MsATL: a Tool for SAT-Based ATL Satisfiability Checking

We present MsATL: the first tool for deciding the satisfiability of Alternating-time Temporal Logic (ATL) with imperfect information. MsATL combines SAT Modulo Monotonic Theories solvers with existing ATL model checkers: MCMAS and STV. The tool can deal with various semantics of ATL, including perfect and imperfect information, and can handle additional practical requirements. MsATL can be applied for synthesis of games that conform to a given specification, with the synthesised game often being minimal

Complexity and Expressivity of Branching- and Alternating-Time Temporal Logics with Finitely Many Variables

We show that Branching-time temporal logics CTL and CTL*, as well as Alternating-time temporal logics ATL and ATL*, are as semantically expressive in the language with a single propositional variable as they are in the full language, i.e., with an unlimited supply of propositional variables. It follows that satisfiability for CTL, as well as for ATL, with a single variable is EXPTIME-complete, while satisfiability for CTL*, as well as for ATL*, with a single variable is 2EXPTIME-complete,--i.e., for these logics, the satisfiability for formulas with only one variable is as hard as satisfiability for arbitrary formulas.Comment: Prefinal version of the published pape

Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

On the Complexity of ATL and ATL* Module Checking

Module checking has been introduced in late 1990s to verify open systems, i.e., systems whose behavior depends on the continuous interaction with the environment. Classically, module checking has been investigated with respect to specifications given as CTL and CTL* formulas. Recently, it has been shown that CTL (resp., CTL*) module checking offers a distinctly different perspective from the better-known problem of ATL (resp., ATL*) model checking. In particular, ATL (resp., ATL*) module checking strictly enhances the expressiveness of both CTL (resp., CTL*) module checking and ATL (resp. ATL*) model checking. In this paper, we provide asymptotically optimal bounds on the computational cost of module checking against ATL and ATL*, whose upper bounds are based on an automata-theoretic approach. We show that module-checking for ATL is EXPTIME-complete, which is the same complexity of module checking against CTL. On the other hand, ATL* module checking turns out to be 3EXPTIME-complete, hence exponentially harder than CTL* module checking.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

Satisfiability in Strategy Logic can be Easier than Model Checking

In the design of complex systems, model-checking and satisfiability arise as two prominent decision problems. While model-checking requires the designed system to be provided in advance, satisfiability allows to check if such a system even exists. With very few exceptions, the second problem turns out to be harder than the first one from a complexity-theoretic standpoint. In this paper, we investigate the connection between the two problems for a non-trivial fragment of Strategy Logic (SL, for short). SL extends LTL with first-order quantifications over strategies, thus allowing to explicitly reason about the strategic abilities of agents in a multi-agent system. Satisfiability for the full logic is known to be highly undecidable, while model-checking is non-elementary.The SL fragment we consider is obtained by preventing strategic quantifications within the scope of temporal operators. The resulting logic is quite powerful, still allowing to express important game-theoretic properties of multi-agent systems, such as existence of Nash and immune equilibria, as well as to formalize the rational synthesis problem. We show that satisfiability for such a fragment is PSPACE-COMPLETE, while its model-checking complexity is 2EXPTIME-HARD. The result is obtained by means of an elegant encoding of the problem into the satisfiability of conjunctive-binding first-order logic, a recently discovered decidable fragment of first-order logic

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

Synthesis of Cost-Optimal Multi-Agent Systems for Resource Allocation

Multi-agent systems for resource allocation (MRAs) have been introduced as a concept for modelling competitive resource allocation problems in distributed computing. An MRA is composed of a set of agents and a set of resources. Each agent has goals in terms of allocating certain resources. For MRAs it is typically of importance that they are designed in a way such that there exists a strategy that guarantees that all agents will achieve their goals. The corresponding model checking problem is to determine whether such a winning strategy exists or not, and the synthesis problem is to actually build the strategy. While winning strategies ensure that all goals will be achieved, following such strategies does not necessarily involve an optimal use of resources. In this paper, we present a technique that allows to synthesise cost-optimal solutions to distributed resource allocation problems. We consider a scenario where system components such as agents and resources involve costs. A multi-agent system shall be designed that is cost-minimal but still capable of accomplishing a given set of goals. Our approach synthesises a winning strategy that minimises the cumulative costs of the components that are required for achieving the goals. The technique is based on a propositional logic encoding and a reduction of the synthesis problem to the maximum satisfiability problem (Max-SAT). Hence, a Max-SAT solver can be used to perform the synthesis. From a truth assignment that maximises the number of satisfied clauses of the encoding a cost-optimal winning strategy as well as a cost-optimal system can be immediately derived.Comment: In Proceedings FROM 2022, arXiv:2209.0920

Taming Strategy Logic: Non-Recurrent Fragments

Strategy Logic (SL for short) is one of the prominent languages for reasoning about the strategic abilities of agents in a multi-agent setting. This logic extends LTL with first-order quantifiers over the agent strategies and encompasses other formalisms, such as ATL* and CTL*. The model-checking problem for SL and several of its fragments have been extensively studied. On the other hand, the picture is much less clear on the satisfiability front, where the problem is undecidable for the full logic. In this work, we study two fragments of One-Goal SL, where the nesting of sentences within temporal operators is constrained. We show that the satisfiability problem for these logics, and for the corresponding fragments of ATL* and CTL*, is ExpSpace and PSpace-complete, respectively