Giuseppe Perelli

Temporal logics are extensively used for the specification of on-going behaviors of computer systems. Two significant developments in this area are the extension of traditional temporal logics with modalities that enable the specification of on-going strategic behaviors in multi-agent systems, and the transition of temporal logics to a quantitative setting, where different satisfaction values enable the specifier to formalize concepts such as certainty or quality. In the first class, SL (Strategy Logic) is one of the most natural and expressive logics describing strategic behaviors. In the second class, a notable logic is LTL[F], which extends LTL with quality operators. In this work we introduce and study SL[F], which enables the specification of quantitative strategic behaviors. The satisfaction value of an SL[F] formula is a real value in [0,1], reflecting “how much” or “how well” the strategic on-going objectives of the underlying agents are satisfied. We demonstrate the applications of \FSL in quantitative reasoning about multi-agent systems, showing how it can express and measure concepts like stability in multi-agent systems, and how it generalizes some fuzzy temporal logics. We also provide a model-checking algorithm for SL[F], based on a quantitative extension of Quantified CTL*. Our algorithm provides the first decidability result for a quantitative extension of Strategy Logic. In addition, it can be used for synthesizing strategies that maximize the quality of the systems’ behavior.

In the context of multi-agent systems, the \textitrational verification problem is concerned with checking which temporal logic properties will hold in a system when its constituent agents are assumed to behave rationally and strategically in pursuit of individual objectives. Typically, those objectives are expressed as temporal logic formulae which the relevant agent desires to see satisfied. Unfortunately, rational verification is computationally complex, and requires specialised techniques in order to obtain practically useable implementations. In this paper, we present such a technique. This technique relies on a reduction of the rational verification problem to the solution of a collection of parity games. Our approach has been implemented in the \emphEquilibrium \emphVerification \emphEnvironment (EVE) system. The EVE system takes as input a model of a concurrent/multi-agent system represented using the Simple Reactive Modules Language (SRML), where agent goals are represented as Linear Temporal Logic (LTL) formulae, together with a claim about the equilibrium behaviour of the system, also expressed as an LTL formula. EVE can then check whether the LTL claim holds on some (or every) computation of the system that could arise through agents choosing Nash equilibrium strategies; it can also check whether a system has a Nash equilibrium, and synthesise individual strategies for players in the multi-player game. After presenting our basic framework, we describe our new technique and prove its correctness. We then describe our implementation in the EVE system, and present experimental results which show that EVE performs favourably in comparison to other existing tools that support rational verification.

In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between agents and express the correctness of the system no matter how the environment behaves. An important contribution in this context is given by modal logics for strategic ability, in the setting of multiagent games, such as Atl, Atl*, and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which we denote here by CHP-Sl, with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-Sl is obtained by using first-order quantifications over strategies and has been investigated in the very specific setting of two-agents turned-based games, where a nonelementary model-checking algorithm has been provided. While CHP-Sl is a very expressive logic, we claim that it does not fully capture the strategic aspects of multiagent systems. \{In this article, we introduce and study a more general strategy logic, denoted Sl, for reasoning about strategies in multiagent concurrent games. As a key aspect, strategies in Sl are not intrinsically glued to a specific agent, but an explicit binding operator allows an agent to bind to a strategy variable. This allows agents to share strategies or reuse one previously adopted. We prove that Sl strictly includes CHP-Sl, while maintaining a decidable model-checking problem. In particular, the algorithm we propose is computationally not harder than the best one known for CHP-Sl. Moreover, we prove that such a problem for Sl is NonElementary. This negative result has spurred us to investigate syntactic fragments of Sl, strictly subsuming Atl*, with the hope of obtaining an elementary model-checking problem. Among others, we introduce and study the sublogics Sl[ng], Sl[bg], and Sl[1g]. They encompass formulas in a special prenex normal form having, respectively, nested temporal goals, Boolean combinations of goals, and, a single goal at a time. Intuitively, for a goal, we mean a sequence of bindings, one for each agent, followed by an Ltl formula. We prove that the model-checking problem for Sl[1g] is 2ExpTime-complete, thus not harder than the one for Atl*. In contrast, Sl[ng] turns out to be NonElementary-hard, strengthening the corresponding result for Sl. Regarding Sl[bg], we show that it includes CHP-Sl and its model-checking is decidable with a 2ExpTimelower-bound. It is worth enlightening that to achieve the positive results about Sl[1g], we introduce a fundamental property of the semantics of this logic, called behavioral, which allows to strongly simplify the reasoning about strategies. Indeed, in a nonbehavioral logic such as Sl[bg] and the subsuming ones, to satisfy a formula, one has to take into account that a move of an agent, at a given moment of a play, may depend on the moves taken by any agent in another counterfactual play.

We introduce Incentive Design: a new class of problems for equilibrium verification in multi-agent systems. In our model, agents attempt to maximize their utility functions, which are expressed as formulae in LTL[F], a quantitative extension of Linear Temporal Logic with functions computable in polynomial time. We assume agents are rational, in the sense that they adopt strategies consistent with game theoretic solution concepts such as Nash equilibrium. For each solution concept we consider, we analyze the problems of verifying whether an incentive scheme achieves a societal objective and finding one that does so, whether it be social welfare or any other aggregate measure of collective well-being. We study both static and dynamic incentive schemes, showing that the latter are more powerful than the former. Finally, we solve the incentive verification and synthesis problems for all the solution concepts we consider, and analyze their complexity.

Norms have been widely proposed to coordinate and regulate behaviour in multi-agent systems (MAS). We consider the problem of synthesising and revising the set of norms in a normative MAS to satisfy a design objective expressed in Alternating Time Temporal Logic (ATL*). ATL* is a well-established language for strategic reasoning, which allows the specification of norms that constrain the strategic behaviour of agents. We focus on dynamic norms, that is, norms corresponding to Mealy machines, that allow us to place different constraints on the agents’ behaviour depending on the state of the norm and the state of the underlying MAS. We show that synthesising dynamic norms is (k + 1)-EXPTIME where k is the alternation depth of quantifiers in the ATL* specification. Note that for typical cases of interest, k is either 1 or 2. We also study the problem of removing existing norms to satisfy a new objective, which we show to be 2EXPTIME-complete.

We introduce and investigate Normative Synthesis: a new class of problems for the equilibrium verification that counters the absence of equilibria by purposely constraining multi-agent systems. We show that norms are powerful enough to ensure a positive answer to every instance of the equilibrium verification problem. Subsequently, we focus on two optimization versions, that aim at providing a solution in compliance with implementation costs. We show that the complexities of our procedures range between 2ExpTime and 3ExpTime, thus that the problems are no harder than the corresponding equilibrium verification ones.