Strategy Synthesis for Multi-Dimensional Quantitative Objectives

Chatterjee, Krishnendu; Randour, Mickaël; Raskin, Jean-François

doi:10.1007/978-3-642-32940-1_10

Cited by 31 publications

(59 citation statements)

References 49 publications

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“…A close relationship has been established between weighted automata under probabilistic semantics and weighted Markov chains [10]. For a weighted automaton A and a Markov chain M representing the distribution over words, the probabilistic problems for A and M coincide with the probabilistic problem of the weighted Markov chain A × M. Weighted Markov chains have been intensively studied with single and multiple quantitative objectives [4,13,18,29]. The above reduction does not extend to non-deterministic weighted automata [12,Example 30].…”

Section: Introductionmentioning

confidence: 99%

Non-deterministic weighted automata evaluated over Markov chains

Michaliszyn

Otop

2020

Journal of Computer and System Sciences

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We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables.The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose approximation algorithms for the probabilistic questions, which work in exponential time in the size of the automaton and polynomial time in the size of the Markov chain and the given precision. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric. ACM Subject ClassificationTheory of computation → Automata over infinite objects; Theory of computation → Quantitative automata Some quantitative-model-checking frameworks [11] are based on the universality problem for non-deterministic automata, which asks whether all words have the value below a given threshold. Unfortunately, the universality problem is undecidable for weighted automata with the sum or the limit average values functions. The distribution question can be considered as a computationally-attractive variant of universality, i.e., we ask whether almost all words have value below some given threshold. We show that if the threshold can be approximated, the distribution question can be computed effectively. Weighted automata have been used to formally study online algorithms [2]. Online algorithms have been modeled by deterministic weighted automata, which make choices based solely on the past, while offline algorithms have been modeled by non-deterministic weighted automata. Relating deterministic and non-deterministic models allowed for formal verification of the worst-case competitiveness ratio of online algorithms. Using the result from our paper, we can extend the analysis from [2] to the average-case competitiveness. Related workThe problem considered in this paper is related to the following areas from the literature. Probabilistic verification of qualitative properties. Probabilistic verification asks for the probability of the set of traces satisfying a given property. For non-weighted automata, it has been extensively studied [33,15,4] and implemented [26,22]. The prevalent approach in this area is to work with deterministic automata, and apply determinisation as needed. To obtain better complexity bounds, the probabilistic verification problem has been directly studied for unambiguous Büchi automata in [5]; the authors explain there the potential pitfalls in the probabilistic analysis of non-deterministic automata.Weighted automata under probabilistic semantics. Probabilistic verification of weighted automata and their extensions has been studied in [12]. All automata considered there are deterministic. (MDPs). MDPs are a classical exte...

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Section: Introductionmentioning

confidence: 99%

Non-deterministic weighted automata evaluated over Markov chains

Michaliszyn

Otop

2020

Journal of Computer and System Sciences

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“…Games on VASS with inhibitor arcs are studied in [4] and decidability is obtained in the case where one of the players can only increment counters and the other player can not test for zero value in counters. In [7], energy games are studied, which are games on counter systems and the goal of the game is to play for ever without any counter going below zero in addition to satisfying parity conditions on the control states that are visited infinitely often. Energy games are further studied in [1], where they are related to single-sided VASS games, which restrict one of the players to not make any changes to the counters.…”

