Bounds on the Automata Size for Presburger Arithmetic

Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the number of states of the minimal deterministic automaton for a Presburger arithmetic formula. This bound depends on the length of the formula and the quantifiers occurring in the formula. The upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that our bound is tight, even for nondeterministic automata. Moreover, we provide optimal automata constructions for linear equations and inequations

The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of structures. A prominent instance is the class of infinite words over which the alternation-free fragment is already as expressive as the full mu-calculus. Our current understanding of when and why the mu-calculus alternation hierarchy is not strict is limited. This paper makes progress in answering these questions by showing that the alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure's graph decomposes it into graphs in which all paths are of finite length. Our collapse results are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Don't care words with an application totheautomata-based approach for real addition

Automata have proved to be a useful tool in infinite-state model checking, since they can represent infinite sets of integers and reals. However, analogous to the use of binary decision diagrams (bdds) to represent finite sets, the sizes of the automata are an obstacle in the automata-based set representation. In this article, we generalize the notion of "don't cares” for bdds to word languages as a means to reduce the automata sizes. We show that the minimal weak deterministic Büchi automaton (wdba) with respect to a given don't care set, under certain restrictions, is uniquely determined and can be efficiently constructed. We apply don't cares to improve the efficiency of a decision procedure for the first-order logic over the mixed linear arithmetic over the integers and the reals based on wdba

Runtime Monitoring of Metric First-order Temporal Properties

We introduce a novel approach to the runtime monitoring of complex system properties. In particular, we present an online algorithm for a safety fragment of metric first-order temporal logic that is considerably more expressive than the logics supported by prior monitoring methods. Our approach, based on automatic structures, allows the unrestricted use of negation, universal and existential quantification over infinite domains, and the arbitrary nesting of both past and bounded future operators. Moreover, we show how to optimize our approach for the common case where structures consist of only finite relations, over possibly infinite domains. Under an additional restriction, we prove that the space consumed by our monitor is polynomially bounded by the cardinality of the data appearing in the processed prefix of the temporal structure being monitored

On regular temporal logics with past

The IEEE standardized Property Specification Language, PSL for short, extends the well-known linear-time temporal logic LTL with so-called semi-extended regular expressions. PSL and the closely related SystemVerilog Assertions, SVA for short, are increasingly used in many phases of the hardware design cycle, from specification to verification. In this article, we extend the common core of these specification languages with past operators. We name this extension PPSL. Although all ω-regular properties are expressible in PSL, SVA, and PPSL, past operators often allow one to specify properties more naturally and concisely. In fact, we show that PPSL is exponentially more succinct than the cores of PSL and SVA. On the star-free properties, PPSL is double exponentially more succinct than LTL. Furthermore, we present a translation of PPSL into language-equivalent nondeterministic Büchi automata, which is based on novel constructions for 2-way alternating automata. The upper bound on the size of the resulting nondeterministic Büchi automata obtained by our translation is almost the same as the upper bound for the nondeterministic Büchi automata obtained from existing translations for PSL and SVA. Consequently, the satisfiability problem and the model-checking problem for PPSL fall into the same complexity classes as the corresponding problems for PSL and SV

Ehrenfeucht–Fraïssé goes automatic for real addition

AbstractThe decision problem of various logical theories can be decided by automata-theoretic methods. Notable examples are Presburger arithmetic FO(Z,+,<) and the linear arithmetic over the reals FO(R,+,<). Despite the practical use of automata to solve the decision problem of such logical theories, many research questions are still only partly answered in this area. One of these questions is the complexity of the automata-based decision procedures and the related question about the minimal size of the automata of the languages that can be described by formulas in the respective logic. In this article, we establish a double exponential upper bound on the automata size for FO(R,+,<) and an exponential upper bound for the first-order theory of the discrete order over the integers FO(Z,<). The proofs of these upper bounds are based on Ehrenfeucht–Fraïssé games. The application of this mathematical tool has a similar flavor as in computational complexity theory, where it can often be used to establish tight upper bounds of the decision problem for logical theories

Decision procedures for inductive Boolean functions based on alternating automata

AbstractWe show how alternating automata provide decision procedures for the equality of inductively defined Boolean functions and present applications to reasoning about parameterized families of circuits. We use alternating word automata to formalize families of linearly structured circuits and alternating tree automata to formalize families of tree structured circuits. We provide complexity bounds for deciding the equality of function (or circuit) families and show how our decision procedures can be implemented using BDDs. In comparison to previous work, our approach is simpler, has better complexity bounds, and, in the case of tree-structured families, is more general

