Omega-Regular Model Checking

Regular model checking is a generic technique for verification of infinite-state and/or parametrised systems which uses finite word automata or finite tree automata to finitely represent potentially infinite sets of reachable configurations of the systems being verified. The problems addressed by regular model checking are typically undecidable. In order to facilitate termination in as many cases as possible, acceleration is needed in the incremental computation of the set of reachable configurations in regular model checking. In this work, we describe how various incrementally refinable abstractions on finite (word and tree) automata can be used for this purpose. Moreover, the use of abstraction does not only increase chances of the technique to terminate, but it also significantly reduces the problem of an explosion in the number of states of the automata that are generated by regular model checking. We illustrate the efficiency of abstract regular (tree) model checking in verification of simple systems with various sources of infinity such as unbounded counters, queues, stacks, and parameters. We then show how abstract regular tree model checking can be used for verification of programs manipulating tree-like dynamic data structures. Even more complex data structures can be handled using a suitable tree-like encoding.

Tree regular model checking: A simulation-based approach

We consider the framework of regular tree model checking where sets of configurations of a system are represented by regular tree languages and its dynamics is modeled by a term rewriting system (or a regular tree transducer). We focus on the computation of the reachability set R*(L) where R is a regular tree transducer and L is a regular tree language. The construction of this set is not possible in general. Therefore, we present a general acceleration technique, called regular tree widening which allows to speed up the convergence of iterative fixpoint computations in regular tree model checking. This technique can be applied uniformly to various kinds of transformations. We show the application of our framework to different analysis contexts: verification of parameterized tree networks and data-flow analysis of multithreaded programs. Parametrized networks are modeled by relabeling tree transducers, and multithreaded programs are modeled by term rewriting rules encoding transformations on control structures. We prove that our widening technique can emulate many existing algorithms for special classes of transformations and we show that it can deal with transformations beyond the scope of these algorithms.

Regular model checking is a form of symbolic model checking technique for systems whose states can be represented as finite words over a finite alphabet, where regular sets are used as symbolic representation. A major problem in symbolic model checking of parameterized and infinite-state systems is that fixpoint computations to generate the set of reachable states or the set of reachable loops do not terminate in general. Therefore, acceleration techniques have been developed, which calculate the effect of arbitrarily long sequences of transitions generated by some action. We present a systematic method for using acceleration in regular model checking, for the case where each transition changes at most one position in the word; this includes many parameterized algorithms and algorithms on data structures. The method extracts a maximal (in a certain sense) set of actions from a transition relation. These actions, and systematically obtained compositions of them, are accelerated to speed up a fixpoint computation. The extraction can be done on any representation of the transition relation, e.g., as a union of actions or as a single monolithic transducer. Using this approach, we are for the first time able to verify completely automatically both safety and absence of starvation properties for a collection of parameterized synchronization protocols from the literature; for some protocols, we obtain significant improvements in verification time. The results show that symbolic state-space exploration, without using abstractions, is a viable alternative for verification of parameterized systems with a linear topology.KeywordsModel CheckTransition RelationReachable StateLiveness PropertySymbolic Model CheckThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Within the context of the verification of infinite-statesystems, (ω)-Regular model checking is the name ofa family of techniques in which states are represented by words,sets of states by finite automata on these objects, and transitionsby finite automata operating on pairs of state encodings, i.e.finite-state transducers. If the states are encoded by finitewords, then sets of (pairs of) states can be represented byfinite-word automata. This setting can be used to represent variousclasses of infinite-state systems, [17], including parametricsystems, FIFO-queue systems, and systems manipulating integervariables (those defined in Presburger arithmetic). When the statesare encoded by infinite words, sets of (pairs of) states arerepresented by deterministic weak Buchi automata. This settingis mainly used to represent systems involving both integer and realvariables [4,6], such as linear hybrid systems with a constantderivative.

