On the Structure and Complexity of Rational Sets of Regular Languages

In a recent thread of papers, we have introduced FQL, a precise specification language for test coverage, and developed the test case generation engine FShell for ANSI C. In essence, an FQL test specification amounts to a set of regular languages, each of which has to be matched by at least one test execution. To describe such sets of regular languages, the FQL semantics uses an automata-theoretic concept known as rational sets of regular languages (RSRLs). RSRLs are automata whose alphabet consists of regular expressions. Thus, the language accepted by the automaton is a set of regular expressions. In this paper, we study RSRLs from a theoretic point of view. More specifically, we analyze RSRL closure properties under common set theoretic operations, and the complexity of membership checking, i.e., whether a regular language is an element of a RSRL. For all questions we investigate both the general case and the case of finite sets of regular languages. Although a few properties are left as open problems, the paper provides a systematic semantic foundation for the test specification language FQL

On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces

We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions considered. For example, the logic with the interior-connectedness predicate (and without contact) is undecidable over polygons or regular closed sets in the Euclidean plane, NP-complete over regular closed sets in three-dimensional Euclidean space, and ExpTime-complete over polyhedra in three-dimensional Euclidean space.Comment: Accepted for publication in the IJCAI 2011 proceeding

On some class of Borel measurable maps and absolute Borel topological spaces

AbstractWe introduce a class of Borel measurable maps between topological spaces which is stable under usual operations. We characterize those completely regular topological spaces which are Borel sets in every regular embedding

A Program That Simplifies Regular Expressions (Tool paper)

This paper presents the main features of a system that aims to transform regular expressions into shorter equivalent expressions. The system is also capable of computing other operations useful for simplification, such as checking the inclusion of regular languages. The main novelty of this work is that it combines known but distinct ways of representing regular languages into a global unified data structure that makes the operations more efficient. In addition, representations of regular languages are dynamically reduced as operations are performed on them. Expressions are normalized and represented by a unique identifier (an integer). Expressions found to be equivalent (i.e. denoting the same regular language) are grouped into equivalence classes from which a shortest representative is chosen. The article briefly describes the main algorithms working on the global data structure. Some of them are direct adaptations of well-known algorithms, but most of them incorporate new ideas, which are really necessary to make the system efficient. Finally, to show its usefulness, the system is applied to some examples from the literature. Statistics on randomly generated sets of expressions are also provided.Comment: rejected at ATVA 202

Free Kleene algebras with domain

First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with domain operations; that is, we add union and reflexive transitive closure to the signature. In this second case, elements of the free algebras are 'regular' sets of the trees of the first case. As a corollary, the axioms of domain semirings provide a finite quasiequational axiomatisation of the equational theory of algebras of binary relations for the intermediate signature of composition, union, and domain. Next we note that our regular sets of trees are not closed under complement, but prove that they are closed under intersection. Finally, we prove that under relational semantics the equational validities of Kleene algebras with domain form a decidable set.Comment: 22 pages. Some proofs expande