The Covering Problem

An important endeavor in computer science is to understand the expressive power of logical formalisms over discrete structures, such as words. Naturally, "understanding" is not a mathematical notion. This investigation requires therefore a concrete objective to capture this understanding. In the literature, the standard choice for this objective is the membership problem, whose aim is to find a procedure deciding whether an input regular language can be defined in the logic under investigation. This approach was cemented as the right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on first-order logic and has been in use since then. However, membership questions are hard: for several important fragments, researchers have failed in this endeavor despite decades of investigation. In view of recent results on one of the most famous open questions, namely the quantifier alternation hierarchy of first-order logic, an explanation may be that membership is too restrictive as a setting. These new results were indeed obtained by considering more general problems than membership, taking advantage of the increased flexibility of the enriched mathematical setting. This opens a promising research avenue and efforts have been devoted at identifying and solving such problems for natural fragments. Until now however, these problems have been ad hoc, most fragments relying on a specific one. A unique new problem replacing membership as the right one is still missing. The main contribution of this paper is a suitable candidate to play this role: the Covering Problem. We motivate this problem with 3 arguments. First, it admits an elementary set theoretic formulation, similar to membership. Second, we are able to reexplain or generalize all known results with this problem. Third, we develop a mathematical framework and a methodology tailored to the investigation of this problem

Analytic aspects of the shuffle product

There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of D-finite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models D-finite generating functions

Combinatorial specification of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positiveand algebraic. It also yields a uniform random sampler of permutations in C. The method presentedis fully algorithmic