We propose abstract regular model checking as a new generic technique for verification of parametric and infinite-state systems. The technique combines the two approaches of regular model checking and verification by abstraction. We propose a general framework of the method as well as several concrete ways of abstracting automata or transducers, which we use for modelling systems and encoding sets of their configurations as usual in regular model checking. The abstraction is based on collapsing states of automata (or transducers) and its precision is being incrementally adjusted by analysing spurious counterexamples. We illustrate the technique on verification of a wide range of systems including a novel application of automata-based techniques to an example of systems with dynamic linked data structures.

Regular model checking is a technique for the verification of infinite-state systems whose configurations can be represented as finite words over a suitable alphabet. The form we are studying applies to systems whose set of initial configurations is regular, and whose transition relation is captured by a length-preserving transducer. To verify safety properties, regular model checking iteratively computes automata recognizing increasingly larger regular sets of reachable configurations, and checks if they contain unsafe configurations. Since this procedure often does not terminate, acceleration, abstraction, and widening techniques have been developed to compute a regular superset of the reachable configurations. In this paper, we develop a complementary procedure. Instead of approaching the set of reachable configurations from below, we start with the set of all configurations and approach it from above. We use that the set of reachable configurations is equal to the intersection of all inductive invariants of the system. Since this intersection is non-regular in general, we introduce b-invariants, defined as those representable by CNF-formulas with at most b clauses. We prove that, for every $b\geq0$, the intersection of all inductive b-invariants is regular, and we construct an automaton recognizing it. We show that whether this automaton accepts some unsafe configuration is in EXPSPACE for every $b\geq0$, and PSPACE-complete for b=1. Finally, we study how large must b be to prove safety properties of a number of benchmarks.

We revisit the classic problem of proving safety over parameterised concurrent systems, i.e., an infinite family of finite-state concurrent systems that are represented by some finite (symbolic) means. An example of such an infinite family is a dining philosopher protocol with any number n of processes (n being the parameter that defines the infinite family). Regular model checking is a well-known generic framework for modelling parameterised concurrent systems, where an infinite set of configurations (resp. transitions) is represented by a regular set (resp. regular transducer). Although verifying safety properties in the regular model checking framework is undecidable in general, many sophisticated semi-algorithms have been developed in the past fifteen years that can successfully prove safety in many practical instances. In this paper, we propose a simple solution to synthesise regular inductive invariants that makes use of Angluin's classic L∗ algorithm (and its variants). We provide a termination guarantee when the set of configurations reachable from a given set of initial configurations is regular. We have tested L∗ algorithm on standard (as well as new) examples in regular model checking including the dining philosopher protocol, the dining cryptographer protocol, and several mutual exclusion protocols (e.g. Bakery, Burns, Szymanski, and German). Our experiments show that, despite the simplicity of our solution, it can perform at least as well as existing semi-algorithms.

In this book we focused on established decidability results in parameterized model checking. As this topic is a lively field, we cannot make any claim about completeness of our survey. In particular, we left out the large body of research that includes invariant-based techniques, regular model checking, symbolic methods, techniques for counter automata, and abstraction-based methods. In this chapter we give a brief overview of these methods and give further references to the interested reader. Our classification of the techniques is unavoidably subjective, as except for several cornerstone papers, many techniques adopted multiple different ideas.

We address the problem of computing simulation relations over tree automata. In particular, we consider downward and upward simulations on tree automata, which are, loosely speaking, analogous to forward and backward relations over word automata. We provide simple and efficient algorithms for computing these relations based on a reduction to the problem of computing simulations on labelled transition systems. Furthermore, we show that downward and upward relations can be combined to get relations compatible with the tree language equivalence, which can subsequently be used for an efficient size reduction of nondeterministic tree automata. This is of a very high interest, for instance, for symbolic verification methods such as regular model checking, which use tree automata to represent infinite sets of reachable configurations. We provide experimental results showing the efficiency of our algorithms on examples of tree automata taken from regular model checking computations.

