Richard Lassaigne

Approximate Probabilistic Model Checking

Symbolic model checking methods have been extended recently to the verification of probabilistic systems. However, the representation of the transition matrix may be expensive for very large systems and may induce a prohibitive cost for the model checking algorithm. In this paper, we propose an approximation method to verify quantitative properties on discrete Markov chains. We give a randomized algorithm to approximate the probability that a property expressed by some positive LTL formula is satisfied with high confidence by a probabilistic system. Our randomized algorithm requires only a succinct representation of the system and is based on an execution sampling method. We also present an implementation and a few classical examples to demonstrate the effectiveness of our approach.

Probabilistic Model Checking of the CSMA/CD Protocol Using PRISM and APMC

Carrier Sense Multiple Access/Collision Detection (CSMA/CD) is the protocol for carrier transmission access in Ethernet networks (international standard IEEE 802.3). On Ethernet, any Network Interface Card (NIC) can try to send a packet in a channel at any time. If another NIC tries to send a packet at the same time, a collision is said to occur and the packets are discarded. The CSMA/CD protocol was designed to avoid this problem, more precisely to allow a NIC to send its packet without collision. This is done by way of a randomized exponential backoff process. In this paper, we analyse the correctness of the CSMA/CD protocol, using techniques from probabilistic model checking and approximate probabilistic model checking. The tools that we use are PRISM and APMC. Moreover, we provide a quantitative analysis of some CSMA/CD properties.

APMC 3.0: Approximate Verification of Discrete and Continuous Time Markov Chains

In this paper, we give a brief overview of APMC (approximate probabilistic model checker). APMC implements approximate probabilistic verification of probabilistic systems. It is based on Monte-Carlo method and the theory of randomized approximation schemes and allows to verify extremely large models without explicitly representing the global transition system. To avoid the state-space explosion phenomenon, APMC gives an accurate approximation of the satisfaction probability of the property instead of the exact value, but using only a very small amount of memory. The version of APMC we present can handle efficiently both discrete and continuous time probabilistic systems

Approximate Verification of Probabilistic Systems

General methods have been proposed [2,4] for the model checking of probabilistic systems, where the verification of a probabilistic statement is reduced to the solution of a linear system over the system’s state space. To overcome the state space explosion problem, some probabilistic model checkers, such as PRISM [3], use MTBDDs. We propose a different solution, in which we use a Monte-Carlo algorithm [6] to approximate Prob[ψ], the probability that a temporal formula is true. We show how to obtain a randomized estimator of Prob[ψ] for a fragment of LTL formulas. This fragment is sufficient to express interesting properties such as reachability and liveness.

Probabilistic abstraction for model checking: An approach based on property testing

The goal of model checking is to verify the correctness of a given program, on all its inputs. The main obstacle, in many cases, is the intractably large size of the programs transition system. Property testing is a randomized method to verify whether some fixed property holds on individual inputs, by looking at a small random part of that input. We join the strengths of both approaches by introducing a new notion of probabilistic abstraction, and by extending the framework of model checking to include the use of these abstractions. Our abstractions map transition systems associated with large graphs to small transition systems associated with small random subgraphs. This reduces the original transition system to a family of small, even constant-size, transition systems. We prove that with high probability, “sufficiently” incorrect programs will be rejected (ϵ-robustness). We also prove that under a certain condition (exactness), correct programs will never be rejected (soundness). Our work applies to programs for graph properties such as bipartiteness, k-colorability, or any ∃∀ first order graph properties. Our main contribution is to show how to apply the ideas of property testing to syntactic programs for such properties. We give a concrete example of an abstraction for a program for bipartiteness. Finally, we show that the relaxation of the test alone does not yield transition systems small enough to use the standard model checking method. More specifically, we prove, using methods from communication complexity, that the OBDD size remains exponential for approximate bipartiteness.

Cell Assisted APMC

In this paper, we give an overview of APMC-CA (cell assisted approximate probabilistic model checker). APMC-CA is a new version of APMC dedicated to the cell processor. We show that using the cell architecture, we achieve better performances than APMC 3.0.

Recursion and decidability

The objective of this chapter is to obtain mathematical definitions for the class of computable functions and decidable problems. The class of recursive functions is first considered and we prove that it coincides with the class of functions computable by a Turing machine. Other characterizations are given: recursive systems and functions represented by a term of the lambda-calculus. What is fundamental is that all these different definitions are equivalent, as they characterize the same class of functions.

Completeness of first order logic

The deduction system we will use to show the completeness of first order logic is an extension of the natural deduction systems, presented for propositional logic. There are essentially two other sorts of formal system for mathematical reasoning. Historically, the first one was Hilbert’s system, based on axiom schemes and deduction rules. The second one, due to G. Gentzen, is the sequent calculus which has the advantage of using symmetric deduction rules.

Models of parallel computations

Sequential models such as the Turing machine or the RAM execute one transition or instruction per unit of time. In the models we introduce in this chapter, a polynomial number of transitions are followed per unit of time. We consider two models: boolean circuits and the PRAM (Parallel Random Access Machine), although there are many other possible models. Both circuits and PRAM assume synchronized elements which realize concurrent operations. Other models do not make this assumption and deal with distributed components.

First-order logic

Propositional logic allows only for the description of extremely simple language constructions: boolean operations with propositions. It is not powerful enough for representing many constructions used in computer science, linguistics, mathematics or for formalizing significant fragments of reasoning in action, as for example: • certain students attend all courses; • no student attends an uninteresting course; • can we conclude that all courses are interesting?