Propositional Dynamic Logic for Hyperproperties

Information security properties of reactive systems like non-interference often require relating different executions of the system to each other and following them simultaneously. Such hyperproperties can also be useful in other contexts, e.g., when analysing properties of distributed systems like linearizability. Since common logics like LTL, CTL, or the modal ?-calculus cannot express hyperproperties, the hyperlogics HyperLTL and HyperCTL^* were developed to cure this defect. However, these logics are not able to express arbitrary ?-regular properties. In this paper, we introduce HyperPDL-?, an adaptation of the Propositional Dynamic Logic of Fischer and Ladner for hyperproperties, in order to remove this limitation. Using an elegant automata-theoretic framework, we show that HyperPDL-? model checking is asymptotically not more expensive than HyperCTL^* model checking, despite its vastly increased expressive power. We further investigate fragments of HyperPDL-? with regard to satisfiability checking

A Temporal Logic for Hyperproperties

Hyperproperties, as introduced by Clarkson and Schneider, characterize the correctness of a computer program as a condition on its set of computation paths. Standard temporal logics can only refer to a single path at a time, and therefore cannot express many hyperproperties of interest, including noninterference and other important properties in security and coding theory. In this paper, we investigate an extension of temporal logic with explicit path variables. We show that the quantification over paths naturally subsumes other extensions of temporal logic with operators for information flow and knowledge. The model checking problem for temporal logic with path quantification is decidable. For alternation depth 1, the complexity is PSPACE in the length of the formula and NLOGSPACE in the size of the system, as for linear-time temporal logic

Logical methods for the hierarchy of hyperlogics

In this thesis, we develop logical methods for reasoning about hyperproperties. Hyperproperties describe relations between multiple executions of a system. Unlike trace properties, hyperproperties comprise relational properties like noninterference, symmetry, and robustness. While trace properties have been studied extensively, hyperproperties form a relatively new concept that is far from fully understood. We study the expressiveness of various hyperlogics and develop algorithms for their satisfiability and synthesis problems. In the first part, we explore the landscape of hyperlogics based on temporal logics, first-order and second-order logics, and logics with team semantics. We establish that first-order/second-order and temporal hyperlogics span a hierarchy of expressiveness, whereas team logics constitute a radically different way of specifying hyperproperties. Furthermore, we introduce the notion of temporal safety and liveness, from which we obtain fragments of HyperLTL (the most prominent hyperlogic) with a simpler satisfiability problem. In the second part, we develop logics and algorithms for the synthesis of smart contracts. We introduce two extensions of temporal stream logic to express (hyper)properties of infinite-state systems. We study the realizability problem of these logics and define approximations of the problem in LTL and HyperLTL. Based on these approximations, we develop algorithms to construct smart contracts directly from their specifications.In dieser Arbeit beschreiben wir logische Methoden, um über Hypereigenschaften zu argumentieren. Hypereigenschaften beschreiben Relationen zwischen mehreren Ausführungen eines Systems. Anders als pfadbasierte Eigenschaften können Hypereigenschaften relationale Eigenschaften wie Symmetrie, Robustheit und die Abwesenheit von Informationsfluss ausdrücken. Während pfadbasierte Eigenschaften in den letzten Jahrzehnten ausführlich erforscht wurden, sind Hypereigenschaften ein relativ neues Konzept, das wir noch nicht vollständig verstehen. Wir untersuchen die Ausdrucksmächtigkeit verschiedener Hyperlogiken und entwickeln ausführbare Algorithmen, um deren Erfüllbarkeits- und Syntheseproblem zu lösen. Im ersten Teil erforschen wir die Landschaft der Hyperlogiken basierend auf temporalen Logiken, Logiken erster und zweiter Ordnung und Logiken mit Teamsemantik. Wir stellen fest, dass temporale Logiken und Logiken erster und zweiter Ordnung eine Hierarchie an Ausdrucksmächtigkeit aufspannen. Teamlogiken hingegen spezifieren Hypereigenschaften auf eine radikal andere Art. Wir führen außerdem das Konzept von temporalen Sicherheits- und Lebendigkeitseigenschaften ein, durch die Fragmente der bedeutensten Logik HyperLTL entstehen, für die das Erfüllbarkeitsproblem einfacher ist. Im zweiten Teil entwickeln wir Logiken und Algorithmen für die Synthese digitaler Verträge. Wir führen zwei Erweiterungen temporaler Stromlogik ein, um (Hyper)eigenschaften in unendlichen Systemen auszudrücken. Wir untersuchen das Realisierungsproblem dieser Logiken und definieren Approximationen des Problems in LTL und HyperLTL. Basierend auf diesen Approximationen entwickeln wir Algorithmen, die digitale Verträge direkt aus einer Spezifikation erstellen

