bounded ness. We prove that bounded ness for the guarded negation fragment is decidable in elementary time, and, making use of an unpublished result of Colcombet, even 2EXPTIME-complete. Our proof extends the connection between guarded logics and automata, reducing bounded ness for guarded logics to a question about cost automata on trees, a type of automaton with counters that assigns a natural number to each input rather than just a boolean.

International audience; Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed-point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

Cost functions are defined as mappings from a domain like words or trees to $\mathbb{N} \cup \left\{{\infty}\right\}$, modulo an equivalence relation ≈ which ignores exact values but preserves boundedness properties. Cost logics, in particular cost monadic second-order logic, and cost automata, are different ways to define such functions. These logics and automata have been studied by Colcombet et al. as part of a "theory of regular cost functions", an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability. We develop this theory over infinite words, and show that the classical results FO = LTL and MSO = WMSO also hold in this cost setting (where the equivalence is now up to ≈). We also describe connections with forms of weak alternating automata with counters.