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38
Weighted automata and weighted logics
 In Automata, Languages and Programming – 32nd International Colloquium, ICALP 2005
, 2005
"... Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speechtotext processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We g ..."
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Cited by 39 (7 self)
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Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speechtotext processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi’s and Elgot’s fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with our weighted logic. We also consider weighted firstorder logic and show that aperiodic series coincide with the firstorder definable ones, if the semiring is locally finite, commutative and has some aperiodicity property. 1
Quantitative languages
"... Quantitative generalizations of classical languages, which assign to each word a real number instead of a boolean value, have applications in modeling resourceconstrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitativ ..."
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Cited by 36 (14 self)
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Quantitative generalizations of classical languages, which assign to each word a real number instead of a boolean value, have applications in modeling resourceconstrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitative languages over finite and infinite words; in particular, the real value of an infinite run is computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We define the classical decision problems of automata theory (emptiness, universality, language inclusion, and language equivalence) in the quantitative setting and study their computational complexity. As the decidability of the languageinclusion problem remains open for some classes of weighted automata, we introduce a notion of quantitative simulation that is decidable and implies language inclusion. We also give a complete characterization of the expressive power of the various classes of weighted automata. In particular, we show that most classes of weighted
Better quality in synthesis through quantitative objectives
 In CoRR, abs/0904.2638
, 2009
"... Abstract. Most specification languages express only qualitative constraints. However, among two implementations that satisfy a given specification, one may be preferred to another. For example, if a specification asks that every request is followed by a response, one may prefer an implementation tha ..."
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Cited by 22 (9 self)
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Abstract. Most specification languages express only qualitative constraints. However, among two implementations that satisfy a given specification, one may be preferred to another. For example, if a specification asks that every request is followed by a response, one may prefer an implementation that generates responses quickly but does not generate unnecessary responses. We use quantitative properties to measure the “goodness ” of an implementation. Using games with corresponding quantitative objectives, we can synthesize “optimal ” implementations, which are preferred among the set of possible implementations that satisfy a given specification. In particular, we show how automata with lexicographic meanpayoff conditions can be used to express many interesting quantitative properties for reactive systems. In this framework, the synthesis of optimal implementations requires the solution of lexicographic meanpayoff games (for safety requirements), and the solution of games with both lexicographic meanpayoff and parity objectives (for liveness requirements). We present algorithms for solving both kinds of novel graph games. 1
Linear and Branching Metrics for Quantitative Transition Systems
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming
, 2004
"... We extend the basic system relations of trace inclusion, trace equivalence, simulation, and bisimulation to a quantitative setting in which propositions are interpreted not as boolean values, but as real values in the interval [0; 1]. Trace inclusion and equivalence give rise to asymmetrical and ..."
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Cited by 20 (1 self)
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We extend the basic system relations of trace inclusion, trace equivalence, simulation, and bisimulation to a quantitative setting in which propositions are interpreted not as boolean values, but as real values in the interval [0; 1]. Trace inclusion and equivalence give rise to asymmetrical and symmetrical linear distances, while simulation and bisimulation give rise to asymmetrical and symmetrical branching distances. We study the relationships among these distances, and we provide a full logical characterization of the distances in terms of quantitative versions of LTL and calculus. We show that, while trace inclusion (resp. equivalence) coincides with simulation (resp. bisimulation) for deterministic boolean transition systems, linear and branching distances do not coincide for deterministic quantitative transition systems. Finally, we provide algorithms for computing the distances, together with matching lower and upper complexity bounds.
Games where you can play optimally without any memory
 In CONCUR 2005, LNCS
, 2005
"... Abstract. Reactive systems are often modelled as two person antagonistic games where one player represents the system while his adversary represents the environment. Undoubtedly, the most popular games in this context are parity games and their cousins (Rabin, Streett and Muller games). Recently how ..."
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Cited by 13 (4 self)
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Abstract. Reactive systems are often modelled as two person antagonistic games where one player represents the system while his adversary represents the environment. Undoubtedly, the most popular games in this context are parity games and their cousins (Rabin, Streett and Muller games). Recently however also games with other types of payments, like discounted or meanpayoff [5,6], previously used only in economic context, entered into the area of system modelling and verification. The most outstanding property of parity, meanpayoff and discounted games is the existence of optimal positional (memoryless) strategies for both players. This observation raises two questions: (1) can we characterise the family of payoff mappings for which there always exist optimal positional strategies for both players and (2) are there other payoff mappings with practical or theoretical interest and admitting optimal positional strategies. This paper provides a complete answer to the first question by presenting a simple necessary and sufficient condition on payoff mapping guaranteeing the existence of optimal positional strategies. As a corollary to this result we show the following remarkable property of payoff mappings: if both players have optimal positional strategies when playing solitary oneplayer games then also they have optimal positional strategies for twoplayer games.
Measuring and synthesizing systems in probabilistic environments
 CoRR
"... Abstract. Often one has a preference order among the different systems that satisfy a given specification. Under a probabilistic assumption about the possible inputs, such a preference order is naturally expressed by a weighted automaton, which assigns to each word a value, such that a system is pre ..."
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Cited by 10 (6 self)
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Abstract. Often one has a preference order among the different systems that satisfy a given specification. Under a probabilistic assumption about the possible inputs, such a preference order is naturally expressed by a weighted automaton, which assigns to each word a value, such that a system is preferred if it generates a higher expected value. We solve the following optimalsynthesis problem: given an omegaregular specification, a Markov chain that describes the distribution of inputs, and a weighted automaton that measures how well a system satisfies the given specification under the given input assumption, synthesize a system that optimizes the measured value. For safety specifications and measures that are defined by meanpayoff automata, the optimalsynthesis problem amounts to finding a strategy in a Markov decision process (MDP) that is optimal for a longrun average reward objective, which can be done in polynomial time. For general omegaregular specifications, the solution rests on a new, polynomialtime algorithm for computing optimal strategies in MDPs with meanpayoff parity objectives. We present some experimental results showing optimal systems that were automatically generated in this way. 1
Game relations and metrics
 In LICS’07
, 2007
"... We consider twoplayer games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. ..."
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Cited by 9 (4 self)
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We consider twoplayer games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal (e.g., “reach a target state”), the question of winning is thus a probabilistic one: “what is the maximal probability of winning from a given state?”. On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state “similarity”. We introduce equivalences and metrics for twoplayer game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative µcalculus. The quantitative µcalculus can express a large set of goals, including reachability, safety, and ωregular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation.
Model Checking Games for the Quantitive µcalculus
 IN DANS PROCEEDINGS OF THE 25TH ANNUAL SYMPOSIUM ON THE THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS 2008
, 2008
"... We investigate quantitative extensions of modal logic and the modal µcalculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative µcalculus is defined in an ..."
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Cited by 8 (0 self)
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We investigate quantitative extensions of modal logic and the modal µcalculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative µcalculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative µcalculus.