Section: Introductionmentioning

confidence: 99%

Playing with Repetitions in Data Words Using Energy Games

Figueira

Praveen

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We introduce two-player games which build words over infinite alphabets, and we study the problem of checking the existence of winning strategies. These games are played by two players, who take turns in choosing valuations for variables ranging over an infinite data domain, thus generating multi-attributed data words. The winner of the game is specified by formulas in the Logic of Repeating Values, which can reason about repetitions of data values in infinite data words. We prove that it is undecidable to check if one of the players has a winning strategy, even in very restrictive settings. However, we prove that if one of the players is restricted to choose valuations ranging over the Boolean domain, the games are effectively equivalent to single-sided games on vector addition systems with states (in which one of the players can change control states but cannot change counter values), known to be decidable and effectively equivalent to energy games.Previous works have shown that the satisfiability problem for various variants of the logic of repeating values is equivalent to the reachability and coverability problems in vector addition systems. Our results raise this connection to the level of games, augmenting further the associations between logics on data words and counter systems.We denote by Z the set of integers and by N the set of non-negative integers. We let N + denote the set of integers that are strictly greater than 0. For any set S, we denote by S * (resp. S ω ) the set of all finite (resp. countably infinite) sequences of elements in S. For a sequence σ ∈ S * , we denote its length by |σ|. We denote by P(S) (resp. P + (S)) the set of all subsets (resp. non-empty subsets) of S. Logic of repeating valuesWe recall the syntax and semantics of the logic of repeating values from [10,11]. This logic extends the usual propositional linear temporal logic with the ability to reason about repetitions of data values from an infinite domain. We let this logic use both Boolean variables (i.e., propositions) and data variables ranging over an infinite data domain D. The Boolean variables can be simulated by data variables. However, we need to consider fragments of the logic, for which explicitly having Boolean variables is convenient. Let BVARS = {q, t, . . .} be a countably infinite set of Boolean variables ranging over { , ⊥}, and let DVARS = {x, y, . . .} be a countably infinite set of 'data' variables ranging over D. We denote by LRV the logic whose formulas are defined as follows: 1A valuation is the union of a mapping from BVARS to { , ⊥} and a mapping from DVARS to D. A model is a finite or infinite sequence of valuations. We use σ to denote models and σ(i) denotes the i th valuation in σ, where i ∈ N + . For any model σ and position i ∈ N + , the satisfaction relation |= is defined inductively as follows. The semantics of temporal operators next (X), previous (X −1 ), until (U), since (S) and the Boolean connectives are defined in the usual way, but for the sake of completeness we provide their 1 In a pr...

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“…Open Questions. However, these recent advances do not settle the case of multi-dimensional energy parity games [7], where Player 1 must ensure that, in addition to the quantitative energy objective (specifying resource consumption and replenishment), she also complies with a qualitative ωregular objective in the form of a parity condition (specifying functional requirements). These games with arbitrary initial credit are still coNPcomplete as a consequence of [7,Lemma 4].…”

Section: Introductionmentioning

confidence: 99%

“…However, these recent advances do not settle the case of multi-dimensional energy parity games [7], where Player 1 must ensure that, in addition to the quantitative energy objective (specifying resource consumption and replenishment), she also complies with a qualitative ωregular objective in the form of a parity condition (specifying functional requirements). These games with arbitrary initial credit are still coNPcomplete as a consequence of [7,Lemma 4]. With given initial credit, they were first proven decidable by Abdulla, Mayr, Sangnier, and Sproston [1], and used to decide both the model-checking problem for a suitable fragment of the µ-calculus against Petri net executions and the weak simulation problem between a finite state system and a Petri net; they also allow to decide the model-checking problem for the resource logic RB±ATL * [2].…”

Section: Introductionmentioning

confidence: 99%

“…Firstly, we show that perfect half space games can be easily translated to the lexicographic energy games of Colcombet and Niwiński [8]. The translation amounts to normalising the edge weights with respect to the current perfect half spaces, and inserting another d dimensions in which EnPar: multi-dimensional energy parity games [7] ExtEn: extended multi-dimensional energy games [5] Bnd: bounding games [12] PHS: perfect half space games [this paper] LexEn: lexicographic energy games [8] MP: mean-payoff games [15,9] we encode appropriate penalties for Player 2 that are imposed whenever he changes the perfect half space (cf. Section 3.2).…”

Section: Introductionmentioning

confidence: 99%

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Perfect half space games

Colcombet

Jurdziński

Lazić

et al. 2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We introduce perfect half space games, in which the goal of Player 2 is to make the sums of encountered multi-dimensional weights diverge in a direction which is consistent with a chosen sequence of perfect half spaces (chosen dynamically by Player 2). We establish that the bounding games of Jurdziński et al. (ICALP 2015) can be reduced to perfect half space games, which in turn can be translated to the lexicographic energy games of Colcombet and Niwiński, and are positionally determined in a strong sense (Player 2 can play without knowing the current perfect half space). We finally show how perfect half space games and bounding games can be employed to solve multidimensional energy parity games in pseudo-polynomial time when both the numbers of energy dimensions and of priorities are fixed, regardless of whether the initial credit is given as part of the input or existentially quantified. This also yields an optimal 2-EXPTIME complexity with given initial credit, where the best known upper bound was non-elementary.

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Strategy Synthesis for Multi-Dimensional Quantitative Objectives

Cited by 31 publications

References 49 publications

Non-deterministic weighted automata evaluated over Markov chains

Non-deterministic weighted automata evaluated over Markov chains

Playing with Repetitions in Data Words Using Energy Games

Perfect half space games

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