Automata-theoretic Techniques Research Articles

An abstraction-refinement framework for verifying strategic properties in multi-agent systems with imperfect information

We investigate the verification of Multi-Agent Systems against strategic properties expressed in Alternating-time Temporal Logic under the assumptions of imperfect information and perfect recall. To this end, we develop a three-valued semantics for concurrent game structures upon which we define an abstraction method. We prove that concurrent game structures with imperfect information admit perfect information abstractions that preserve three-valued satisfaction. Furthermore, to deal with cases in which the value of a specification is undefined, we develop a novel automata-theoretic technique for the linear-time logic (LTL), then apply it to finding “failure” states. The latter can then be fed into a refinement procedure, thus providing a sound, albeit incomplete, verification method. We illustrate the overall procedure in a variant of the Train Gate Controller scenario and a simple voting protocol under imperfect information and perfect recall. We also present an implementation of our procedure and provide preliminary experimental results.

Learning formulas in finite variable logics

We consider grammar-restricted exact learning of formulas and terms in finite variable logics. We propose a novel and versatile automata-theoretic technique for solving such problems. We first show results for learning formulas that classify a set of positively- and negatively-labeled structures. We give algorithms for realizability and synthesis of such formulas along with upper and lower bounds. We also establish positive results using our technique for other logics and variants of the learning problem, including first-order logic with least fixed point definitions, higher-order logics, and synthesis of queries and terms with recursively-defined functions.

Co-Design of Arbitrated Network Control Systems With Overrun Strategies

This paper addresses co-design of platform and control of multiple control applications in a network control system. Limited and shared resources among control and non-control applications introduce delays in transmitted messages. These delays in turn can degrade system performance and cause instabilities. In this paper, we propose an overrun framework together with a co-design to achieve both optimal control performance and efficient resource utilization. The starting point for this framework is an Arbitrated Network Control System (ANCS) approach, where flexibility and transparency in the network are utilized to arbitrate control messages. Using a two-parameter model for delays experienced by control messages that classifies them as nominal, medium, and large, we propose a controller that switches between nominal, skip and abort strategies. An automata-theoretic technique is introduced to derive analytical bounds on the abort and skip rates. A co-design algorithm is proposed to optimize the selection of the overrun parameters. A case study is presented that demonstrates the ANCS approach, the overrun framework and the overall co-design.

Algorithmic games for full ground references

We present a full classification of decidable and undecidable cases for contextual equivalence in a finitary ML-like language equipped with full ground storage (both integers and reference names can be stored). The simplest undecidable type is mathsf {unit}rightarrow mathsf {unit}rightarrow mathsf {unit}. At the technical level, our results marry game semantics with automata-theoretic techniques developed to handle infinite alphabets. On the automata-theoretic front, we show decidability of the emptiness problem for register pushdown automata extended with fresh-symbol generation.

How Hard Is Positive Quantification?

We show that the constructive predicate logic with positive (covariant) quantification is hard for doubly exponential universal time, that is, for the class co- 2-N exptime . Our approach is to represent proof-search as computation of an alternating automaton. The memory of the automaton is structured in a way that strictly corresponds to scopes of the binders used in the constructed proof. This provides an application of automata-theoretic techniques in proof theory.

Knowledge base exchange: The case of OWL 2 QL

In this article, we define and study the problem of exchanging knowledge between a source and a target knowledge base (KB), connected through mappings. Differently from the traditional database exchange setting, which considers only the exchange of data, we are interested in exchanging implicit knowledge. As representation formalism we use Description Logics (DLs), thus assuming that the source and target KBs are given as a DL TBox+ABox, while the mappings have the form of DL TBox assertions. We define a general framework of KB exchange, and study the problem of translating the knowledge in the source KB according to the mappings expressed in OWL 2 QL, the profile of the standard Web Ontology Language OWL 2 based on the description logic DL-LiteR. We develop novel game- and automata-theoretic techniques, and we provide complexity results that range from NLogSpace to ExpTime.

Symbolic models for stochastic switched systems: A discretization and a discretization-free approach

Stochastic switched systems are a relevant class of stochastic hybrid systems with probabilistic evolution over a continuous domain and control-dependent discrete dynamics over a finite set of modes. In the past few years several different techniques have been developed to assist in the stability analysis of stochastic switched systems. However, more complex and challenging objectives related to the verification of and the controller synthesis for logic specifications have not been formally investigated for this class of systems as of yet. With logic specifications we mean properties expressed as formulae in linear temporal logic or as automata on infinite strings. This paper addresses these complex objectives by constructively deriving approximately equivalent (bisimilar) symbolic models of stochastic switched systems. More precisely, this paper provides two different symbolic abstraction techniques: one requires state space discretization, but the other one does not require any space discretization which can be potentially more efficient than the first one when dealing with higher dimensional stochastic switched systems. Both techniques provide finite symbolic models that are approximately bisimilar to stochastic switched systems under some stability assumptions on the concrete model. This allows formally synthesizing controllers (switching signals) that are valid for the concrete system over the finite symbolic model, by means of mature automata-theoretic techniques in the literature. The effectiveness of the results are illustrated by synthesizing switching signals enforcing logic specifications for two case studies including temperature control of a six-room building.

