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30
Alternatingtime Temporal Logic
 Journal of the ACM
, 1997
"... Temporal logic comes in two varieties: lineartime temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branchingtime temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general var ..."
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Cited by 448 (47 self)
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Temporal logic comes in two varieties: lineartime temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branchingtime temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternatingtime temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While lineartime and branchingtime logics are natural specification languages for closed systems, alternatingtime logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. Also the problems of receptiveness, realizability, and controllability can be formulated as modelchecking problems for alternatingtime formulas.
Optimal strategies in priced timed game automata
 In FSTTCS 04, LNCS 3328
, 2004
"... Abstract. Priced timed (game) automata extend timed (game) automata with costs on both locations and transitions. In this paper we focus on reachability games for priced timed game automata and prove that the optimal cost for winning such a game is computable under conditions concerning the nonzeno ..."
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Cited by 49 (23 self)
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Abstract. Priced timed (game) automata extend timed (game) automata with costs on both locations and transitions. In this paper we focus on reachability games for priced timed game automata and prove that the optimal cost for winning such a game is computable under conditions concerning the nonzenoness of cost and we prove that it is decidable. Under stronger conditions (strictness of constraints) we prove that in case an optimal strategy exists, we can compute a statebased winning optimal strategy. 1
On optimal timed strategies
 In FORMATS 05, LNCS 3829
, 2005
"... Abstract. In this paper, we study timed games played on weighted timed automata. In this context, the reachability problem asks if, given a set T of locations and a cost C, Player 1 has a strategy to force the game into T with a cost less than C no matter how Player 2 behaves. Recently, this problem ..."
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Cited by 23 (4 self)
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Abstract. In this paper, we study timed games played on weighted timed automata. In this context, the reachability problem asks if, given a set T of locations and a cost C, Player 1 has a strategy to force the game into T with a cost less than C no matter how Player 2 behaves. Recently, this problem has been studied independently by Alur et al and by Bouyer et al. In those two works, a semialgorithm is proposed to solve the reachability problem, which is proved to terminate under a condition imposing the nonzenoness of cost. In this paper, we show that in the general case the existence of a strategy for Player 1 to win the game with a bounded cost is undecidable. Our undecidability result holds for weighted timed game automata with five clocks. On the positive side, we show that if we restrict the number of clocks to one and we limit the form of the cost on locations, then the semialgorithm proposed by Bouyer et al always terminates. 1
K.G.: Optimal infinite scheduling for multipriced timed automata
 Formal Methods in System Design 32
, 2008
"... Abstract. This paper is concerned with the derivation of infinite schedules for timed automata that are in some sense optimal. To cover a wide class of optimality criteria we start out by introducing an extension of the (priced) timed automata model that includes both costs and rewards as separate m ..."
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Cited by 19 (4 self)
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Abstract. This paper is concerned with the derivation of infinite schedules for timed automata that are in some sense optimal. To cover a wide class of optimality criteria we start out by introducing an extension of the (priced) timed automata model that includes both costs and rewards as separate modelling features. A precise definition is then given of what constitutes optimal infinite behaviours for this class of models. We subsequently show that the derivation of optimal nonterminating schedules for such doublepriced timed automata is computable. This is done by a reduction of the problem to the determination of optimal meancycles in finite graphs with weighted edges. This reduction is obtained by introducing the socalled cornerpoint abstraction, a powerful abstraction technique of which we show that it preserves optimal schedules. 1
Infinite Runs in Weighted Timed Automata with Energy Constraints
"... Abstract. We study the problems of existence and construction of infinite schedules for finite weighted automata and oneclock weighted timed automata, subject to boundary constraints on the accumulated weight. More specifically, we consider automata equipped with positive and negative weights on tr ..."
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Cited by 18 (5 self)
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Abstract. We study the problems of existence and construction of infinite schedules for finite weighted automata and oneclock weighted timed automata, subject to boundary constraints on the accumulated weight. More specifically, we consider automata equipped with positive and negative weights on transitions and locations, corresponding to the production and consumption of some resource (e.g. energy). We ask the question whether there exists an infinite path for which the accumulated weight for any finite prefix satisfies certain constraints (e.g. remains between 0 and some given upperbound). We also consider a game version of the above, where certain transitions may be uncontrollable. 1
On the optimal reachability problem on weighted timed automata
, 2007
"... We study the costoptimal reachability problem for weighted timed automata such that positive and negative costs are allowed on edges and locations. By optimality, we mean an infimum cost as well as a supremum cost. We show that this problem is PSPACECOMPLETE. Our proof uses techniques of linear p ..."
