Decidability in Parameterized Verification

Bloem, Roderick; Jacobs, Swen; Khalimov, Ayrat; Konnov, Igor; Rubin, Sasha; Veith, Helmut; Widder, Josef

doi:10.1145/2951860.2951873

Cited by 73 publications

(111 citation statements)

References 24 publications

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“…Other methods identify systems with well-structured transition relations [29,1,27]. An exhaustive chart of decidability results for verification of parameterized systems is drawn in [12]. When decidability is not of concern, over-approximating and semi-algorithmic techniques such as regular model checking [33,2], SMT-based bounded model checking [4,19], abstraction [10,14] and automata learning [17] can be used to deal with more general classes of systems.The efficiency of a verification method crucially relies on its ability to synthesize an inductive safety invariant, i.e., an infinite set of configurations that contains the initial configurations, is closed under the transition relation, and excludes the error configurations.…”

mentioning

confidence: 99%

Structural Invariants for the Verification of Systems with Parameterized Architectures

Bozga

Esparza

Iosif

et al. 2020

Tools and Algorithms for the Construction and Analysis of Systems

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We consider parameterized concurrent systems consisting of a finite but unknown number of components, obtained by replicating a given set of finite state automata. Components communicate by executing atomic interactions whose participants update their states simultaneously. We introduce an interaction logic to specify both the type of interactions (e.g. rendez-vous, broadcast) and the topology of the system (e.g. pipeline, ring). The logic can be easily embedded in monadic second order logic of κ ≥ 1 successors (WSκS), and is therefore decidable.Proving safety properties of such a parameterized system, like deadlock freedom or mutual exclusion, requires to infer an inductive invariant that contains all reachable states of all system instances, and no unsafe state. We present a method to automatically synthesize inductive invariants directly from the formula describing the interactions, without costly fixed point iterations. We experimentally prove that this invariant is strong enough to verify safety properties of a large number of systems including textbook examples (dining philosophers, synchronization schemes), classical mutual exclusion algorithms, cache-coherence protocols and self-stabilization algorithms, for an arbitrary number of components. arXiv:2002.07672v1 [cs.DC] 18 Feb 2020correctness for at most c processes implies correctness for any number of processes [16,25,24,7,31]. Other methods identify systems with well-structured transition relations [29,1,27]. An exhaustive chart of decidability results for verification of parameterized systems is drawn in [12]. When decidability is not of concern, over-approximating and semi-algorithmic techniques such as regular model checking [33,2], SMT-based bounded model checking [4,19], abstraction [10,14] and automata learning [17] can be used to deal with more general classes of systems.The efficiency of a verification method crucially relies on its ability to synthesize an inductive safety invariant, i.e., an infinite set of configurations that contains the initial configurations, is closed under the transition relation, and excludes the error configurations. In general, automatically synthesizing invariants requires computationally expensive fixpoint iterations [20]. In the particular case of parameterized systems, invariants can be either global, relating the local states of all processes [21], or modular, relating the local states of a few processes whose identity is irrelevant [35,18]. Our Contributions. The novelty of the approach described in this paper is three-fold:1. The architecture of the system is not fixed a priori, but given as a parameter of the verification problem. In fact, we describe parameterized systems using the Behavior-Interaction-Priorities (BIP) framework [9], in which processes are instances of finite-state component types, whose interfaces are sets of ports, labeling transitions between local states, and interactions are sets of strongly synchronizing ports, described by formulae of an interaction logic. An interaction formula captur...

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mentioning

confidence: 99%

Structural Invariants for the Verification of Systems with Parameterized Architectures

Bozga

Esparza

Iosif

et al. 2020

Tools and Algorithms for the Construction and Analysis of Systems

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“…We discuss our approach on the VASS V run , stated in Figure 1, which will serve as running example. The VASS has dimension 3 (i.e., the vectors annotating the transitions have dimension 3) and the four states s 1 , s 2 , s 3 , s 4 . In this paper we will always represent vectors using a set of variables Var , whose cardinality |Var | equals the dimension of the VASS.…”

Section: Overview and Illustration Of Resultsmentioning

confidence: 99%

“…VASSs over a parameterized initial configuration naturally arise in two areas: 1) For concurrent systems the number of system processes is often not known in advance, and thus the system is designed such that a template process can be instantiated an arbitrary number of times. The parameterized verification problem, i.e., the problem of analyzing the concurrent system for all possible system sizes, is a common theme in the literature [9,8,1,11,4,2,3]. 2) VASSs have been used as backend for the computational complexity analysis of programs [18,19,20].…”

Section: Introductionmentioning

confidence: 99%

The Polynomial Complexity of Vector Addition Systems with States

Zuleger

2020

Lecture Notes in Computer Science

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Vector addition systems are an important model in theoretical computer science and have been used in a variety of areas. In this paper, we consider vector addition systems with states over a parameterized initial configuration. For these systems, we are interested in the standard notion of computational complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states depending on the size of the initial configuration. We show that the asymptotic complexity of a given vector addition system with states is either Θ(N k ) for some computable integer k, where N is the size of the initial configuration, or at least exponential. We further show that k can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that 1 ≤ k ≤ 2 n , where n is the dimension of the considered vector addition system.

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“…Pairwise-rendezvous isPMCP with all the mentioned communication primitives, as well as others, is summarised in [14]. (ii) Specifications of parameterized systems are typically expressed in indexed temporal logic [15] which allows one to quantify over processes.…”

Section: System Modelmentioning

confidence: 99%

“…Although the PMCP is undecidable in general (see [28,52]) it becomes decidable for some combinations of communication primitives, network topologies, and specification languages, e.g., [1,8,14,21,22,30,51]. Often, it is proved decidable by a reduction to model checking finitely many finite-state systems [2,16,24,28,36].…”

Section: Introductionmentioning

confidence: 99%

Parameterized model checking of rendezvous systems

et al. 2017

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Parameterized model checking is the problem of deciding if a given formula holds irrespective of the number of participating processes. A standard approach for solving the parameterized model checking problem is to reduce it to model checking finitely many finite-state systems. This work considers the theoretical power and limitations of this technique. We focus on concurrent systems in which processes communicate via pairwise rendezvous, as well as the special cases of disjunctive guards and token passing; specifications are expressed in indexed temporal logic without the next operator; and the underlying network topologies are generated by suitable formulas and graph operations. First, we settle the exact computational complexity of the parameterized model checking problem for some of our concurrent systems, and establish new decidability results for others. Second, we consider the cases where model checking the parameterized system can be reduced to model checking some fixed number of processes, the number is known as a cutoff. We provide many cases for when such cutoffs can be computed, establish lower bounds on the size of such cutoffs, and identify cases where no cutoff exists. Third, we consider cases for which the parameterized system is equivalent to a single finite-state system (more precisely a Büchi word automaton), and establish tight bounds on the sizes of such automata.

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Decidability in Parameterized Verification

Cited by 73 publications

References 24 publications

Structural Invariants for the Verification of Systems with Parameterized Architectures

Structural Invariants for the Verification of Systems with Parameterized Architectures

The Polynomial Complexity of Vector Addition Systems with States

Parameterized model checking of rendezvous systems

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