Improved Algorithms for Parity and Streett objectives

The computation of the winning set for parity objectives and for Streett objectives in graphs as well as in game graphs are central problems in computer-aided verification, with application to the verification of closed systems with strong fairness conditions, the verification of open systems, checking interface compatibility, well-formedness of specifications, and the synthesis of reactive systems. We show how to compute the winning set on $n$ vertices for (1) parity-3 (aka one-pair Streett) objectives in game graphs in time $O(n^{5/2})$ and for (2) k-pair Streett objectives in graphs in time $O(n^2 + nk \log n)$. For both problems this gives faster algorithms for dense graphs and represents the first improvement in asymptotic running time in 15 years

Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter

A model of computation that is widely used in the formal analysis of reactive systems is symbolic algorithms. In this model the access to the input graph is restricted to consist of symbolic operations, which are expensive in comparison to the standard RAM operations. We give lower bounds on the number of symbolic operations for basic graph problems such as the computation of the strongly connected components and of the approximate diameter as well as for fundamental problems in model checking such as safety, liveness, and co-liveness. Our lower bounds are linear in the number of vertices of the graph, even for constant-diameter graphs. For none of these problems lower bounds on the number of symbolic operations were known before. The lower bounds show an interesting separation of these problems from the reachability problem, which can be solved with $O(D)$ symbolic operations, where $D$ is the diameter of the graph. Additionally we present an approximation algorithm for the graph diameter which requires $\tilde{O}(n \sqrt{D})$ symbolic steps to achieve a $(1+\epsilon)$-approximation for any constant $\epsilon > 0$. This compares to $O(n \cdot D)$ symbolic steps for the (naive) exact algorithm and $O(D)$ symbolic steps for a 2-approximation. Finally we also give a refined analysis of the strongly connected components algorithms of Gentilini et al., showing that it uses an optimal number of symbolic steps that is proportional to the sum of the diameters of the strongly connected components

Conditionally Optimal Algorithms for Generalized B\"uchi Games

Games on graphs provide the appropriate framework to study several central problems in computer science, such as the verification and synthesis of reactive systems. One of the most basic objectives for games on graphs is the liveness (or B\"uchi) objective that given a target set of vertices requires that some vertex in the target set is visited infinitely often. We study generalized B\"uchi objectives (i.e., conjunction of liveness objectives), and implications between two generalized B\"uchi objectives (known as GR(1) objectives), that arise in numerous applications in computer-aided verification. We present improved algorithms and conditional super-linear lower bounds based on widely believed assumptions about the complexity of (A1) combinatorial Boolean matrix multiplication and (A2) CNF-SAT. We consider graph games with $n$ vertices, $m$ edges, and generalized B\"uchi objectives with $k$ conjunctions. First, we present an algorithm with running time $O(k \cdot n^2)$, improving the previously known $O(k \cdot n \cdot m)$ and $O(k^2 \cdot n^2)$ worst-case bounds. Our algorithm is optimal for dense graphs under (A1). Second, we show that the basic algorithm for the problem is optimal for sparse graphs when the target sets have constant size under (A2). Finally, we consider GR(1) objectives, with $k_1$ conjunctions in the antecedent and $k_2$ conjunctions in the consequent, and present an $O(k_1 \cdot k_2 \cdot n^{2.5})$-time algorithm, improving the previously known $O(k_1 \cdot k_2 \cdot n \cdot m)$-time algorithm for $m > n^{1.5}$

Faster Algorithms for Computing Maximal 2-Connected Subgraphs in Sparse Directed Graphs

Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected (resp., 2-vertex-connected) if the removal of any edge (resp., vertex) leaves the graph strongly connected. In this paper we present improved algorithms for computing the maximal 2-edge- and 2-vertex-connected subgraphs of a given directed graph. These problems were first studied more than 35 years ago, with $\widetilde{O}(mn)$ time algorithms for graphs with m edges and n vertices being known since the late 1980s. In contrast, the same problems for undirected graphs are known to be solvable in linear time. Henzinger et al. [ICALP 2015] recently introduced $O(n^2)$ time algorithms for the directed case, thus improving the running times for dense graphs. Our new algorithms run in time $O(m^{3/2})$, which further improves the running times for sparse graphs. The notion of 2-connectivity naturally generalizes to k-connectivity for $k>2$. For constant values of k, we extend one of our algorithms to compute the maximal k-edge-connected in time $O(m^{3/2} \log{n})$, improving again for sparse graphs the best known algorithm by Henzinger et al. [ICALP 2015] that runs in $O(n^2 \log n)$ time.Comment: Revised version of SODA 2017 paper including details for k-edge-connected subgraph

