Probabilistic Reachability for Parametric Markov Models

Abstract. Given a parametric Markov model, we consider the problem of computing the formula expressing the probability of reaching a given set of states. To attack this principal problem, Daws has suggested to first convert the Markov chain into a finite automaton, from which a regular expression is computed. Afterwards, this expression is evaluated to a closed form expression representing the reachability probability. This paper investigates how this idea can be turned into an effective procedure. It turns out that the bottleneck lies in an exponential growth of the regular expression relative to the number of states. We therefore proceed differently, by tightly intertwining the regular expression computation with its evaluation. This allows us to arrive at an effective method that avoids the exponential blow up in most practical cases. We give a detailed account of the approach, also extending to parametric models with rewards and with non-determinism. Experimental evidence is provided, illustrating that our implementation provides meaningful insights on non-trivial models.

Probabilistic Reachability for Parametric Markov Models

Abstract. Given a parametric Markov model, we consider the problem of computing the rational function expressing the probability of reaching a given set of states. To attack this principal problem, Daws has suggested to first convert the Markov chain into a finite automaton, from which a regular expression is computed. Afterwards, this expression is evaluated to a closed form function representing the reachability probability. This paper investigates how this idea can be turned into an effective procedure. It turns out that the bottleneck lies in the growth of the regular expression relative to the number of states (nΘ(logn)). We therefore proceed differently, by tightly intertwining the regular expression computation with its evaluation. This allows us to arrive at an effective method that avoids this blow up in most practical cases. We give a detailed account of the approach, also extending to parametric models with rewards and with non-determinism. Experimental evidence is provided, illustrating that our implementation provides meaningful insights on non-trivial models.

www.elsevier.com/locate/jsc Telescoping in the context of symbolic summation

S.A. Abramov a,J.J.Carette b,K.O.Geddes c,H.Q.Le c,

Laurent-Pade approximants to four kinds of Chebyshev polynomial expansion. Part 2. Clenshaw-Lord type approximants

Laurent–Padé (Chebyshev) rational approximants P m (w,w –1)/Q n (w,w –1) of Clenshaw–Lord type [2,1] are defined, such that the Laurent series of P m /Q n matches that of a given function f(w,w –1) up to terms of order w ±(m+n), based only on knowledge of the Laurent series coefficients of f up to terms in w ±(m+n). This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series of P m matches that of Q n f up to terms of order w ±(m+n), but based on knowledge of the series coefficients of f up to terms in w ±(m+2n). The Clenshaw–Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé–Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for all m0, n0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé–Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw–Lord type methods, thus validating the use of either

Symbolic Analysis of Imperative Programming Languages

Abstract. We present a generic symbolic analysis framework for imperative programming languages. Our framework is capable of computing all valid variable bindings of a program at given program points. This information is invaluable for domain-specific static program analyses such as memory leak detection, program parallelisation, and the detection of superfluous bound checks, variable aliases and task deadlocks. We employ path expression algebra to model the control flow information of programs. A homomorphism maps path expressions into the symbolic domain. At the center of the symbolic domain is a compact algebraic structure called supercontext. A supercontext contains the complete control and data flow analysis information valid at a given program point. Our approach to compute supercontexts is based purely on algebra and is fully automated. This novel representation of program semantics closes the gap between program analysis and computer algebra systems, which makes supercontexts an ideal intermediate representation for all domainspecific static program analyses. Our approach is more general than existing methods because it can derive solutions for arbitrary (even intra-loop) nodes of reducible and irreducible control flow graphs. We prove the correctness of our symbolic analysis method. Our experimental results show that the problem sizes arising from real-world applications such as the SPEC95 benchmark suite are tractable for our symbolic analysis framework.