Logical and deep learning methods for temporal reasoning

In this thesis, we study logical and deep learning methods for the temporal reasoning of reactive systems. In Part I, we determine decidability borders for the satisfiability and realizability problem of temporal hyperproperties. Temporal hyperproperties relate multiple computation traces to each other and are expressed in a temporal hyperlogic. In particular, we identify decidable fragments of the highly expressive hyperlogics HyperQPTL and HyperCTL*. As an application, we elaborate on an enforcement mechanism for temporal hyperproperties. We study explicit enforcement algorithms for specifications given as formulas in universally quantified HyperLTL. In Part II, we train a (deep) neural network on the trace generation and realizability problem of linear-time temporal logic (LTL). We consider a method to generate large amounts of additional training data from practical specification patterns. The training data is generated with classical solvers, which provide one of many possible solutions to each formula. We demonstrate that it is sufficient to train on those particular solutions such that the neural network generalizes to the semantics of the logic. The neural network can predict solutions even for formulas from benchmarks from the literature on which the classical solver timed out. Additionally, we show that it solves a significant portion of problems from the annual synthesis competition (SYNTCOMP) and even out-of-distribution examples from a recent case study.Diese Arbeit befasst sich mit logischen Methoden und mehrschichtigen Lernmethoden für das zeitabhängige Argumentieren über reaktive Systeme. In Teil I werden die Grenzen der Entscheidbarkeit des Erfüllbarkeits- und des Realisierbarkeitsproblem von temporalen Hypereigenschaften bestimmt. Temporale Hypereigenschaften setzen mehrere Berechnungsspuren zueinander in Beziehung und werden in einer temporalen Hyperlogik ausgedrückt. Insbesondere werden entscheidbare Fragmente der hochexpressiven Hyperlogiken HyperQPTL und HyperCTL* identifiziert. Als Anwendung wird ein Enforcement-Mechanismus für temporale Hypereigenschaften erarbeitet. Explizite Enforcement-Algorithmen für Spezifikationen, die als Formeln in universell quantifiziertem HyperLTL angegeben werden, werden untersucht. In Teil II wird ein (mehrschichtiges) neuronales Netz auf den Problemen der Spurgenerierung und Realisierbarkeit von Linear-zeit Temporallogik (LTL) trainiert. Es wird eine Methode betrachtet, um aus praktischen Spezifikationsmustern große Mengen zusätzlicher Trainingsdaten zu generieren. Die Trainingsdaten werden mit klassischen Solvern generiert, die zu jeder Formel nur eine von vielen möglichen Lösungen liefern. Es wird gezeigt, dass es ausreichend ist, an diesen speziellen Lösungen zu trainieren, sodass das neuronale Netz zur Semantik der Logik generalisiert. Das neuronale Netz kann Lösungen sogar für Formeln aus Benchmarks aus der Literatur vorhersagen, bei denen der klassische Solver eine Zeitüberschreitung hatte. Zusätzlich wird gezeigt, dass das neuronale Netz einen erheblichen Teil der Probleme aus dem jährlichen Synthesewettbewerb (SYNTCOMP) und sogar Beispiele außerhalb der Distribution aus einer aktuellen Fallstudie lösen kann

