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

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

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

A Formal Model of Ambiguity and its Applications in Machine Translation

Systems that process natural language must cope with and resolve ambiguity. In this dissertation, a model of language processing is advocated in which multiple inputs and multiple analyses of inputs are considered concurrently and a single analysis is only a last resort. Compared to conventional models, this approach can be understood as replacing single-element inputs and outputs with weighted sets of inputs and outputs. Although processing components must deal with sets (rather than individual elements), constraints are imposed on the elements of these sets, and the representations from existing models may be reused. However, to deal efficiently with large (or infinite) sets, compact representations of sets that share structure between elements, such as weighted finite-state transducers and synchronous context-free grammars, are necessary. These representations and algorithms for manipulating them are discussed in depth in depth. To establish the effectiveness and tractability of the proposed processing model, it is applied to several problems in machine translation. Starting with spoken language translation, it is shown that translating a set of transcription hypotheses yields better translations compared to a baseline in which a single (1-best) transcription hypothesis is selected and then translated, independent of the translation model formalism used. More subtle forms of ambiguity that arise even in text-only translation (such as decisions conventionally made during system development about how to preprocess text) are then discussed, and it is shown that the ambiguity-preserving paradigm can be employed in these cases as well, again leading to improved translation quality. A model for supervised learning that learns from training data where sets (rather than single elements) of correct labels are provided for each training instance and use it to learn a model of compound word segmentation is also introduced, which is used as a preprocessing step in machine translation

Robustly Complete Temporal Logic Control Synthesis for Nonlinear Systems

Modern systems such as spacecrafts and autonomous vehicles are complex yet safety-critical, and therefore the control methods that can deal with different dynamics and constraints while being provably correct are sought after. Formal methods are rigorous techniques originally used for developing and verifying finite-state systems with respect to specifications in formal languages. This thesis is concerned with using formal methods in control synthesis for nonlinear systems, which can guarantee the correctness of the resulting control strategies. For nonlinear continuous-state dynamical systems, formal control synthesis relies on finite abstractions of the original system by discretizing the system state space and over approximating system transitions. Without further assumptions, control synthesis is usually not complete in the way that no control strategies can be found even if there exists one. To deal with this problem, this thesis proposes a formal control synthesis approach that is sound and robustly complete in the sense that correct control strategies can be found whenever the specifications can be realized for the system with additional disturbance. Fundamental to the soundness and robust completeness is a fixed-point characterization of the winning set of the system with respect to a given specification, which is the set of initial conditions that can be controlled to satisfy the specification. Regarding discrete-time systems, such characterizations are first presented by using iterative computation of predecessors for basic linear temporal logic (LTL) specifications, including invariance, reachability and reach-and-stay. A more general class of LTL formulas, which can be translated into deterministic B\"uchi automata (DBA), is also considered, and an algorithm guided by the graph structure of the LTL-equivalent DBA is proposed for characterizing the winning set in this situation. It is then shown that the computational complexity of the algorithm can be reduced by using a pre-processing procedure to the graphs of the DBA. Because of the general nonlinearity, exact computation of winning sets is currently almost impossible. In this work, the conditions for set approximations are derived so that control synthesis is robustly complete. To meet such conditions, the proposed approach adopts interval arithmetic and a subdivision scheme in the approximation of predecessors. Under such a scheme, the system state space is adaptively partitioned with respect to both the given dynamics and specification and set approximation can be made arbitrarily precise to satisfy the robust completeness conditions. The proposed method is also shown applicable to sampled-data systems by computing validated solutions over one sampling period based on high-order Taylor expansion. Applications such as converter voltage regulation, parallel parking, and reactive locomotion planning problems are studied to show the effectiveness and efficiency of the proposed approach

