Distributed graph-based state space generation

LTSMIN provides a framework in which state space generation can be distributed easily over many cores on a single compute node, as well as over multiple compute nodes. The tool works on the basis of a vector representation of the states; the individual cores are assigned the task of computing all successors of states that are sent to them. In this paper we show how this framework can be applied in the case where states are essentially graphs interpreted up to isomorphism, such as the ones we have been studying for GROOVE. This involves developing a suitable vector representation for a canonical form of those graphs. The canonical forms are computed using a third tool called BLISS. We combined the three tools to form a system for distributed state space generation based on graph grammars. We show that the time performance of the resulting system scales well (i.e., close to linear) with the number of cores. We also report surprising statistics on the memory\ud consumption, which imply that the vector representation used to store graphs in LTSMIN is more compact than the representation used in GROOVE

Alcove path and Nichols-Woronowicz model of the equivariant KK-theory of generalized flag varieties

Fomin and Kirillov initiated a line of research into the realization of the cohomology and $K$-theory of generalized flag varieties $G/B$ as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the $T$-equivariant $K$-theory of a generalized flag variety $K_T(G/B)$ in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for $K_T(G/B)$ due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach

Parameterized Synthesis

We study the synthesis problem for distributed architectures with a parametric number of finite-state components. Parameterized specifications arise naturally in a synthesis setting, but thus far it was unclear how to detect realizability and how to perform synthesis in a parameterized setting. Using a classical result from verification, we show that for a class of specifications in indexed LTL\X, parameterized synthesis in token ring networks is equivalent to distributed synthesis in a network consisting of a few copies of a single process. Adapting a well-known result from distributed synthesis, we show that the latter problem is undecidable. We describe a semi-decision procedure for the parameterized synthesis problem in token rings, based on bounded synthesis. We extend the approach to parameterized synthesis in token-passing networks with arbitrary topologies, and show applicability on a simple case study. Finally, we sketch a general framework for parameterized synthesis based on cutoffs and other parameterized verification techniques.Comment: Extended version of TACAS 2012 paper, 29 page

Dynamical systems of type (m,n) and their C*-algebras

Given positive integers n and m, we consider dynamical systems in which n copies of a topological space is homeomorphic to m copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra we denote by O_{mn}, which in turn is obtained as a quotient of the well known Leavitt C*-algebra L_{mn}, a process meant to transform the generating set of partial isometries of L{mn} into a tame set. Describing O_{mn} as the crossed-product of the universal (m,n)-dynamical system by a partial action of the free group F_{m+n}, we show that O_{mn} is not exact when n and m are both greater than or equal to 2, but the corresponding reduced crossed-product, denoted O_{mn}^r, is shown to be exact and non-nuclear. Still under the assumption that m,n>=2, we prove that the partial action of F_{m+n} is topologically free and that O_{mn}^r satisfies property (SP) (small projections). We also show that O_{mn}^r admits no finite dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.Comment: 38 page

Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic $\ell$-adic representations of \GQ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups E_{8}(\FF_{\ell}) are Galois groups over \QQ for large enough primes $\ell$.Comment: 40 page