Expressiveness and Decidability of Temporal Logics for Asynchronous Hyperproperties

Hyperproperties are properties of systems that relate different executions traces, with many applications from security to symmetry, consistency models of concurrency, etc. In recent years, different linear-time logics for specifying asynchronous hyperproperties have been investigated. Though model checking of these logics is undecidable, useful decidable fragments have been identified with applications e.g. for asynchronous security analysis. In this paper, we address expressiveness and decidability issues of temporal logics for asynchronous hyperproperties. We compare the expressiveness of these logics together with the extension S1S[E] of S1S with the equal-level predicate by obtaining an almost complete expressiveness picture. We also study the expressive power of these logics when interpreted on singleton sets of traces. We show that for two asynchronous extensions of HyperLTL, checking the existence of a singleton model is already undecidable, and for one of them, namely Context HyperLTL (HyperLTL_C), we establish a characterization of the singleton models in terms of the extension of standard FO[<] over traces with addition. This last result generalizes the well-known equivalence between FO[<] and LTL. Finally, we identify new boundaries on the decidability of model checking HyperLTL_C

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

Symbolic reactive synthesis

In this thesis, we develop symbolic algorithms for the synthesis of reactive systems. Synthesis, that is the task of deriving correct-by-construction implementations from formal specifications, has the potential to eliminate the need for the manual—and error-prone—programming task. The synthesis problem can be formulated as an infinite two-player game, where the system player has the objective to satisfy the specification against all possible actions of the environment player. The standard synthesis algorithms represent the underlying synthesis game explicitly and, thus, they scale poorly with respect to the size of the specification. We provide an algorithmic framework to solve the synthesis problem symbolically. In contrast to the standard approaches, we use a succinct representation of the synthesis game which leads to improved scalability in terms of the symbolically represented parameters. Our algorithm reduces the synthesis game to the satisfiability problem of quantified Boolean formulas (QBF) and dependency quantified Boolean formulas (DQBF). In the encodings, we use propositional quantification to succinctly represent different parts of the implementation, such as the state space and the transition function. We develop highly optimized satisfiability algorithms for QBF and DQBF. Based on a counterexample-guided abstraction refinement (CEGAR) loop, our algorithms avoid an exponential blow-up by using the structure of the underlying symbolic encodings. Further, we extend the solving algorithms to extract certificates in the form of Boolean functions, from which we construct implementations for the synthesis problem. Our empirical evaluation shows that our symbolic approach significantly outperforms previous explicit synthesis algorithms with respect to scalability and solution quality.In dieser Dissertation werden symbolische Algorithmen für die Synthese von reaktiven Systemen entwickelt. Synthese, d.h. die Aufgabe, aus formalen Spezifikationen korrekte Implementierungen abzuleiten, hat das Potenzial, die manuelle und fehleranfällige Programmierung überflüssig zu machen. Das Syntheseproblem kann als unendliches Zweispielerspiel verstanden werden, bei dem der Systemspieler das Ziel hat, die Spezifikation gegen alle möglichen Handlungen des Umgebungsspielers zu erfüllen. Die Standardsynthesealgorithmen stellen das zugrunde liegende Synthesespiel explizit dar und skalieren daher schlecht in Bezug auf die Größe der Spezifikation. Diese Arbeit präsentiert einen algorithmischen Ansatz, der das Syntheseproblem symbolisch löst. Im Gegensatz zu den Standardansätzen wird eine kompakte Darstellung des Synthesespiels verwendet, die zu einer verbesserten Skalierbarkeit der symbolisch dargestellten Parameter führt. Der Algorithmus reduziert das Synthesespiel auf das Erfüllbarkeitsproblem von quantifizierten booleschen Formeln (QBF) und abhängigkeitsquantifizierten booleschen Formeln (DQBF). In den Kodierungen verwenden wir propositionale Quantifizierung, um verschiedene Teile der Implementierung, wie den Zustandsraum und die Übergangsfunktion, kompakt darzustellen. Wir entwickeln hochoptimierte Erfüllbarkeitsalgorithmen für QBF und DQBF. Basierend auf einer gegenbeispielgeführten Abstraktionsverfeinerungsschleife (CEGAR) vermeiden diese Algorithmen ein exponentielles Blow-up, indem sie die Struktur der zugrunde liegenden symbolischen Kodierungen verwenden. Weiterhin werden die Lösungsalgorithmen um Zertifikate in Form von booleschen Funktionen erweitert, aus denen Implementierungen für das Syntheseproblem abgeleitet werden. Unsere empirische Auswertung zeigt, dass unser symbolischer Ansatz die bisherigen expliziten Synthesealgorithmen in Bezug auf Skalierbarkeit und Lösungsqualität deutlich übertrifft

