Regular Cost Functions, Part I: Logic and Algebra over Words

The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values "inside" and "outside". This theory is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be -- as in the classical theory of regular languages -- equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. These equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.Comment: 47 page

A Burnside Approach to the Termination of Mohri’s Algorithm for Polynomially Ambiguous Min-Plus-Automata

We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm

Distance Desert Automata and Star Height Substitutions

We introduce the notion of nested distance desert automata as a joint generalization and further development of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 22O(n2) space whether the language accepted by an n-state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration). We also show some decidability results for some substitution problems for recognizable languages