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones.

Specialization Slicing

This paper defines a new variant of program slicing, called specialization slicing , and presents an algorithm for the specialization-slicing problem that creates an optimal output slice. An algorithm for specialization slicing is polyvariant : for a given procedure р, the algorithm may create multiple specialized copies of р. In creating specialized procedures, the algorithm must decide for which patterns of formal parameters a given procedure should be specialized and which program elements should be included in each specialized procedure. We formalize the specialization-slicing problem as a partitioning problem on the elements of the possibly infinite unrolled program. To manipulate possibly infinite sets of program elements, the algorithm makes use of automata-theoretic techniques originally developed in the model-checking community. The algorithm returns a finite answer that is optimal (with respect to a criterion defined in the article). In particular, (i) each element replicated by the specialization-slicing algorithm provides information about specialized patterns of program behavior that are intrinsic to the program, and (ii) the answer is of minimal size (i.e., among all possible answers with property (i), there is no smaller one). The specialization-slicing algorithm provides a new way to create executable slices. Moreover, by combining specialization slicing with forward slicing, we obtain a method for removing unwanted features from a program. While it was previously known how to solve the feature-removal problem for single-procedure programs, it was not known how to solve it for programs with procedure calls.

Algorithmic program synthesis: introduction

Program synthesis is a process of producing an executable program from a specification. Algorithmic synthesis produces the program automatically, without an intervention from an expert. While classical compilation falls under the definition of algorithmic program synthesis, with the source program being the specification, the synthesis literature is typically concerned with producing programs that cannot be (easily) obtained with the deterministic transformations of a compiler. To this end, synthesis algorithms often perform a search, either in a space of candidate programs or in a space of transformations that might be composed to transform the specification into a desired program. In this introduction to the special journal issue, we survey the history of algorithmic program synthesis and introduce the contributed articles. We divide the field into reactive synthesis, which is concerned with automata-theoretic techniques for controllers that handle an infinite stream of requests, and functional synthesis, which produces programs consuming finite input. Contributed articles are divided analogously. We also provide pointers to synthesis work outside these categories and list many applications of synthesis.

Fixed points of endomorphisms of certain free products

The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.

Separation of Test-Free Propositional Dynamic Logics over Context-Free Languages

For a class L of languages let PDL[L] be an extension of Propositional Dynamic Logic which allows programs to be in a language of L rather than just to be regular. If L contains a non-regular language, PDL[L] can express non-regular properties, in contrast to pure PDL. For regular, visibly pushdown and deterministic context-free languages, the separation of the respective PDLs can be proven by automata-theoretic techniques. However, these techniques introduce non-determinism on the automata side. As non-determinism is also the difference between DCFL and CFL, these techniques seem to be inappropriate to separate PDL[DCFL] from PDL[CFL]. Nevertheless, this separation is shown but for programs without test operators.

Model-checking trace-based information flow properties*

In this paper we consider the problem of verifying trace-based information flow properties for different classes of system models. We begin by proposing an automata-theoretic technique for model-checking trace-based information flow properties for fi

Regular expression containment

We present a new sound and complete axiomatization of regular expression containment. It consists of the conventional axiomatization of concatenation, alternation, empty set and (the singleton set containing) the empty string as an idempotent semiring, the fixed- point rule E * = 1 + E × E * for Kleene-star, and a general coinduction rule as the only additional rule. Our axiomatization gives rise to a natural computational interpretation of regular expressions as simple types that represent parse trees, and of containment proofs as coercions . This gives the axiom- atization a Curry-Howard-style constructive interpretation: Containment proofs do not only certify a language-theoretic contain- ment, but, under our computational interpretation, constructively transform a membership proof of a string in one regular expression into a membership proof of the same string in another regular expression. We show how to encode regular expression equivalence proofs in Salomaa's, Kozen's and Grabmayer's axiomatizations into our containment system, which equips their axiomatizations with a computational interpretation and implies completeness of our axiomatization. To ensure its soundness, we require that the computational interpretation of the coinduction rule be a hereditarily total function. Hereditary totality can be considered the mother of syn- tactic side conditions: it "explains" their soundness, yet cannot be used as a conventional side condition in its own right since it turns out to be undecidable. We discuss application of regular expressions as types to bit coding of strings and hint at other applications to the wide-spread use of regular expressions for substring matching, where classical automata-theoretic techniques are a priori inapplicable. Neither regular expressions as types nor subtyping interpreted coercively are novel per se . Somewhat surprisingly, this seems to be the first investigation of a general proof-theoretic framework for the latter in the context of the former, however.

