Abstract. We show that a subclass of infinite-state probabilistic programs that can be modeled by probabilistic one-counter automata (pOC) admits an efficient quantitative analysis. In particular, we show that the expected termination time can be approximated up to an arbitrarily small relative error with polynomially many arithmetic operations, and the same holds for the probability of all runs that satisfy a given ω-regular property. Further, our results establish a powerful link between pOC and martingale theory, which leads to fundamental observations about quantitative properties of runs in pOC. In particular, we provide a "divergence gap theorem", which bounds a positive non-termination probability in pOC away from zero.

Abstract. This article presents a novel generic technique for solving dataflow equations in interprocedural dataflow analysis. The technique is obtained by generalizing Newton's method for computing a zero of a differentiable function to ω-continuous semirings. Complete semilattices, the common program analysis framework, are a special class of ω-continuous semirings. We show that our generalized method always converges to the solution, and requires at most as many iterations as current methods based on Kleene's fixed-point theorem. We also show that, contrary to Kleene's method, Newton's method always terminates for arbitrary idempotent and commutative semirings. More precisely, in the latter setting the number of iterations required to solve a system of n equations is at most n.

Given Markov chains and Markov decision processes (MDPs) whose transitions are labelled with non-negative integer costs, we study the computational complexity of deciding whether the probability of paths whose accumulated cost satisfies a Boolean combination of inequalities exceeds a given threshold. For acyclic Markov chains, we show that this problem is PP-complete, whereas it is hard for the POSSLP problem and in PSPACE for general Markov chains. Moreover, for acyclic and general MDPs, we prove PSPACE-and EXP-completeness, respectively. Our results have direct implications on the complexity of computing reward quantiles in succinctly represented stochastic systems.

We show that a subclass of infinite-state probabilistic programs that can be modeled by probabilistic one-counter automata (pOC) admits an efficient quantitative analysis. In particular, we show that the expected termination time can be approximated up to an arbitrarily small relative error with polynomially many arithmetic operations, and the same holds for the probability of all runs that satisfy a given ω-regular property. Further, our results establish a powerful link between pOC and martingale theory, which leads to fundamental observations about quantitative properties of runs in pOC. In particular, we provide a "diver-gence gap theorem", which bounds a positive non-termination probability in pOC away from zero.

Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations X1 = f1(X1, . . . , Xn), . . . , Xn = fn(X1, . . . , Xn) where each fi is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE X = f (X) arises naturally in the analysis of stochastic context-free grammars, recursive Markov chains, and probabilistic pushdown automata. While the Kleene sequence f (0), f (f (0)), . . . always converges to the least solution µf , if it exists, the number of iterations needed to compute the first i bits of µf may grow exponentially in i. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs and proved that the Newton sequence converges at least as fast as the Kleene sequence and exponentially faster in many cases. They conjecture that, given an MSPE of size m, the number of Newton iterations needed to obtain i accurate bits of µf grows polynomially in i and m. In this paper we show that the number of iterations grows linearly in i for strongly connected MSPEs and may grow exponentially in m for general MSPEs.

Abstract. Proving programs terminating is a fundamental computer science challenge. Recent research has produced powerful tools that can check a wide range of programs for termination. The analog for probabilistic programs, namely termination with probability one ("almostsure termination"), is an equally important property for randomized algorithms and probabilistic protocols. We suggest a novel algorithm for proving almost-sure termination of probabilistic programs. Our algorithm exploits the power of state-of-the-art model checkers and termination provers for nonprobabilistic programs: it calls such tools within a refinement loop and thereby iteratively constructs a "terminating pattern", which is a set of terminating runs with probability one. We report on various case studies illustrating the effectiveness of our algorithm. As a further application, our algorithm can improve lower bounds on reachability probabilities.

Abstract. In this paper, we propose a new randomised algorithm for deciding language equivalence for probabilistic automata. This algorithm is based on polynomial identity testing and thus returns an answer with an error probability that can be made arbitrarily small. We implemented our algorithm, as well as deterministic algorithms of Tzeng and Doyen et al., optimised for running time whilst adequately handling issues of numerical stability. We conducted extensive benchmarking experiments, including the verification of randomised anonymity protocols, the outcome of which establishes that the randomised algorithm significantly outperforms the deterministic ones in a majority of our test cases. Finally, we also provide fine-grained analytical bounds on the complexity of these algorithms, accounting for the differences in performance.

Parikh's theorem states that the Parikh image of a context-free language is semilinear or, equivalently, that every context-free language has the same Parikh image as some regular language. We present a very simple construction that, given a context-free grammar, produces a finite automaton recognizing such a regular language.