Expressiveness and Decidability of Temporal Logics for Asynchronous Hyperproperties

Hyperproperties are properties of systems that relate different executions traces, with many applications from security to symmetry, consistency models of concurrency, etc. In recent years, different linear-time logics for specifying asynchronous hyperproperties have been investigated. Though model checking of these logics is undecidable, useful decidable fragments have been identified with applications e.g. for asynchronous security analysis. In this paper, we address expressiveness and decidability issues of temporal logics for asynchronous hyperproperties. We compare the expressiveness of these logics together with the extension S1S[E] of S1S with the equal-level predicate by obtaining an almost complete expressiveness picture. We also study the expressive power of these logics when interpreted on singleton sets of traces. We show that for two asynchronous extensions of HyperLTL, checking the existence of a singleton model is already undecidable, and for one of them, namely Context HyperLTL (HyperLTLC), we establish a characterization of the singleton models in terms of the extension of standard FO[<] over traces with addition. This last result generalizes the well-known equivalence between FO[<] and LTL. Finally, we identify new boundaries on the decidability of model checking HyperLTL

Monitoring and Enforcement of Safety Hyperproperties

Certain important security policies such as information flow characterize system-wide behaviors and are not properties of individual executions. It is known that such security policies cannot be expressed in trace-based specification languages such as linear-time temporal logic (LTL). However, formalisms such as hyperproperties and the associated logic HyperLTL allow us to specify such policies. In this thesis, we concentrate on the static enforcement and runtime verification of safety hyperproperties expressed in HyperLTL. For static enforcement of safety hyperproperties, we incorporate program repair techniques, where an input program is modified to satisfy certain properties while preserving its existing specifications. Assuming finite state space for the input program, we show that the complexity of program repair for safety hyperproperties is in general NP-hard. However, there are certain cases in which the problem can be solved in polynomial time. We identify such cases and give polynomial-time algorithms for them. In the context of runtime verification, we make two contributions: we (1) analyze the complexity of decision procedures for verifying safety hyperproperties, (2) provide a syntactic fragment in HyperLTL to express certain k-safety hyperproperties, and (3) develop a general runtime verification technique for HyperLTL k-safety formulas, for cases where verification at run time can be done in polynomial time. Our technique is based on runtime formula progression as well as on-the-fly monitor synthesis across multiple executions

Computation Tree Logic for Synchronization Properties

We present a logic that extends CTL (Computation Tree Logic) with operators that express synchronization properties. A property is synchronized in a system if it holds in all paths of a certain length. The new logic is obtained by using the same path quantifiers and temporal operators as in CTL, but allowing a different order of the quantifiers. This small syntactic variation induces a logic that can express non-regular properties for which known extensions of MSO with equality of path length are undecidable. We show that our variant of CTL is decidable and that the model-checking problem is in Delta_3^P = P^{NP^{NP}}, and is hard for the class of problems solvable in polynomial time using a parallel access to an NP oracle. We analogously consider quantifier exchange in extensions of CTL, and we present operators defined using basic operators of CTL* that express the occurrence of infinitely many synchronization points. We show that the model-checking problem remains in Delta_3^P. The distinguishing power of CTL and of our new logic coincide if the Next operator is allowed in the logics, thus the classical bisimulation quotient can be used for state-space reduction before model checking