Symbolic Algorithms for Graphs and Markov Decision Processes with Fairness Objectives

Given a model and a specification, the fundamental model-checking problem asks for algorithmic verification of whether the model satisfies the specification. We consider graphs and Markov decision processes (MDPs), which are fundamental models for reactive systems. One of the very basic specifications that arise in verification of reactive systems is the strong fairness (aka Streett) objective. Given different types of requests and corresponding grants, the objective requires that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All $\omega$-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. To handle the state-space explosion, symbolic algorithms are required that operate on a succinct implicit representation of the system rather than explicitly accessing the system. While explicit algorithms for graphs and MDPs with Streett objectives have been widely studied, there has been no improvement of the basic symbolic algorithms. The worst-case numbers of symbolic steps required for the basic symbolic algorithms are as follows: quadratic for graphs and cubic for MDPs. In this work we present the first sub-quadratic symbolic algorithm for graphs with Streett objectives, and our algorithm is sub-quadratic even for MDPs. Based on our algorithmic insights we present an implementation of the new symbolic approach and show that it improves the existing approach on several academic benchmark examples.Comment: Full version of the paper. To appear in CAV 201

Improved set-based symbolic algorithms for parity games

Graph games with omega-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the verification of branching time properties). Parity conditions are canonical forms to specify omega-regular winning conditions. Graph games with parity conditions are equivalent to mu-calculus model checking, and thus a very important algorithmic problem. Symbolic algorithms are of great significance because they provide scalable algorithms for the analysis of large finite-state systems, as well as algorithms for the analysis of infinite-state systems with finite quotient. A set-based symbolic algorithm uses the basic set operations and the one-step predecessor operators. We consider graph games with n vertices and parity conditions with c priorities (equivalently, a mu-calculus formula with c alternations of least and greatest fixed points). While many explicit algorithms exist for graph games with parity conditions, for set-based symbolic algorithms there are only two algorithms (notice that we use space to refer to the number of sets stored by a symbolic algorithm): (a) the basic algorithm that requires O(n^c) symbolic operations and linear space; and (b) an improved algorithm that requires O(n^{c/2+1}) symbolic operations but also O(n^{c/2+1}) space (i.e., exponential space). In this work we present two set-based symbolic algorithms for parity games: (a) our first algorithm requires O(n^{c/2+1}) symbolic operations and only requires linear space; and (b) developing on our first algorithm, we present an algorithm that requires O(n^{c/3+1}) symbolic operations and only linear space. We also present the first linear space set-based symbolic algorithm for parity games that requires at most a sub-exponential number of symbolic operations

Improved algorithms and conditional lower bounds for problems in formal verification and reactive synthesis

