Solving Parity Games in Scala

Parity games are two-player games, played on directed graphs, whose nodes are labeled with priorities. Along a play, the maximal priority occurring infinitely often determines the winner. In the last two decades, a variety of algorithms and successive optimizations have been proposed. The majority of them have been implemented in PGSolver, written in OCaml, which has been elected by the community as the de facto platform to solve efficiently parity games as well as evaluate their performance in several specific cases. PGSolver includes the Zielonka Recursive Algorithm that has been shown to perform better than the others in randomly generated games. However, even for arenas with a few thousand of nodes (especially over dense graphs), it requires minutes to solve the corresponding game. In this paper, we deeply revisit the implementation of the recursive algorithm introducing several improvements and making use of Scala Programming Language. These choices have been proved to be very successful, gaining up to two orders of magnitude in running time

Monomial right ideals and the Hilbert series of noncommutative modules

In this paper we present a procedure for computing the rational sum of the Hilbert series of a finitely generated monomial right module $N$ over the free associative algebra $K\langle x_1,\ldots,x_n \rangle$. We show that such procedure terminates, that is, the rational sum exists, when all the cyclic submodules decomposing $N$ are annihilated by monomial right ideals whose monomials define regular formal languages. The method is based on the iterative application of the colon right ideal operation to monomial ideals which are given by an eventual infinite basis. By using automata theory, we prove that the number of these iterations is a minimal one. In fact, we have experimented efficient computations with an implementation of the procedure in Maple which is the first general one for noncommutative Hilbert series.Comment: 15 pages, to appear in Journal of Symbolic Computatio

BeSpaceD: Towards a Tool Framework and Methodology for the Specification and Verification of Spatial Behavior of Distributed Software Component Systems

In this report, we present work towards a framework for modeling and checking behavior of spatially distributed component systems. Design goals of our framework are the ability to model spatial behavior in a component oriented, simple and intuitive way, the possibility to automatically analyse and verify systems and integration possibilities with other modeling and verification tools. We present examples and the verification steps necessary to prove properties such as range coverage or the absence of collisions between components and technical details

Unified Analysis of Collapsible and Ordered Pushdown Automata via Term Rewriting

We model collapsible and ordered pushdown systems with term rewriting, by encoding higher-order stacks and multiple stacks into trees. We show a uniform inverse preservation of recognizability result for the resulting class of term rewriting systems, which is obtained by extending the classic saturation-based approach. This result subsumes and unifies similar analyses on collapsible and ordered pushdown systems. Despite the rich literature on inverse preservation of recognizability for term rewrite systems, our result does not seem to follow from any previous study.Comment: in Proc. of FRE

Multigraded Hilbert Series of noncommutative modules

In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory of regular languages, we provide conditions when the methods are effective and hence the sum of the Hilbert series is a rational function. Moreover, a characterization of finite-dimensional algebras is obtained in terms of the nilpotency of a key matrix involved in the computations. Using this result, efficient variants of the methods are also developed for the computation of Hilbert series of truncated infinite-dimensional algebras whose (non-truncated) Hilbert series may not be rational functions. We consider some applications of the computation of multigraded Hilbert series to algebras that are invariant under the action of the general linear group. In fact, in this case such series are symmetric functions which can be decomposed in terms of Schur functions. Finally, we present an efficient and complete implementation of (standard) graded and multigraded Hilbert series that has been developed in the kernel of the computer algebra system Singular. A large set of tests provides a comprehensive experimentation for the proposed algorithms and their implementations.Comment: 28 pages, to appear in Journal of Algebr

Small-world behavior in a system of mobile elements

We analyze the propagation of activity in a system of mobile automata. A number r L^d of elements move as random walkers on a lattice of dimension d, while with a small probability p they can jump to any empty site in the system. We show that this system behaves as a Dynamic Small-World (DSW) and present analytic and numerical results for several quantities. Our analysis shows that the persistence time T* (equivalent to the persistence size L* of small-world networks) scales as T* ~ (r p)^(-t), with t = 1/(d+1).Comment: To appear in Europhysics Letter