Commutative Languages and their Composition by Consensual Methods

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP languages. The paper contributes to the knowledge of the CREG family, and introduces novel techniques for language composition, based on arithmetic congruences that act as language signatures. Open problems are listed.Comment: In Proceedings AFL 2014, arXiv:1405.527

The submonoid and rational subset membership problems for graph groups

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids. We also show that the rational subset membership problem is decidable for a graph group if and only if the independence graph is a transitive forest, answering a question of Kambites, Silva, and the second author. Finally we prove that for certain amalgamated free products and HNN-extensions the rational subset and submonoid membership problems are recursively equivalent. In particular, this applies to finitely generated groups with two or more ends that are either torsion-free or residually finite

Learning semilinear sets from examples and via queries

AbstractSemilinear sets play an important role in parallel computation models such as matrix grammars, commutative grammars, and Petri nets. In this paper, we consider the problems of learning semilinear sets from examples and via queries. We shall show that (1) the family of semilinear sets is not learnable only from positive examples, while the family of linear sets is learnable only from positive examples, although the problem of learning linear sets from positive examples seems to be computationally intractable; (2) if for any unknown semilinear set Su and any conjectured semilinear set S′, queries whether or not Su⊆S′ and queries whether or not S′⊆Su can be made, there exists a learning procedure which identifies any semilinear set and halts, although the procedure is time-consuming; (3) however, under the same condition, for each fixed dimension, there exist meaningful subfamilies of semilinear sets learnable in polynomial time of the minimum size of representations and, in particular, for any variable dimension, if for any unknown linear set Lu and any conjectured semilinear set S′, queries whether or not Lu⊆S′ can be made, the family of linear sets is learnable in polynomial time of the minimum size of representations and the dimension

Algebraic Principles for Rely-Guarantee Style Concurrency Verification Tools

We provide simple equational principles for deriving rely-guarantee-style inference rules and refinement laws based on idempotent semirings. We link the algebraic layer with concrete models of programs based on languages and execution traces. We have implemented the approach in Isabelle/HOL as a lightweight concurrency verification tool that supports reasoning about the control and data flow of concurrent programs with shared variables at different levels of abstraction. This is illustrated on two simple verification examples