Die Modellprüfung ist ein vollautomatisches Verfahren zur formalen Verifikation, die entweder die Korrektheit eines Systems zeigt oder einen Fehler findet. Sie ist ein essentieller und oft verwendeter Bestandteil im schrittweisen Design von Systemen, wie zum Beispiel von Mikroprozessoren. Im Gegensatz zu schrittweisem Design verlangt das Syntheseproblem von Church die automatische Generierung eines korrekten Systems aus einer vorgegebenen Spezifikation. Reaktive Sythese ist die Synthese von reaktiven Systemen, welche laufend mit ihrer Umgebung interagieren. Für die formale Verifikation und Synthese werden mathematische Modelle von Systemen und ihrem Verhalten benötigt. Gerichtete Graphen sind ein grundlegendes Modell von Systemen. Markow-Entscheidungsprozesse (MEPs) können zusätzlich zufallsgesteuertes Verhalten abbilden, zum Beispiel von randomisierten parallelen Systemen und von Kommunikationsprotokollen. Ein Modell für reaktive Systeme sind Spielgraphen, bei denen die Knoten des Graphens zwischen einer Spielerin, die die kontrollierbaren Eingaben repräsentiert, und ihrem Gegenspieler, der die unkontrollierbaren Eingaben repräsentiert, aufgeteilt sind. Der Automaten-basierte Ansatz zur Modellprüfung und Synthese ist eine anerkannte Methode um das erwünschte Verhalten von Systemen mit Hilfe von omega-regulären Zielvorgaben wie Büchi-, Paritäts- oder Streett-Zielvorgaben formal zu beschreiben. Zusätzlich können quantitative Eigenschaften wie Ressourcenverbrauch durch Mittelwerts-Zielvorgaben ausgedrückt werden. In dieser Arbeit entwickeln wir Algorithmen mit verbesserter Laufzeit für mehrere Probleme auf Graphen, MEPs, und Spielgraphen mit omega-regulären Zielvorgaben und Mittelwerts-Zielvorgaben. Zusätzlich zeigen wir die ersten super-linearen bedingten unteren Schranken für Polynomialzeitprobleme in diesem Gebiet. Konkret präsentieren wir die folgenden Ergebnisse: * Für Mittelwerts-Zielvorgaben auf Graphen den ersten Approximationsalgorithmus, der für dichte Graphen die lange bekannten Laufzeitschranken für exakte Algorithmen durchbricht. * Für Streett-Zielvorgaben den ersten Algorithmus mit weniger als quadratischer Laufzeit sowie verbesserte Algorithmen für dichte MEPs und Graphen. * Für Paritätsspiele den ersten sub-kubischen Algorithmus für drei Prioritäten sowie verbesserte symbolische Algorithmen für den allgemeinen Fall. * Neue Algorithmen und super-lineare bedingte untere Schranken für Konjunktionen und Disjunktionen von einfachen omega-regulären Zielvorgaben. Diese Ergebnisse zeigen zum ersten Mal, dass es unter weitverbreiteten Annahmen für MEPs strikt höhere Laufzeitschranken als für Graphen (``Modell-Separierung'') und für manche Zielvorgaben strikt höhere Laufzeitschranken als für nah verwandte Zielvorgaben (``Zielvorgaben-Separierung'') gibt. * Für verallgemeinerte Büchi Spiele einen neuen Algorithmus und passende bedingte untere Schranken, die eine Modellseparierung zwischen MEPs und Spielgraphen implizieren, sowie für GR(1) Spiele einen schnelleren Algorithmus auf dichten Graphen.Model checking is a fully automated approach in formal verification to either prove a system's correctness or find an error. It is an essential and widely-used component in the iterative design of systems such as microprocessors. In contrast to iterative design, Church's synthesis problem asks to automatically generate a correct system from its specification. Reactive synthesis is the synthesis of reactive systems that are systems that repeatedly interact with their environment. For formal verification and synthesis mathematical models of systems and their behaviors are needed. Directed graphs are a fundamental model of systems. Markov decision processes (MDPs) additionally incorporate probabilistic behavior of, for example, randomized concurrent systems or communication protocols. A model for reactive systems are game graphs, where the vertices of the graph are partitioned between two players and one player represents controllable inputs and the other uncontrollable inputs. The automata-theoretic approach to model-checking and synthesis is a canonical way to formally specify the desired behaviors of a system using omega-regular objectives such as Büchi, parity, and Streett objectives. Additionally, mean-payoff objectives allow for expressing quantitative properties of systems such as resource consumption. In this thesis we develop algorithms with improved worst-case running times for several problems on graphs, MDPs, and game graphs with omega-regular and mean-payoff objectives. Additionally, we show the first super-linear conditional lower bounds for polynomial-time problems in this area. In particular we present the following results: * For mean-payoff objectives on graphs the first approximation algorithm that improves for dense graphs upon the long-standing running time bounds for exact algorithms. * For Streett objectives the first sub-quadratic time algorithm for MDPs and a faster algorithm for dense MDPs and graphs. * For parity games the first sub-cubic time algorithm for three priorities and improved symbolic algorithms for the general case. * New algorithms and super-linear conditional lower bounds for conjunctions and disjunctions of basic omega-regular objectives. These results show for the first time that, under popular assumptions, there exist problems with strictly higher running times on MDPs than on graphs (``model separation'') and that for each graph and MDPs there exist objectives with strictly higher running times compared to closely related objectives (``objective separation''). * For generalized Büchi games a new upper and tight conditional lower bounds that imply a model separation between MDPs and game graphs, and a faster algorithm for dense GR(1) games

Improved algorithms for one-pair and k-pair Streett objectives

The computation of the winning set for one-pair Streett objectives and for k-pair Streett objectives in (standard) graphs as well as in game graphs are central problems in computer-aided verification, with application to the verification of closed systems with strong fairness conditions, the verification of open systems, checking interface compatibility, well-formed ness of specifications, and the synthesis of reactive systems. We give faster algorithms for the computation of the winning set for (1) one-pair Streett objectives (aka parity-3 problem) in game graphs and (2) for k-pair Streett objectives in graphs. For both problems this represents the first improvement in asymptotic running time in 15 years