Noomene Ben Henda

Eager markov chains

We consider infinite-state discrete Markov chains which are eager: the probability of avoiding a defined set of final states for more than n steps is bounded by some exponentially decreasing function f(n). We prove that eager Markov chains include those induced by Probabilistic Lossy Channel Systems, Probabilistic Vector Addition Systems with States, and Noisy Turing Machines, and that the bounding function f(n) can be effectively constructed for them. Furthermore, we study the problem of computing the expected reward (or cost) of runs until reaching the final states, where rewards are assigned to individual runs by computable reward functions. For eager Markov chains, an effective path exploration scheme, based on forward reachability analysis, can be used to approximate the expected reward up-to an arbitrarily small error.

Regular model checking without transducers (on efficient verification of parameterized systems)

We give a simple and efficient method to prove safety properties for parameterized systems with linear topologies. A process in the system is a finite-state automaton, where the transitions are guarded by both local and global conditions. Processes may communicate via broadcast, rendez-vous and shared variables. The method derives an over-approximation of the induced transition system, which allows the use of a simple class of regular expressions as a symbolic representation. Compared to traditional regular model checking methods, the analysis does not require the manipulation of transducers, and hence its simplicity and efficiency. We have implemented a prototype which works well on several mutual exclusion algorithms and cache coherence protocols.

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, all Markov chains induced by probabilistic lossy channel systems (PLCS) con- tain a finite attractor and are thus decisive. Furthermore, all globally coarse Markov chains are decisive. The class of globally coarse Markov chains includes, e.g., those induced by probabilistic vector addition systems (PVASS) with upward-closed sets F, and all Markov chains induced by probabilistic noisy Turing machines (PNTM) (a generalization of the noisy Turing machines (NTM) of Asarin and Collins). We consider both safety and liveness problems for decisive Markov chains. Safety: What is the probability that a given set of states F is eventually reached. Liveness: What is the probability that a given set of states is reached infinitely often. There are three variants of these questions. (1) The qualitative problem, i.e., deciding if the probability is one (or zero); (2) the approximate quantitative problem, i.e., computing the probability up-to arbitrary precision; (3) the exact quantitative problem, i.e., computing probabilities exactly. 1. We express the qualitative problem 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 can be used to solve 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 (unlike for probabilistic pushdown automata).

Handling parameterized systems with non-atomic global conditions

We consider verification of safety properties for parameterized systems with linear topologies. A process in the system is an extended automaton, where the transitions are guarded by both local and global conditions. The global conditions are non-atomic, i.e., a process allows arbitrary interleavings with other transitions while checking the states of all (or some) of the other processes. We translate the problem into model checking of infinite transition systems where each configuration is a labeled finite graph. We derive an over-approximation of the induced transition system, which leads to a symbolic scheme for analyzing safety properties. We have implemented a prototype and run it on several nontrivial case studies, namely non-atomic versions of Burns protocol, Dijkstras protocol, the Bakery algorithm, Lamports distributed mutual exclusion protocol, and a two-phase commit protocol used for handling transactions in distributed systems. As far as we know, these protocols have not previously been verified in a fully automated framework.

Verifying infinite Markov chains with a finite attractor or the global coarseness property

We consider infinite Markov chains which either have a finite attractor or satisfy the global coarseness property. Markov chains derived from probabilistic lossy channel systems (PLCS) or probabilistic vector addition systems with states (PVASS) are classic examples for these types, respectively. We consider three different variants of the reachability problem and the repeated reachability problem: the qualitative problem, i.e., deciding if the probability is one (or zero); the approximate quantitative problem, i.e., computing the probability up-to arbitrary precision; the exact quantitative problem, i.e., computing probabilities exactly. We express the qualitative problem in abstract terms for Markov chains with a finite attractor and for globally coarse Markov chains, and show an almost complete picture of its decidability of PLCS and PVASS. We also show that the path enumeration algorithm of (P. Iyer et al., 1997) terminates for our types of Markov chain and can thus be used to solve the approximate quantitative reachability problem. Furthermore, a modified variant of this algorithm can solve the approximate quantitative repeated reachability problem for Markov chains with a finite attractor. Finally, we show that the exact probability of (repeated) reachability cannot be effectively expressed in the first-order theory of the reals (R,+,*,/spl les/) for either PLCS or PVASS (unlike for other probabilistic models, e.g., probabilistic pushdown automata (J. Esparza et al., 2004, K. Etessami et al., 2005, J. Esparza et al., 2004).

MONOTONIC ABSTRACTION (ON EFFICIENT VERIFICATION OF PARAMETERIZED SYSTEMS)

We introduce the simple and efficient method of monotonic abstraction to prove safety properties for parameterized systems with linear topologies. A process in the system is a finite-state automato ...

Stochastic games with lossy channels

We consider turn-based stochastic games on infinite graphs induced by game probabilistic lossy channel systems (GPLCS), the game version of probabilistic lossy channel systems (PLCS). We study games with Buchi (repeated reachability) objectives and almost-sure winning conditions. These games are pure memoryless determined and, under the assumption that the target set is regular, a symbolic representation of the set of winning states for each player can be effectively constructed. Thus, turn-based stochastic games on GPLCS are decidable. This generalizes the decidability result for PLCS-induced Markov decision processes in [10].

Parameterized Tree Systems

Several recent works have considered parameterized verification, i.e. automatic verification of systems consisting of an arbitrary number of finite-state processes organized in a linear array. The aim of this paper is to extend these works by giving a simple and efficient method to prove safety properties for systems with tree-likearchitectures. A process in the system is a finite-state automaton and a transition is performed jointly by a process and its parent and children processes. The method derives an over-approximation of the induced transition system, which allows the use of finite trees as symbolic representations of infinite sets of configurations. Compared to traditional methods for parameterized verification of systems with tree topologies, our method does not require the manipulation of tree transducers, hence its simplicity and efficiency. We have implemented a prototype which works well on several nontrivial tree-based protocols.

Limiting Behavior of Markov Chains with Eager Attractors

We consider discrete infinite-state Markov chains which contain an eager finite attractor. A finite attractor is a finite subset of states that is eventually reached with probability 1 from every other state, and the eagerness condition requires that the probability of avoiding the attractor in n or more steps after leaving it is exponentially bounded in n. Examples of such Markov chains are those induced by probabilistic lossy channel systems and similar systems. We show that the expected residence time (a generalization of the steady state distribution) exists for Markov chains with eager attractors and that it can be effectively approximated to arbitrary precision. Furthermore, arbitrarily close approximations of the limiting average expected reward, with respect to state-based bounded reward functions, are also computable