P4P_4-free Partition and Cover Numbers and Application

$P_4$-free graphs-- also known as cographs, complement-reducible graphs, or hereditary Dacey graphs--have been well studied in graph theory. Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite $P_4$-free graphs. For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant. Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems. Interestingly, our covering problem is closely related to the well-studied problem of product/Prague dimension of loopless undirected graphs, which allows us to employ algebraic lower-bounding techniques for the product/Prague dimension. We prove that computing these numbers is \npol-complete, even for bipartite graphs. We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size-$N$ partite sets such that these numbers are at least ${\epsilon\cdot N^{1-2\epsilon}}$, for $\epsilon\in\{1/3,1/4,1/5,\dotsc\}$. Finally, we accurately estimate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality. For applications in information theory and communication \& cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the participants, Alice and Bob, their portion from this joint sample. Alice and Bob\u27s objective is to non-interactively establish a shared key and extract the left-over entropy from their portion of the samples as independent private randomness. A genie, who observes the joint sample, provides appropriate assistance to help Alice and Bob with their objective. Lower bounds to the minimum size of the genie\u27s assistance translate into communication and cryptographic lower bounds. We show that (the $\log_2$ of) the $P_4$-free partition number of a graph encoding the joint distribution that the setup uses is equivalent to the size of the genie\u27s assistance. Consequently, the joint distributions corresponding to the bipartite graphs constructed above with high $P_4$-free partition numbers correspond to joint distributions requiring more assistance from the genie. As a representative application in non-deterministic communication complexity, we study the communication complexity of nondeterministic protocols augmented by access to the equality oracle at the output. We show that (the $\log_2$ of) the $P_4$-free cover number of the bipartite graph encoding a Boolean function $f$ is equivalent to the minimum size of the nondeterministic input required by the parties (referred to as the communication complexity of $f$ in this model). Consequently, the functions corresponding to the bipartite graphs with high $P_4$-free cover numbers have high communication complexity. Furthermore, there are functions with communication complexity close to the \naive protocol where the nondeterministic input reveals a party\u27s input. Finally, the access to the equality oracle reduces the communication complexity of computing set disjointness by a constant factor in contrast to the model where parties do not have access to the equality oracle. To compute the inequality function, we show an exponential reduction in the communication complexity, and this bound is optimal. On the other hand, access to the equality oracle is (nearly) useless for computing set intersection

Logics on data words

We investigate logics on data words, i.e., words where each position is labelled by some propositions from a finite set and by some data values from an infinite domain. A basic motivation for the study of these logics, called data logics in this work, is that data words are a suitable model to represent traces of concurrent systems with unboundedly many interacting processes. In such representations data values stand for process IDs. Thus, data logics can be used to formulate requirements on such traces. We first study the expressivity and complexity of the satisfiability problem for these logics. Then, we investigate suitable models for concurrent systems with unboundedly many processes. Finally, we analyse the model checking problem for such systems in the case that data logics are used to specify system requirements. One of our main results is that, despite the bad properties of data logics with respect to satisfiability, there are important cases in which model checking with data logics has moderate complexity. Hence, our results motivate for further investigations with the aim to find interesting models and data logics which can be used in practical model checking tools

Model counting for reactive systems

Model counting is the problem of computing the number of solutions for a logical formula. In the last few years, it has been primarily studied for propositional logic, and has been shown to be useful in many applications. In planning, for example, propositional model counting has been used to compute the robustness of a plan in an incomplete domain. In information-flow control, model counting has been applied to measure the amount of information leaked by a security-critical system. In this thesis, we introduce the model counting problem for linear-time properties, and show its applications in formal verification. In the same way propositional model counting generalizes the satisfiability problem for propositional logic, counting models for linear-time properties generalizes the emptiness problem for languages over infinite words to one that asks for the number of words in a language. The model counting problem, thus, provides a foundation for quantitative extensions of model checking, where not only the existence of computations that violate the specification is determined, but also the number of such violations. We solve the model counting problem for the prominent class of omega-regular properties. We present algorithms for solving the problem for different classes of properties, and show the advantages of our algorithms in comparison to indirect approaches based on encodings into propositional logic. We further show how model counting can be used for solving a variety of quantitative problems in formal verification, including probabilistic model checking, quantitative information-flow in security-critical systems, and the synthesis of approximate implementations for reactive systems.Das Modellzählproblem fragt nach der Anzahl der Lösungen einer logischen Formel, und wurde in den letzten Jahren hauptsächlich für Aussagenlogik untersucht. Das Zählen von Modellen aussagenlogischer Formeln hat sich in vielen Anwendungen als nützlich erwiesen. Im Bereich der künstlichen Intelligenz wurde das Zählen von Modellen beispielsweise verwendet, um die Robustheit eines Plans in einem unvollständigen Weltmodell zu bewerten. Das Zählen von Modellen kann auch verwendet werden, um in sicherheitskritischen Systemen die Menge an enthüllten vertraulichen Daten zu messen. Diese Dissertation stellt das Modellzählproblem für Linearzeiteigenschaften vor, und untersucht dessen Rolle in der Welt der formalen Verifikation. Das Zählen von Modellen für Linearzeiteigenschaften führt zu neuen quantitativen Erweiterungen klassischer Verifikationsprobleme, bei denen nicht nur die Existenz eines Fehlers in einem System zu überprüfen ist, sondern auch die Anzahl solcher Fehler. Wir präsentieren Algorithmen zur Lösung des Modellzählproblems für verschiedene Klassen von Linearzeiteigenschaften und zeigen die Vorteile unserer Algorithmen im Vergleich zu indirekten Ansätzen, die auf Kodierungen der untersuchten Probleme in Aussagenlogik basieren. Darüberhinaus zeigen wir wie das Zählen von Modellen zur Lösung einer Vielzahl quantitativer Probleme in der formalen Verifikation verwendet werden kann. Dies beinhaltet unter anderem die Analyse probabilistischer Modelle, die Kontrolle quantitativen Informationsflusses in sicherheitskritischen Systemen, und die Synthese von approximativen Implementierungen für reaktive Systeme

