Model Checking Probabilistic Pushdown Automata

We consider the model checking problem for probabilistic pushdown automata (pPDA) and properties expressible in various probabilistic logics. We start with properties that can be formulated as instances of a generalized random walk problem. We prove that both qualitative and quantitative model checking for this class of properties and pPDA is decidable. Then we show that model checking for the qualitative fragment of the logic PCTL and pPDA is also decidable. Moreover, we develop an error-tolerant model checking algorithm for PCTL and the subclass of stateless pPDA. Finally, we consider the class of omega-regular properties and show that both qualitative and quantitative model checking for pPDA is decidable

Probabilistic regular graphs

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. We present here an algorithm to perform approximate verification of PCTL formulae. Moreover, we prove that the exact model-checking problem for PCTL on probabilistic regular graphs is undecidable, unless restricting to qualitative properties. Our results generalise those of EKM06, on probabilistic pushdown automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611

Decisive Markov Chains

We consider qualitative and quantitative verification problems for infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, this holds for probabilistic lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains are decisive. This class includes probabilistic vector addition systems (PVASS) and probabilistic noisy Turing machines (PNTM). We consider both safety and liveness problems for decisive Markov chains, i.e., the probabilities that a given set of states F is eventually reached or reached infinitely often, respectively. 1. We express the qualitative problems in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm solves the approximate quantitative liveness problem. 3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM.Comment: 32 pages, 0 figure

Undecidability of model-checking branching-time properties of stateless probabilistic pushdown process

In this paper, we settle a problem in probabilistic verification of infinite--state process (specifically, {\it probabilistic pushdown process}). We show that model checking {\it stateless probabilistic pushdown process} (pBPA) against {\it probabilistic computational tree logic} (PCTL) is undecidable.Comment: Author's comments on referee's report added, Interestin

Model-checking branching-time properties of probabilistic automata and probabilistic one-counter automata

This paper studies the problem of model-checking of probabilistic automaton and probabilistic one-counter automata against probabilistic branching-time temporal logics (PCTL and PCTL$^*$). We show that it is undecidable for these problems. We first show, by reducing to emptiness problem of probabilistic automata, that the model-checking of probabilistic finite automata against branching-time temporal logics are undecidable. And then, for each probabilistic automata, by constructing a probabilistic one-counter automaton with the same behavior as questioned probabilistic automata the undecidability of model-checking problems against branching-time temporal logics are derived, herein.Comment: Comments are welcom

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)

10252 Abstracts Collection -- Game Semantics and Program Verification

From 20th to 25th June 2010, the Dagstuhl Seminar "Game Semantics and Program Verification\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

08171 Abstracts Collection -- Beyond the Finite: New Challenges in Verification and Semistructured Data

From 20.04. to 25.04.2008, the Dagstuhl Seminar 08171 ``Beyond the Finite: New Challenges in Verification and Semistructured Data\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations $X_1 = f_1(X_1, ..., X_n),$ $..., X_n = f_n(X_1, ..., X_n)$ where each $f_i$ is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE $\vec X = \vec f(\vec X)$ arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and back-button processes. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs. In a previous paper we have proved the existence of a threshold $k_{\vec f}$ for strongly connected MSPEs, such that after $k_{\vec f}$ iterations of Newton's method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for $k_{\vec f}$ as a function of the minimal component of the least fixed-point $\mu\vec f$ of $\vec f(\vec X)$. Using this result we show that $k_{\vec f}$ is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from back-button processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least $1/w2^h$ new bits of the solution, where $w$ and $h$ are the width and height of the DAG of strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS conference. Two minor mistakes correcte

A Counting Logic for Structure Transition Systems

Quantitative questions such as "what is the maximum number of tokens in a place of a Petri net?" or "what is the maximal reachable height of the stack of a pushdown automaton?" play a significant role in understanding models of computation. To study such problems in a systematic way, we introduce structure transition systems on which one can define logics that mix temporal expressions (e.g. reachability) with properties of a state (e.g. the height of the stack). We propose a counting logic Qmu[#MSO] which allows to express questions like the ones above, and also many boundedness problems studied so far. We show that Qmu[#MSO] has good algorithmic properties, in particular we generalize two standard methods in model checking, decomposition on trees and model checking through parity games, to this quantitative logic. These properties are used to prove decidability of Qmu[#MSO] on tree-producing pushdown systems, a generalization of both pushdown systems and regular tree grammars