Rational subsets of partially reversible monoids

A class of monoids that can model partial reversibility allowing simultaneously instances of two-sided reversibility, one-sided reversibility and no reversibility is considered. Some of the basic decidability problems involving their rational subsets, syntactic congruences and characterization of recognizability, are solved using purely automata-theoretic techniques, giving further insight into the structure of recognizable languages.

Complexity results on branching-time pushdown model checking

The model checking problem of pushdown systems (PMC problem, for short) against standard branching temporal logics has been intensively studied in the literature. In particular, for the modal μ -calculus, the most powerful branching temporal logic used for verification, the problem is known to be Exptime-complete (even for a fixed formula). The problem remains Exptime-complete also for the logic CTL, which corresponds to a fragment of the alternation-free modal μ -calculus. For the logic CTL ∗ , the problem is known to be in 2Exptime. In this paper, we show that the complexity of the PMC problem for CTL ∗ is in fact 2Exptime-complete. Moreover, we give a new optimal algorithm to solve this problem based on automata theoretic techniques. Finally, we prove that the program complexity of the PMC problem against CTL (i.e., the complexity of the problem in terms of the size of the system) is Exptime-complete.

An automata-theoretic approach to constraint LTL

We consider an extension of linear-time temporal logic (LTL) with constraints interpreted over a concrete domain. We use a new automata-theoretic technique to show PSPACE decidability of the logic for the constraint systems ( Z , < , = ) and ( N , < , = ). Along the way, we give an automata-theoretic proof of a result of Balbiani and Condotta when the constraint system satisfies the completion property. Our decision procedures extend easily to handle extensions of the logic with past-time operators and constants, as well as an extension of the temporal language itself to monadic second order logic. Finally we show that the logic becomes undecidable when one considers constraint systems that allow a counting mechanism.

Decidable containment of recursive queries

One of the most important reasoning tasks on queries is checking containment, i.e., verifying whether one query yields necessarily a subset of the result of another one. Query containment is crucial in several contexts, such as query optimization, query reformulation, knowledge-base verification, information integration, integrity checking, and cooperative answering. Containment is undecidable in general for Datalog, the fundamental language for expressing recursive queries. On the other hand, it is known that containment between monadic Datalog queries and between Datalog queries and unions of conjunctive queries are decidable. It is also known that containment between unions of conjunctive two-way regular path queries, which are queries used in the context of semistructured data models containing a limited form of recursion in the form of transitive closure, is decidable. In this paper, we combine the automata-theoretic techniques at the base of these two decidability results to show that containment of Datalog in union of conjunctive two-way regular path queries is decidable in 2EXPTIME. By sharpening a known lower bound result for containment of Datalog in union of conjunctive queries we show also a matching lower bound.

A Generic Approach to the Static Analysis of Concurrent Programs with Procedures

We present a generic aproach to the static analysis of concurrent programs with procedures. We model programs as communicating pushdown systems. It is known that typical dataflow problems for this model are undecidable, because the emptiness problem for the intersection of context-free languages, which is undecidable, can be reduced to them. In this paper we propose an algebraic framework for defining abstractions (upper approximations) of context-free languages. We consider two classes of abstractions: finite-chain abstractions, which are abstractions whose domains do not contain any infinite chains, and commutative abstractions corresponding to classes of languages that contain a word if and only if they contain all its permutations. We show how to compute such approximations by combining automata theoretic techniques with algorithms for solving systems of polynomial inequations in Kleene algebras.

A generic approach to the static analysis of concurrent programs with procedures

We present a generic aproach to the static analysis of concurrent programs with procedures. We model programs as communicating pushdown systems. It is known that typical dataflow problems for this model are undecidable, because the emptiness problem for the intersection of context-free languages, which is undecidable, can be reduced to them. In this paper we propose an algebraic framework for defining abstractions (upper approximations) of context-free languages. We consider two classes of abstractions: finite-chain abstractions, which are abstractions whose domains do not contain any infinite chains, and commutative abstractions corresponding to classes of languages that contain a word if and only if they contain all its permutations. We show how to compute such approximations by combining automata theoretic techniques with algorithms for solving systems of polynomial inequations in Kleene algebras.