Julian Gutierrez

Iterated Boolean games

Iterated games are well-known in the game theory literature. We study iterated Boolean games. These are games in which players repeatedly choose truth values for Boolean variables they have control over. Our model of iterated Boolean games assumes that players have goals given by formulae of Linear Temporal Logic (LTL), a formalism for expressing properties of state sequences. In order to represent the strategies of players in such games, we use a finite state machine model. After introducing and formally defining iterated Boolean games, we investigate the computational complexity of their associated game-theoretic decision problems, as well as semantic conditions characterising classes of LTL properties that are preserved by equilibrium points (pure-strategy Nash equilibria) whenever they exist.

Reasoning about equilibria in game-like concurrent systems

Our aim is to develop techniques for reasoning about game-like concurrent systems, where the components of the system act rationally and strategically in pursuit of logically-specified goals. We first present a computational model for such systems, and investigate its properties. We then define and investigate a branching-time logic for reasoning about the equilibrium properties of such systems. The key operator in this logic is a path quantifier [NE]φ, which asserts that p holds on all Nash equilibrium computations of the system.

Model-checking games for fixpoint logics with partial order models

In this paper, we introduce model-checking games that allow local second-order power on sets of independent transitions in the underlying partial order models where the games are played. Since the interleaving semantics of such models is not considered, some problems that may arise when using interleaving representations are avoided and new decidability results for partial order models of concurrency are achieved. The games are shown to be sound and complete, and therefore determined. While in the interleaving case they coincide with the local model-checking games for the @m-calculus, in a partial order setting they verify properties of a number of fixpoint modal logics that can specify, in concurrent systems with partial order semantics, several properties not expressible with the @m-calculus. The games underpin a novel decision procedure for model-checking all temporal properties of a class of infinite and regular event structures, thus improving, in terms of temporal expressive power, previous results in the literature.

Logics and Bisimulation Games for Concurrency, Causality and Conflict

Based on a simple axiomatization of concurrent behaviour we define two ways of observing parallel computations and show that in each case they are dual to conflict and causality, respectively. We give a logical characterization to those dualities and show that natural fixpoint modal logics can be extracted from such a characterization. We also study the equivalences induced by such logics and prove that they are decidable and can be related with well-known bisimulations for interleaving and noninterleaving concurrency. Moreover, by giving a game-theoretical characterization to the equivalence induced by the main logic, which is called Separation Fixpoint Logic (SFL), we show that the equivalence SFL induces is strictly stronger than a history-preserving bisimulation (hpb) and strictly weaker than a hereditary history-preserving bisimulation (hhpb). Our study considers branching-time models of concurrency based on transition systems and petri net structures.

Model-Checking Games for Fixpoint Logics with Partial Order Models

We introduce model-checking games that allow local second-order power on sets of independent transitions in the underlying partial order models where the games are played. Since the one-step interleaving semantics of such models is not considered, some problems that may arise when using interleaving semantics are avoided and new decidability results for partial orders are achieved. The games are shown to be sound and complete , and therefore determined. While in the interleaving case they coincide with the local model-checking games for the μ -calculus (Lμ ), in a noninterleaving setting they verify properties of Separation Fixpoint Logic (SFL), a logic that can specify in partial orders properties not expressible with Lμ . The games underpin a novel decision procedure for model-checking all temporal properties of a class of infinite and regular event structures, thus improving previous results in the literature.

Expresiveness and Complexity Results for Strategic Reasoning.

This paper presents a range of expressiveness and complexity results for the specification, computation, and verification of Nash equilibria in multi-player non-zero-sum concurrent games in which players have goals expressed as temporal logic formulae. Our results are based on a novel approach to the characterisation of equilibria in such games: a semantic characterisation based on winning strategies and memoryful reasoning. This characterisation allows us to obtain a number of other results relating to the analysis of equilibrium properties in temporal logic. We show that, up to bisimilarity, reasoning about Nash equilibria in multi-player non-zero-sum concurrent games can be done in ATL^* and that constructing equilibrium strategy profiles in such games can be done in 2EXPTIME using finite-memory strategies. We also study two simpler cases, two-player games and sequential games, and show that the specification of equilibria in the latter setting can be obtained in a temporal logic that is weaker than ATL^*. Based on these results, we settle a few open problems, put forward new logical characterisations of equilibria, and provide improved answers and alternative solutions to a number of questions.

Equilibria of concurrent games on event structures

Event structures form a canonical model of concurrent behaviour which has a natural game-theoretic interpretation. This game-based interpretation was initially given for zero-sum concurrent games. This paper studies an extension of such games on event structures to include a much wider class of game types and solution concepts. The extension permits modelling scenarios where, for instance, cooperation or independent goal-driven behaviour of computer agents is desired. Specifically, we will define non-zero-sum games on event structures, and give full characterisations---existence and completeness results---of the kinds of games, payoff sets, and strategies for which Nash equilibria and subgame perfect Nash equilibria always exist. The game semantics of various logics and systems are outlined to illustrate the power of this framework.

Imperfect Information in Logic and Concurrent Games

This paper builds on a recent definition of concurrent games as event structures and an application giving a concurrent-game model for predicate calculus. An extension to concurrent games with imperfect information, through the introduction of ‘access levels’ to restrict the allowable strategies, leads to a concurrent-game semantics for a variant of Hintikka and Sandu’s Independence-Friendly (IF) logic.

Concurrent logic games on partial orders

Most games for analysing concurrent systems are played on interleaving models, such as graphs or infinite trees. However, several concurrent systems have partial order models rather than interleaving ones. As a consequence, a potentially algorithmically undesirable translation from a partial order setting to an interleaving one is required before analysing them with traditional techniques. In order to address this problem, this paper studies a game played directly on partial orders and describes some of its algorithmic applications. The game provides a unified approach to system and property verification which applies to different decision problems and models of concurrency. Since this framework uses partial orders to give a uniform representation of concurrent systems, logical specifications, and problem descriptions, it is particularly suitable for reasoning about concurrent systems with partial order semantics, such as Petri nets or event structures. Two applications can be cast within this unified approach: bisimulation and model-checking.

The µ-calculus alternation hierarchy collapses over structures with restricted connectivity

The alternation hierarchy of least and greatest fixpoint operators in the µ-calculus is strict. However, the strictness of the hierarchy does not necessarily carry over when considering restricted classes of structures. For instance, over the class of infinite words the alternation-free fragment of the µ-calculus is already as expressive as the full logic. Our current understanding of when and why the µ-calculus alternation hierarchy is (and is not) strict is limited. This article makes progress in answering these questions by showing that the alternation hierarchy of the µ-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structures graph decomposes it into graphs in which all paths are of finite length. The collapse results herein are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the µ-calculus over restricted classes of structures.