A uniform approach to the complexity and analysis of succinct systems

“ This thesis provides a unifying view on the succinctness of systems: the capability of a modeling formalism to describe the behavior of a system of exponential size using a polynomial syntax. The key theoretical contribution is the introduction of sequential circuit machines as a new universal computation model that focuses on succinctness as the central aspect. The thesis demonstrates that many well-known modeling formalisms such as communicating state machines, linear-time temporal logic, or timed automata exhibit an immediate connection to this machine model. Once a (syntactic) connection is established, many complexity bounds for structurally restricted sequential circuit machines can be transferred to a certain formalism in a uniform manner. As a consequence, besides a far-reaching unification of independent lines of research, we are also able to provide matching complexity bounds for various analysis problems, whose complexities were not known so far. For example, we establish matching lower and upper bounds of the small witness problem and several variants of the bounded synthesis problem for timed automata, a particularly important succinct modeling formalism. Also for timed automata, our complexity-theoretic analysis leads to the identification of tractable fragments of the timed synthesis problem under partial observability. Specifically, we identify timed controller synthesis based on discrete or template-based controllers to be equivalent to model checking. Based on this discovery, we develop a new model checking-based algorithm to efficiently find feasible template instantiations. From a more practical perspective, this thesis also studies the preservation of succinctness in analysis algorithms using symbolic data structures. While efficient techniques exist for specific forms of succinctness considered in isolation, we present a general approach based on abstraction refinement to combine off-the-shelf symbolic data structures. In particular, for handling the combination of concurrency and quantitative timing behavior in networks of timed automata, we report on the tool Synthia which combines binary decision diagrams with difference bound matrices. In a comparison with the timed model checker Uppaal and the timed game solver Tiga running on standard benchmarks from the timed model checking and synthesis domain, respectively, the experimental results clearly demonstrate the effectiveness of our new approach.Diese Dissertation liefert eine vereinheitlichende Sicht auf die Kompaktheit von Systemen: die Fähigkeit eines Modellierungsformalismus, das Verhalten eines Systems exponentieller Größe mit polynomieller Syntax zu beschreiben. Der wesentliche theoretische Beitrag ist die Einführung von sequenziellen Schaltkreis-Maschinen als neues universelles Berechnungsmodell, das sich auf den zentralen Aspekt der Kompaktheit konzentriert. Die Dissertation demonstriert, dass viele bekannte Modellierungsformalismen, wie z.B. kommunizierende Zustandsmaschinen, linear-Zeit temporale Logik (LTL) oder gezeitete Automaten eine direkte Verbindung zu diesem Maschinenmodell aufzeigen. Sobald eine (syntaktische) Verbindung hergestellt ist, können viele Komplexitätsschranken für strukturell beschränkte sequenzielle Schaltkreis-Maschinen für einen bestimmten Formalismus einheitlich übernommen werden. Neben einer weitreichenden Vereinheitlichung unabhängiger Forschungsrichtungen können auch zahlreiche Komplexitätsschranken für Analyse-Probleme etabliert werden, deren genaue Komplexität bisher noch nicht bekannt war. Zum Beispiel werden passende untere und obere Schranken des small witness Problems und mehrere Varianten des Synthese-Problems von Controllern mit beschränkter Größe für gezeitete Automaten bewiesen. Die theoretische Analyse deckt Fragmente geringerer Komplexität des partiell informierten Syntheseproblems für gezeitete Automaten auf. Es wird im Besonderen gezeigt, dass das gezeitete Syntheseproblem für diskrete oder Vorlagen-basierte Controller äquivalent zum Model Checking-Problem ist. Basierend auf dieser Einsicht wird ein neuartiger Model Checking-basierter Algorithmus zur effizienten Synthese von gültigen Instantiierungen von Vorlagen entwickelt. Der praktische Beitrag der Dissertation untersucht die Erhaltung von Kompaktheit in Analyse-Algorithmen durch die Benutzung symbolischer Datenstrukturen. Es wird ein allgemeiner Ansatz zur Kombination von Standard-Datenstrukturen vorgestellt, die jeweils bisher nur in Isolation verwendet werden konnten. Insbesondere wird für die Analyse von Netzwerken von gezeiteten Automaten das Tool Synthia vorgestellt, welches binäre Entscheidungs-Diagramme mit Differenzen-Matrizen verbindet. In einem experimentellen Vergleich mit den Tools Uppaal und Tiga wird klar die Effektivität des neuen Ansatzes belegt