. In this paper, we report on a recent work for the verication of nonrepudiation

We consider concurrent two-player games with reachability objectives. In such games, at each round, player 1 and player 2 independently and simultaneously choose moves, and the two choices determine the next state of the game. The objective of player 1 is to reach a set of target states; the objective of player 2 is to prevent this. These are zero-sum games, and the reachability objective is one of the most basic objectives: determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification. There are three types of winning states, according to the degree of certainty with which player 1 can reach the target. From type-1 states, player 1 has a deterministic strategy to always reach the target. From type-2 states, player 1 has a randomized strategy to reach the target with probability 1. From type-3 states, player 1 has for every real ε> 0 a randomized strategy to reach the target with probability greater than 1 − ε. We show that for finite state spaces, all three sets of winning states can be computed in polynomial time: type-1 states in linear time, and type-2 and type-3 states in quadratic time. The algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies.

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A procedure for the analysis of state spaces is called symbolic if it manipulates not individual states, but sets of states that are represented by constraints. Such a procedure can be used for the analysis of infinite state spaces, provided termination is guaranteed. We present symbolic procedures, and corresponding termination criteria, for the solution of infinite-state games, which occur in the control and modular verification of infinite-state systems. To characterize the termination of symbolic procedures for solving infinite-state games, we classify these game structures into four increasingly restrictive categories: 1. Class 1 consists of infinite-state structures for which all safety and reachability games can be solved...

Cooperation logics have recently begun to attract attention within the multi-agent systems community. Using a cooperation logic, it is possible to represent and reason about the strategic powers of agents and coalitions of agents in game-like multi-agent systems. These powers are generally assumed to be implicitly defined within the structure of the environment, and their origin is rarely discussed. In this paper, we study a cooperation logic in which agents are each assumed to control a set of propositional variables---the powers of agents and coalitions then derive from the allocation of propositions to agents. The basic modal constructs in this Coalition Logic of Propositional Control (CL-PC) allow us to express the fact that a group of agents can cooperate to bring about a certain state of affairs. After motivating and introducing CL-PC, we provide a complete axiom system for the logic, investigate the issue of characterising control in CL-PC with respect to the underlying power structures of the logic, and formally investigate the relationship between CL-PC and Pauly's Coalition Logic. We then show that the model checking and satisfiability problems for CL-PC are both PSPACE-complete, and conclude by discussing our results and how CL-PC sits in relation to other logics of cooperation.

We consider concurrent two-person games played in real time, in which the players decide both which action to play, and when to play it. Such timed games differ from untimed games in two essential ways. First, players can take each other by surprise, because actions are played with delays that cannot be anticipated by the opponent. Second, a player should not be able to win the game by preventing time from diverging. We present a model of timed games that preserves the element of surprise and accounts for time divergence in a way that treats both players symmetrically and applies to all !-regular winning conditions.

We draw parallels between coalition game logics developed in [Pauly, 2000b] and [Pauly, 2000c] on one hand, and alternating-time temporal logics of computations in-troduced in [Alur et al, 97] on the other. In particular, we show equivalence of their semantics, embedding of coalition game logics into alternating-time temporal logic, and propose axiomatic systems for these logics. 1

Temporal logics, first-order logics, and automata over data words have recently attracted considerable attention. A data word is a word over a finite alphabet, together with a datum (an element of an infinite domain) at each position. Examples include timed words and XML documents. To refer to the data, temporal logics are extended with the freeze quantifier, first-order logics with predicates over the data domain, and automata with registers or pebbles. We investigate relative expressiveness and complexity of standard decision problems for LTL with the freeze quantifier (LTL ↓), 2-variable first-order logic (FO 2) over data words, and register automata. The only predicate available on data is equality. Previously undiscovered connections among those formalisms, and to counter automata with incrementing errors, enable us to answer several questions left open in recent literature. We show that the future-time fragment of LTL ↓ which corresponds to FO 2 over finite data words can be extended considerably while preserving decidability, but at the expense of non-primitive recursive complexity, and that most of further extensions are undecidable. We also prove that surprisingly, over infinite data words, LTL ↓ without the ‘until’ operator, as well as nonemptiness of one-way universal register automata, are undecidable even when there is only 1 register. 1.

Abstract. We develop an automata-theoretic framework for reasoning about infinitestate sequential systems. Our framework is based on the observation that states of such systems, which carry a finite but unbounded amount of information, can be viewed as nodes in an infinite tree, and transitions between states can be simulated by finite-state automata. Checking that the system satisfies a temporal property can then be done by an alternating two-way tree automaton that navigates through the tree. As has been the case with finite-state systems, the automatatheoretic framework is quite versatile. We demonstrate it by solving several versions of the model-checking problem for §-calculus specifications and prefixrecognizable systems, and by solving the realizability and synthesis problems for §-calculus specifications with respect to prefix-recognizable environments. 1

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