Automaten-basierende Entscheidungsverfahren für schwache Arithmetik

Around forty years ago, mathematicians such as Büchi and Rabin discovered that automata are a useful mathematical tool for understanding the decidability of different weak systems of arithmetic. A prominent example is the weak monadic second-order logic of one successor, WS1S for short, which is tightly connected to automata over finite words. Nowadays, automata have also emerged as a tool for effectively mechanizing decision procedures for such logical theories. A notable example is Presburger arithmetic for which effective decision procedures can be built using automata. Despite the practical use of automata, many research questions in the automata-based approach to decide weak systems of arithmetic are open. For instance, both lower and upper bounds on the sizes of the automata produced by the automata-based approach for deciding Presburger arithmetic are still unknown. This thesis comprises two parts. In the first part, we analyze the automata-based approach for deciding Presburger arithmetic. We prove that the number of states of the minimal deterministic finite word automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the automata for formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight. Moreover, we provide optimal automata constructions for linear equations and inequations, and present new techniques for mechanizing an automata-based decision procedure for Presburger arithmetic. In the second part of this thesis, we focus on another system of arithmetic and investigate several decision problems for it. More precisely, we look at WS1S extended with linear cardinality constraints of the form |X_1|+...+|X_r|<|Y_1|+...+|Y_s|, where the X_is and Y_js range over finite sets of natural numbers. We delimit the boundary between decidability and undecidability for WS1S with cardinality constraints. Our investigation is based on the fact that the classical connection between automata and WS1S carries over to a fragment of the extension of WS1S and finite word automata with an extended acceptance condition. The extended acceptance condition is based on a generalization of the commutative image of a word. We identify a decidable fragment of WS1S with cardinality constraints, which non-trivially extends WS1S, and give applications for this decidable fragment.Mathematiker wie Büchi und Rabin haben vor ungefähr vierzig Jahren entdeckt, dass Automaten ein nützliches mathematisches Werkzeug sind, um die Entscheidbarkeit von bestimmten Teilsystemen der Arithmetik zu verstehen. Ein bedeutendes Beispiel hierfür ist die schwache monadische Logik zweiter Stufe mit einer Nachfolgerfunktion (WS1S). Diese Logik steht in engem Zusammenhang mit Automaten über endlichen Wörtern. Heutzutage werden Automaten auch als Werkzeug eingesetzt, um Entscheidungsprozeduren für solche logischen Theorien umzusetzen. Ein erwähnenswertes Beispiel ist die Presburger-Arithmetik~(PA), für die es leistungsfähige Automaten-basierte Entscheidungsprozeduren gibt. Trotz des hohen praktischen Nutzens von Automaten in diesem Gebiet sind viele Forschungsfragen bezüglich des auf Automaten beruhenden Ansatzes noch unbeantwortet. Ungeklärt sind zum Beispiel sowohl obere als auch untere Schranken für die Größe der Automaten, die bei einer auf Automaten basierenden Entscheidungsprozedur für PA erzeugt werden. Die vorliegende Dissertation gliedert sich in zwei Teile. Im ersten Teil analysieren wir die auf Automaten beruhende Herangehensweise, um PA zu entscheiden. Wir zeigen, dass die Anzahl der Zustände des minimalen, deterministischen, endlichen Automaten dreifach exponentiell in der Länge der PA-Formel beschränkt ist. Wir beweisen die Existenz dieser oberen Schranke dadurch, dass wir die Automaten für PA-Formeln mit den Automaten vergleichen, die wir aus PA-Formeln erhalten, aus denen die Quantoren durch ein Quantoreneliminationsverfahren entfernt wurden. Des Weiteren zeigen wir, dass diese obere, dreifach exponentielle Schranke scharf ist. Darüber hinaus liefern wir optimale Automatenkonstruktionen für lineare Gleichungen und Ungleichungen und präsentieren neue Techniken, die es erlauben, eine auf Automaten basierende Entscheidungsprozedur für PA zu implementieren. Im zweiten Teil der Arbeit betrachten wir ein anderes System der Arithmetik und widmen uns der Untersuchung von mehreren Entscheidbarkeitsfragen darin. Genauer: Wir erweitern WS1S um lineare Kardinalitätsvergleiche der Form |X_1|+...+|X_r|<|Y_1|+...+|Y_s|. Hierbei sind die X_i und Y_i Variablen zweiter Stufe, die durch endliche Mengen natürlicher Zahlen interpretiert werden. Wir zeigen die Grenze zwischen Entscheidbarkeit und Unentscheidbarkeit in diesem System auf. Unsere Untersuchung von Entscheidbarkeitsfragen beruht auf der Übertragung der bekannten Beziehung zwischen WS1S und endlichen Automaten. Wir führen dazu einen neuen Akzpetanzbegriff für Automaten ein, der auf dem kommutativen Bild von Wörtern basiert. Das sich daraus ergebende erweiterte Automatenmodell entspricht hinsichtlich der Ausdrucksmächtigkeit, wie wir zeigen, einem Fragment von WS1S mit Kardinalitätsvergleichen. Schließlich beweisen wir die Entscheidbarkeit eines Fragments von WS1S mit Kardinalitätsvergleichen und zeigen Anwendungen für eben dieses Fragment auf