Automata for branching and layered temporal structures: An investigation into regularities of infinite transition systems

This manuscript is a revised version of the PhD Thesis I wrote under the supervision of Prof. Angelo Montanari at Udine University. The leitmotif underlying the results herein provided is that, given any infinite complex system (e.g., a computer program) to be verified against a finite set of properties, there often exists a simpler system that satisfies the same properties and, in addition, presents strong regularities (e.g., periodicity) in its structure. Those regularities can then be exploited to decide, in an effective way, which property is satisfied by the system and which is not. Perhaps the most natural and effective way to deal with inherent regularities of infinite systems is through the notion of finite-state automaton. Intuitively, a finite-state automaton is an abstract machine with only a bounded amount of memory at its disposal, which processes an input (e.g., a sequence of symbols) and eventually outputs true or false, depending on the way the machine was designed and on the input itself. The present book focuses precisely on automaton-based approaches that ease the representation of and the reasoning on properties of infinite complex systems. The most simple notion of finite-state automaton, is that of single-string automaton. Such a device outputs true on a single (finite or infinite) sequence of symbols and false on any other sequence. We will show how single-string automata processing infinite sequences of symbols can be successfully applied in various frameworks for temporal representation and reasoning. In particular, we will use them to model single ultimately periodic time granularities, namely, temporal structures that are left-bounded and that, ultimately, periodically group instants of the underlying temporal domain (a simple example of such a structure is given by the partitioning of the temporal domain of days into weeks). The notion of single-string automaton can be further refined by introducing counters in order to compactly represent repeated occurrences of the same subsequence in the given input. By introducing restricted policies of counter update and by exploiting suitable abstractions of the configuration space for the resulting class of automata, we will devise efficient algorithms for reasoning on quasi-periodic time granularities (e.g., the partitioning of the temporal domain of days into years). Similar abstractions can be used when reasoning on infinite branching (temporal) structures. In such a case, one has to consider a generalized notion of automaton, which is able to process labeled branching structures (hereafter called trees), rather than linear sequences of symbols. We will show that sets of trees featuring the same properties can be identified with the equivalence classes induced by a suitable automaton. More precisely, given a property to be verified, one can first define a corresponding automaton that accepts all and only the trees satisfying that property, then introduce a suitable equivalence relation that refines the standard language equivalence and groups all trees being indistinguishable by the automaton, and, finally, exploit such an equivalence to reduce several instances of the verification problem to equivalent simpler instances, which can be eventually decided

Temporal Data Modeling and Reasoning for Information Systems

Temporal knowledge representation and reasoning is a major research field in Artificial Intelligence, in Database Systems, and in Web and Semantic Web research. The ability to model and process time and calendar data is essential for many applications like appointment scheduling, planning, Web services, temporal and active database systems, adaptive Web applications, and mobile computing applications. This article aims at three complementary goals. First, to provide with a general background in temporal data modeling and reasoning approaches. Second, to serve as an orientation guide for further specific reading. Third, to point to new application fields and research perspectives on temporal knowledge representation and reasoning in the Web and Semantic Web

Temporalized logics and automata for time granularity

Suitable extensions of the monadic second-order theory of k successors have been proposed in the literature to capture the notion of time granularity. In this paper, we provide the monadic second-order theories of downward unbounded layered structures, which are infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, and of upward unbounded layered structures, which consist of a finest domain and an infinite number of coarser and coarser domains, with expressively complete and elementarily decidable temporal logic counterparts. We obtain such a result in two steps. First, we define a new class of combined automata, called temporalized automata, which can be proved to be the automata-theoretic counterpart of temporalized logics, and show that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporal logics, transfer from component automata to temporalized ones. Then, we exploit the correspondence between temporalized logics and automata to reduce the task of finding the temporal logic counterparts of the given theories of time granularity to the easier one of finding temporalized automata counterparts of them.Comment: Journal: Theory and Practice of Logic Programming Journal Acronym: TPLP Category: Paper for Special Issue (Verification and Computational Logic) Submitted: 18 March 2002, revised: 14 Januari 2003, accepted: 5 September 200

Supporting Temporal Reasoning by Mapping Calendar Expressions to Minimal Periodic Sets