We present an efficient algorithm to reduce the size of nondeterministic tree automata, while retaining their language. It is based on new transition pruning techniques, and quotienting of the state space w.r.t. suitable equivalences. It uses criteria based on combinations of downward and upward simulation preorder on trees, and the more general downward and upward language inclusions. Since tree-language inclusion is EXPTIME-complete, we describe methods to compute good approximations in polynomial time. We implemented our algorithm as a module of the well-known libvata tree automata library, and tested its performance on a given collection of tree automata from various applications of libvata in regular model checking and shape analysis, as well as on various classes of randomly generated tree automata. Our algorithm yields substantially smaller and sparser automata than all previously known reduction techniques, and it is still fast enough to handle large instances.

In reinforcement learning, an agent incrementally refines a behavioral policy through a series of episodic interactions with its environment. This process can be characterized as explicit reinforcement learning, as it deals with explicit states and concrete transitions. Building upon the concept of symbolic model checking, we propose a symbolic variant of reinforcement learning, in which sets of states are represented through predicates and transitions are represented by predicate transformers. Drawing inspiration from regular model checking, we choose regular languages over the states as our predicates, and rational transductions as predicate transformations. We refer to this framework as regular reinforcement learning, and study its utility as a symbolic approach to reinforcement learning. Theoretically, we establish results around decidability, approximability, and efficient learnability in the context of regular reinforcement learning. Towards practical applications, we develop a deep regular reinforcement learning algorithm, enabled by the use of graph neural networks. We showcase the applicability and effectiveness of (deep) regular reinforcement learning through empirical evaluation on a diverse set of case studies.

An injection structure \({\mathcal A}= (A,f)\) is a set A together with a one-place one-to-one function f. \({\mathcal A}\) is a Finite State Transducer (abbreviated FST) injection structure if A is a regular set, that is, the set of words accepted by some finite automaton, and f is realized by a deterministic finite-state transducer. Automatic relational structures have been well-studied along with the isomorphism problem for automatic structures. For an FST injection structure (A, f), the graph of f is not necessarily automatic. We continue the study of the complexity of FST injection structures by showing that the isomorphism problem for unary FST injection structures is decidable in quadratic time in the size (number of states) of the FST.KeywordsComputability theoryInjection structuresAutomatic structuresFinite state automataFinite state transducers

It is well known that reversible finite automata do not accept all regular languages, that reversible pushdown automata do not accept all deterministic context-free languages, and that reversible queue automata are less powerful than deterministic real-time queue automata. It is of significant interest from both a practical and theoretical point of view to close these gaps. We here extend these reversible models by a preprocessing unit which is basically a reversible injective and length-preserving finite state transducer. It turns out that preprocessing the input using such weak devices increases the computational power of reversible deterministic finite automata to the acceptance of all regular languages, whereas for reversible pushdown automata the accepted family of languages lies strictly in between the reversible deterministic context-free languages and the real-time deterministic context-free languages. For reversible queue automata the preprocessing of the input leads to machines that are stronger than real-time reversible queue automata, but less powerful than real-time deterministic (irreversible) queue automata. Moreover, it is shown that the computational power of all three types of machines is not changed by allowing the preprocessing finite state transducer to work irreversibly. Finally, we examine the closure properties of the family of languages accepted by such machines.

Tree Regular Model Checking (TRMC) is the name of a family of techniques for analyzing infinite-state systems in which states are represented by trees and sets of states by tree automata. The central problem is to decide whether a set of bad states belongs to the set of reachable states. An obstacle is that this set is in general neither regular nor computable in finite time.This paper proposes a new CounterExample Guided Abstraction Refinement (CEGAR) algorithm for TRMC. Our approach relies on a new equational-abstraction based completion algorithm to compute a regular overapproximation of the set of reachable states in finite time. This set is represented by \(\mathcal{R}_{/E}\)-automata, a new extended tree automaton formalism whose structure can be exploited to detect and remove false positives in an efficient manner. Our approach has been implemented in TimbukCEGAR, a new toolset that is capable of analyzing Java programs by exploiting an elegant translation from the Java byte code to term rewriting systems. Experiments show that TimbukCEGAR outperforms existing CEGAR-based completion algorithms. Contrary to existing TRMC toolsets, the answers provided by TimbukCEGAR are certified by Coq, which means that they are formally proved correct.KeywordsReachable StateCritical PairCryptographic ProtocolProof AssistantJava Virtual MachineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.