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Cited by 15 (2 self)
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We study the costoptimal reachability problem for weighted timed automata such that positive and negative costs are allowed on edges and locations. By optimality, we mean an infimum cost as well as a supremum cost. We show that this problem is PSPACECOMPLETE. Our proof uses techniques of linear programming, and thus exploits an important property of optimal runs: their timetransitions use a time τ which is arbitrarily close to an integer. We then propose an extension of the region graph, the weighted discrete graph, whose structure gives light on the way to solve the costoptimal reachability problem. We also give an application of the costoptimal reachability problem in the context of timed games.
Reachabilitytime games on timed automata
 In Proc. of 34th Int. Colloquium on Automata, Languages and Programming (ICALP’07), LNCS 4596
, 2007
"... In a reachabilitytime game, players Min and Max choose moves so that the time to reach a final state in a timed automaton is minimised or maximised, respectively. Asarin and Maler showed decidability of reachabilitytime games on strongly nonZeno timed automata using a value iteration algorithm. T ..."
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Cited by 11 (4 self)
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In a reachabilitytime game, players Min and Max choose moves so that the time to reach a final state in a timed automaton is minimised or maximised, respectively. Asarin and Maler showed decidability of reachabilitytime games on strongly nonZeno timed automata using a value iteration algorithm. This paper complements their work by providing a strategy improvement algorithm for the problem. It also generalizes their decidability result because the proposed strategy improvement algorithm solves reachabilitytime games on all timed automata. The exact computational complexity of solving reachabilitytime games is also established: the problem is EXPTIMEcomplete for timed automata with at least two clocks. 1
Average reward timed games
 In FORMATS 05, LNCS 3829
, 2005
"... Abstract. We consider realtime games where the goal consists, for each player, in maximizing the average reward he or she receives per time unit. We consider zerosum rewards, so that a reward of +r to one player corresponds to a reward of −r to the other player. The games are played on discreteti ..."
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Cited by 10 (2 self)
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Abstract. We consider realtime games where the goal consists, for each player, in maximizing the average reward he or she receives per time unit. We consider zerosum rewards, so that a reward of +r to one player corresponds to a reward of −r to the other player. The games are played on discretetime game structures which can be specified using a twoplayer version of timed automata whose locations are labeled by reward rates. Even though the rewards themselves are zerosum, the games are not, due to the requirement that time must progress along a play of the game. Since we focus on control applications, we define the value of the game to a player to be the maximal average reward per time unit that the player can ensure. We show that, in general, the values to players 1 and 2 do not sum to zero. We provide algorithms for computing the value of the game for either player; the algorithms are based on the relationship between the original, infiniteround game, and a derived game that is played for only finitely many rounds. As memoryless optimal strategies exist for both players in both games, we show that the problem of computing the value of the game is in NP∩coNP. 1
Minimumtime reachability in timed games
 In ICALP 2007, volume 4596 of LNCS
, 2007
"... Abstract. We consider the minimumtime reachability problem in concurrent twoplayer timed automaton game structures. We show how to compute the minimum time needed by a player to reach a location against all possible choices of the opponent We do not put any syntactic restriction on the game struct ..."
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Cited by 9 (0 self)
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Abstract. We consider the minimumtime reachability problem in concurrent twoplayer timed automaton game structures. We show how to compute the minimum time needed by a player to reach a location against all possible choices of the opponent We do not put any syntactic restriction on the game structure, nor do we require any player to guarantee time divergence. We only require players to use physically realizable strategies. The minimal time is computed in part using a fixpoint expression which we show can be used on equivalence classes of a nontrivial extension of the region equivalence relation. 1
Weighted timed automata: modelchecking and games
 Electr. Notes Theor. Comput. Sci
, 2006
"... In this paper, we present weighted/priced timed automata, an extension of timed automaton with costs, and solve several interesting problems on that model. Key words: Weighted/priced timed automata, modelchecking, games. 1 ..."
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Cited by 8 (2 self)
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In this paper, we present weighted/priced timed automata, an extension of timed automaton with costs, and solve several interesting problems on that model. Key words: Weighted/priced timed automata, modelchecking, games. 1