In the recent years several research efforts have focused on the concept of time granularity and its applications. A first stream of research investigated the mathematical models behind the notion of granularity and the algorithms to manage temporal data based on those models. A second stream of research investigated symbolic formalisms providing a set of algebraic operators to define granularities in a compact and compositional way. However, only very limited manipulation algorithms have been proposed to operate directly on the algebraic representation making it unsuitable to use the symbolic formalisms in applications that need manipulation of granularities. This paper aims at filling the gap between the results from these two streams of research, by providing an efficient conversion from the algebraic representation to the equivalent low-level representation based on the mathematical models. In addition, the conversion returns a minimal representation in terms of period length. Our results have a major practical impact: users can more easily define arbitrary granularities in terms of algebraic operators, and then access granularity reasoning and other services operating efficiently on the equivalent, minimal low-level representation. As an example, we illustrate the application to temporal constraint reasoning with multiple granularities. From a technical point of view, we propose an hybrid algorithm that interleaves the conversion of calendar subexpressions into periodical sets with the minimization of the period length. The algorithm returns set-based granularity representations having minimal period length, which is the most relevant parameter for the performance of the considered reasoning services. Extensive experimental work supports the techniques used in the algorithm, and shows the efficiency and effectiveness of the algorithm

Verification of qualitative Z\mathbb{Z} constraints

International audienceWe introduce an LTL-like logic with atomic formulae built over a constraint language interpreting variables in $\mathbb{Z}$. The constraint language includes periodicity constraints, comparison constraints of the form $x = y$ and $x < y$, it is closed under Boolean operations and it admits a restricted form of existential quantification. This is the largest set of qualitative constraints over $\mathbb{Z}$ known so far, shown to admit a decidable LTL extension. Such constraints are those used for instance in calendar formalisms or in abstractions of counter automata by using congruences modulo some power of two. Indeed, various programming languages perform arithmetic operators modulo some integer. We show that the satisfiability and model-checking problems (with respect to an appropriate class of constraint automata) for this logic are decidable in polynomial space improving significantly known results about its strict fragments. As a by-product, LTL model-checking over integral relational automata is proved complete for polynomial space which contrasts with the known undecidability of its CTL counterpart

Unterstützung von Periodizität in Informationssystemen - Herausforderungen und Lösungsansätze

Die systemseitige Unterstützung von Periodizität bzw. periodischen Spezifikationen weist Anforderungen auf, die weit über die temporalen Fähigkeiten heutiger Informationssysteme hinausgehen. Im Allgemeinen charakterisieren periodische Spezifikationen Vorgänge, die aus regelmäßig wiederkehrenden Aktivitäten bestehen. Neben der Ausdrucksstärke ist die größte Herausforderung periodische Spezifikationen miteinander vergleichen zu können. Diese Vergleichbarkeit ist ein wichtiger Aspekt in einer Vielzahl von Anwendungen, etwa um vorausschauend sich eventuell ergebende potentielle Ressourcen- oder Terminkonflikte erkennen zu können. Erschwert wird dieses durch unterschiedliche (zeitliche) Granularitäten sowie Ausnahmen in entsprechenden Spezifikationen. Für den praktischen Einsatz ist es darüber hinaus unumgänglich, periodische Zusammenhänge auch im Kontext einer großen (umfangreichen) Menge periodischer Daten effizient verwalten und auswerten zu können. Der vorliegende Beitrag gibt einen Einblick in die Herausforderungen sowie einen Überblick zu in der aktuellen Literatur vorliegenden Lösungsansätzen einer systemseitigen Unterstützung von periodischen Spezifikationen

Modeling Time in Computing: A Taxonomy and a Comparative Survey

The increasing relevance of areas such as real-time and embedded systems, pervasive computing, hybrid systems control, and biological and social systems modeling is bringing a growing attention to the temporal aspects of computing, not only in the computer science domain, but also in more traditional fields of engineering. This article surveys various approaches to the formal modeling and analysis of the temporal features of computer-based systems, with a level of detail that is suitable also for non-specialists. In doing so, it provides a unifying framework, rather than just a comprehensive list of formalisms. The paper first lays out some key dimensions along which the various formalisms can be evaluated and compared. Then, a significant sample of formalisms for time modeling in computing are presented and discussed according to these dimensions. The adopted perspective is, to some extent, historical, going from "traditional" models and formalisms to more modern ones.Comment: More typos fixe

A visual language for temporal specifications based on Spider diagrams

Spider Diagrams are a well-established visual language to specify sets, their relationships, and constraints on their cardinalities. However, they do not support evolution of specifications, where one wants to state that under certain circumstances a specification becomes invalid and a new one must be used, nor transformation of specifications, where one needs operators to manipulate specifications. In this paper, we attack the first problem by developing a new system of timed Spider Diagrams which allow modellers to indicate the temporal range of validity of a specification. The approach is illustrated with examples of policies for library management