Synthesis of distributed systems

This thesis offers a comprehensive solution of the distributed synthesis problem. It starts with the problem of solving Parity games, which form an integral part of the automata-theoretic synthesis algorithms we use. We improve the known complexity bound for solving parity games with n positions and c colors approximately from O(n^(1/2*c)) to O(n^(1/3*c)), and introduce an accelerated strategy improvement technique that can consider all combinations of local improvements in every update step, selecting the globally optimal combination. We then demonstrate the decidability and finite model property of alternating-time specification languages, and determine the complexity of the satisfiability and synthesis problem for the alternating-time μ-calculus and the temporal logic ATL*. The impact of the architecture, that is, the set of system processes with known (white-box) and unknown (black-box) implementation, and the com- munication structure between them, is determined. We introduce information forks, a simple but comprehensive criterion that characterizes all architectures for which the synthesis problem is undecidable. The information fork crite- rion takes the impact of nondeterminism, the communication topology, and the specification language into account. For decidable architectures, we present an automata-based synthesis algorithm. We introduce bounded synthesis, which deviates from general synthesis by considering only implementations up to a predefined size, and thus avoids the expensive representation of all solutions. We develop a SAT based approach to bounded synthesis, which is nondeterministic quasilinear in the minimal implementation instead of nonelementary in the system specification. We determine the complexity of open synthesis under the assumption of probabilistic or reactive environments. Our automata based approach allows for a seamless integration of the new environment models into the uniform synthesis algorithm. Finally, we study the synthesis problem for asynchronous systems. We show that distributed synthesis remains only decidable for architectures with a single black-box process, and determine the complexity of the synthesis problem for different scheduler types. Furthermore, we combine the undecidability results and synthesis procedures for synchronous and asynchronous systems; systems that are globally asynchronous and locally synchronous are decidable if all black-box components are contained in a single fork-free synchronized component.Diese Dissertation löst das Syntheseproblem für verteilte Systeme. Sie beginnt mit verbesserten Algorithmen zum Lösen von Parity Spielen, die einen integralen Bestandteil der Automaten basierten Synthese bilden. Die bekannte Komplexitätsschranke für das Lösen von Parity Spielen mit n Knoten und c Farben wird von ca. O(n^(1/2*c)) auf ca. O(n^(1/3*c)) verbessert, und es wird eine beschleunigte Strategie Verbesserungsmethode entwickelt, die, in jedem Schritt, die optimale Kombination aller lokalen Verbesserungen findet. Die Entscheidbarkeit alternierender Logiken wird gezeigt, und die Komplexität des Erfüllbarkeits- und Syntheseproblems für das Alternierende µ-Kalkül (EXPTIME-vollständig) und die Temporallogik ATL* (2EXPTIME-vollständig) bestimmt. Der Einfluss der Systemarchitektur, der Spezifikationssprache und, damit verbunden, des Implementierungsmodells (deterministisch vs. nichtdeterministisch) auf die Entscheidbarkeit und Komplexität des Syntheseproblems wird herausgearbeitet. Es wird gezeigt, dass die Klasse der entscheidbaren Architekturen durch die Abwesenheit von Information Forks, einem einfachen und leicht prüfbaren Kriterium auf der Kommunikationsarchitektur, vollständig beschrieben werden kann. Für entscheidbare Architekturen wird ein einheitliches Automaten basiertes Syntheseverfahren entwickelt. Darüber hinaus wird ein SAT basiertes Verfahren entwickelt, dass die Repräsentation aller Lösungen in einem Automaten umgeht. Die Komplexität des SAT basierten Verfahrens ist nichtdeterministisch quasilinear in der Größe des minimalen Modells, statt nicht-elementar in der Größe der Spezifikation. Für probabilistische und reaktive Umgebungen wird die Komplexität des offenen Syntheseproblems bestimmt, und jeweils ein Automaten basiertes Syntheseverfahren entwickelt, dass sich nahtlos in das Syntheseverfahren für verteilte Systeme integrieren lässt. Ferner wird gezeigt, dass verteilte Synthese für asynchrone Systeme nur dann entscheidbar bleibt, wenn lediglich die Implementierung einer Komponente konstruiert werden soll. Schließlich werden die Entscheidbarkeitsresultate und Synthese Algorithmen für synchrone und asynchrone Modelle zusammengeführt: Global asynchrone lokal synchrone Systeme sind entscheidbar, wenn alle zu synthetisierenden Prozesse in der gleichen synchronisierten Komponente liegen, und diese Komponente keine Information Forks enthält

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems