Model Checking: From Tools to Theory

While the theory of languages of words is very mature, our understanding of relations on words is still lagging behind. And yet such relations appear in many new applications such as verification of parameterized systems, querying graph-structured data, and information extraction, for instance. Classes of well-behaved relations typically used in such applications are obtained by adapting some of the equivalent definitions of regularity of words for relations, leading to non-equivalent notions of recognizable, regular, and rational relations. The goal of this paper is to propose a systematic way of defining classes of relations on words, of which these three classes are just natural examples, and to demonstrate its advantages compared to some of the standard techniques for studying word relations. The key idea is that of a synchronization of a pair of words, which is a word over an extended alphabet. Using it, we define classes of relations via classes of regular languages over a fixed alphabet, just {1,2} for binary relations. We characterize some of the standard classes of relations on words via finiteness of parameters of synchronization languages, called shift, lag, and shiftlag. We describe these conditions in terms of the structure of cycles of graphs underlying automata, thereby showing their decidability. We show that for these classes there exist canonical synchronization languages, and every class of relations can be effectively re-synchronized using those canonical representatives. We also give sufficient conditions on synchronization languages, defined in terms of injectivity and surjectivity of their Parikh images, that guarantee closure under intersection and complement of the classes of relations they define.

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of omega-tree languages derived from weak regular omega-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived omega-tree languages to learning the underlying class of omega-word languages, for any class of omega-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [Maler and Pneuli, Inform. Comput., 1995] for the class of weak regular omega-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi omega-word acceptors. A previous claim of an algorithm to learn regular omega-trees due to Jayasrirani, Begam and Thomas [Jayasrirani et al., ICGI, 2008] is unfortunately incorrect, as shown in [Angluin, YALEU/DCS/TR-1528, 2016].

In this research, inductive inference is done with an informant on the class of regular languages. The approach is to evolve formal language accepters which are consistent with a set of sample strings from the language, and a set of sample strings known not to be in the language. Deterministic finite automata (DFA) were chosen as the formal language accepters to alleviate the computational difficulties of nondeterministic constructs such as rewrite grammars. Genetic programming (GP) offers two significant improvements for regular language induction over genetic algorithms. First, GP allows the size of the solution (the DFA) to be determined at run time in response to population pressure. Second, GP's potential for assuring correct dependencies in complex individuals can be exploited to assure that all states in a DFA are reachable from the start state. The contribution of this research is the effective translation of DFAs to S-expressions, the application of renumbering, and of editing to the problem of language induction. DFAs or transition tables form the basis of many problems. By using the techniques found in this paper, many of these problems can be directly translated into the domain of genetic programming. >

Learning from positive data constitutes an important topic in Grammatical Inference since it is believed that the acquisition of grammar by children only needs syntactically correct (i.e. positive) instances. However, classical learning models provide no way to avoid the problem of overgeneralization. In order to overcome this problem, we use here a learning model from simple examples, where the notion of simplicity is defined with the help of Kolmogorov complexity. We show that a general and natural heuristic which allows learning from simple positive examples can be developed in this model. Our main result is that the class of regular languages is probably exactly learnable from simple positive examples.

This paper studies the relaxation of word-order, using dependency grammars (DG’s). We study ‘relax-ability of languages (and DG’s)’ based on the notion of DR-(non)projectivity. We obtain two infinite scales of classes of languages using different degrees of relaxation. The one scale contains the class of context-free languages as its first (minimal) element, the other one starts by the class of regular languages.

Learning approximately regular languages with reversible languages

Inspired by RNA-editing, we study an elementary formalism of string replacement based on insertion and deletion using a fixed finite set of guides, and in which only occurrences of a single symbol are inserted or deleted. While in this replacement mechanism computation lengths are at most exponential by construction, we show that this exponential upper bound is tight. Moreover, we show that both the class of regular languages and the class of context-free languages are not closed under this replacement mechanism.

In this paper, we define three classes of languages of bi-infinite words, namely local bi-ω languages, recognizable bi-ω languages and Buchi local bi-ω languages as subclasses of the class of regular bi-ω languages and prove some basic results. We observe that the class of recognizable bi-ω languages coincides with the well-known class of rational bi-adherence languages and show that the class of regular bi-ω languages is the class of morphic images of Buchi local bi-ω languages. We provide learning algorithms for Buchi local bi-ω languages and regular bi-ω languages.

In temporal-logic model checking, we verify the correctness of a program with respect to a desired behavior by checking whether a structure that models the program satisfies a temporal-logic formula that specifies the behavior. The model-checking problem for the branching-time temporal logic CTL can be solved in linear running time, and model-checking tools for CTL are used successfully in industrial applications. The development of programs that must meet rigid real-time constraints has brought with it a need for real-time temporal logics that enable quantitative reference to time. Early research on real-time temporal logics uses the discrete domain of the integers to model time. Present research on real-time temporal logics focuses on continuous time and uses the dense domain of the reals to model time. There, model checking becomes significantly more complicated. For example, the model-checking problem for TCTL, a continuous-time extension of the logic CTL, is PSPACE-complete.In this paper we suggest a reduction from TCTL model checking to CTL model checking. The contribution of such a reduction is twofold. Theoretically, while it has long been known that model-checking methods for untimed temporal logics can be extended quite easily to handle discrete time, it was not clear whether and how untimed methods can handle the reset quantifier of TCTL, which resets a realvalued clock. Practically, our reduction enables anyone who has a tool for CTL model checking to use it for TCTL model checking. The TCTL model-checking algorithm that follows from our reduction is in PSPACE, matching the known bound for this problem. In addition, it enjoys the wide distribution of CTL model-checking tools and the extensive and fruitful research efforts and heuristics that have been put into these tools.KeywordsModel CheckTemporal LogicAtomic PropositionKripke StructureTemporal ReasoningThese 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.

We define the notion of height-deterministic pushdown automata, a model where for any given input string the stack heights during any (nondeterministic) computation on the input are a priori fixed. Different subclasses of height-deterministic pushdown automata, strictly containing the class of regular languages and still closed under boolean language operations, are considered. Several such language classes have been described in the literature. Here, we suggest a natural and intuitive model that subsumes all the formalisms proposed so far by employing height-deterministic pushdown automata. Decidability and complexity questions are also considered.KeywordsModel CheckRegular LanguageClosure PropertyLabel Transition SystemState AutomatonThese 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.

We show that the class of string languages generated by linear context-free rewriting systems is equal to the class of output languages of deterministic tree-walking transducers. From equivalences that have previously been established we know that this class of languages is also equal to the string languages generated by context-free hypergraph grammars, multicomponent tree-adjoining grammars, and multiple context-free grammars and to the class of yields of images of the regular tree languages under finite-copying top-down tree transducers.

We associate a modal operator with each language belonging to a given class of regular languages and use the (reverse) wreath product of monoids with distinguished generators to characterize the expressive power of the resulting logic.

It is well-known that one-tape Turing machines working in linear time are no more powerful than finite automata, namely they recognize exactly the class of regular languages. We study the costs, in terms of description sizes, of the conversion of nondeterministic finite automata into equivalent linear-time one-tape deterministic machines. We prove a polynomial blowup from two-way nondeterministic finite automata into equivalent weight-reducing one-tape deterministic machines that work in linear time. The blowup remains polynomial if the tape in the resulting machines is restricted to the portion which initially contains the input. However, in this case the machines resulting from our construction are not weight reducing, unless the input alphabet is unary.

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class \(\mathscr{C}\) , the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of \(\mathscr{C}\) as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.