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### Citations

13994 |
Computers and Intractability: A Guide to the Theory of NPcompleteness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...rall also pseudo-polynomial. We prove that no trulypolynomial-time algorithm can be obtained unless P= NP as the one-player AEL problem is NP-hard. We show it by reduction from the subset-sum problem =-=[18]-=-. Memory requirements. Recall that forP2, the situation is simpler and memoryless strategies suffice. By the reduction to AELU , we know that pseudo-polynomial memory suffices for P1. This bound is ti... |

329 |
Variance-penalised Markov decision processes
- Filar, Kallenberg, et al.
- 1989
(Show Context)
Citation Context ...s graph techniques with local linear program P. Bouyer, N. Markey, M. Randour, K.G. Larsen, S. Laursen 3 Game objective 1-player 2-player memory MP in P [26] in NP ∩ coNP [33] memoryless [15] TP in P =-=[17]-=- in NP ∩ coNP [19] memoryless [20] EGL in P [5] in NP ∩ coNP [10, 5] memoryless [10] EGLU PSPACE-complete [16] EXPTIME-complete [5] pseudo-polynomial AE in P in NP ∩ coNP memoryless AELU , polynomial ... |

263 |
A characterization of the minimum cycle mean in a digraph
- Karp
- 1978
(Show Context)
Citation Context ...ts the structure of winning strategies and mixes graph techniques with local linear program P. Bouyer, N. Markey, M. Randour, K.G. Larsen, S. Laursen 3 Game objective 1-player 2-player memory MP in P =-=[26]-=- in NP ∩ coNP [33] memoryless [15] TP in P [17] in NP ∩ coNP [19] memoryless [20] EGL in P [5] in NP ∩ coNP [10, 5] memoryless [10] EGLU PSPACE-complete [16] EXPTIME-complete [5] pseudo-polynomial AE ... |

128 |
Deciding the winner in parity games is
- Jurdzinski
(Show Context)
Citation Context ...er 1 is then to ensure a sufficient payoff with regard to a given threshold value. Seminal classes of quantitative games include mean-payoff (MP), total-payoff (TP) and energy games (EG). In MP games =-=[15, 33, 24]-=-, player 1 has to optimize his long-run average gain per edge taken whereas, in TP games [20, 19], player 1 has to optimize his long-run sum of weights. Energy games [10, 5, 23] model safety-like prop... |

114 |
Positional strategies for mean payoff games
- Ehrenfeucht, Mycielski
- 1979
(Show Context)
Citation Context ...er 1 is then to ensure a sufficient payoff with regard to a given threshold value. Seminal classes of quantitative games include mean-payoff (MP), total-payoff (TP) and energy games (EG). In MP games =-=[15, 33, 24]-=-, player 1 has to optimize his long-run average gain per edge taken whereas, in TP games [20, 19], player 1 has to optimize his long-run sum of weights. Energy games [10, 5, 23] model safety-like prop... |

59 | L.: Energy parity games
- Chatterjee, Doyen
- 2010
(Show Context)
Citation Context ...ary to go beyond them to accurately model practical applications. For example, multi-dimensional games and conjunctions with a parity objective model trade-offs between different quantitative aspects =-=[11, 14, 32]-=-. Similarly, window objectives address the need for strategies ensuring good quantitative behaviors within reasonable time frames [12]. Average-energy games. We study the average-energy (AE) payoff fu... |

20 | Intruder deduction for AClike equational theories with homomorphisms
- Lafourcade, Lugiez, et al.
- 2004
(Show Context)
Citation Context ...the AEL objective, but also keeps the energy below U at all times. Our construction exploits a result of Lafourcade et al. that bounds the value of the counter along a path in a one-counter automaton =-=[28]-=-. We build upon it to define an appropriate transformation leading to the witness path and derive a sufficiently large upper bound U ∈N for the AELU problem. Complexity. Plugging this bound U in the p... |

19 | Games where you can play optimally without any memory
- Gimbert, Zielonka
- 2005
(Show Context)
Citation Context ...r bound U ∈N) on the energy. Results without reference are proved in this paper. solving (Thm. 4). Finally, we lift memoryless determinacy to the two-player case using results by Gimbert and Zielonka =-=[21]-=- and obtain the NP ∩ coNP-membership as a corollary (Thm. 6). C) We establish an EXPTIME algorithm to solve two-player AE games with lower- and upperbounded energy (AELU) with an arbitrary upper bound... |

13 | Half-positional determinacy of infinite games
- Kopczyński
(Show Context)
Citation Context ...E equal to +∞. 3.2 Useful properties of the average-energy Classical sufficient criteria. Various sufficient criteria—or connected approaches—to deduce memoryless determinacy appear in the literature =-=[15, 2, 1, 20, 27]-=-. Unfortunately, they cannot be applied straight out of the box to the AE payoff. Intuitively, a common requirement is for winning objectives to be closed under cyclic permutation and under concatenat... |

13 | The bad match, a total reward stochastic game, Operations Research Spektrum 9
- Thuijsman, Vrieze
- 1987
(Show Context)
Citation Context ...e to formalize the specification desired in a practical application [9], which we detail in the following as a motivating example. Interestingly, it turns out that this payoff first appeared long ago =-=[31]-=-, but it was not subject to a systematic study until very recently: see related work for more discussion. In addition to being meaningful w.r.t. practical applications, AE games also have theoretical ... |

12 |
Deciding the winner in parity games is in UP∩coUP,
- Jurdzinski
- 1998
(Show Context)
Citation Context ...er 1 is then to ensure a sufficient payoff with regard to a given threshold value. Seminal classes of quantitative games include mean-payoff (MP), total-payoff (TP) and energy games (EG). In MP games =-=[15, 33, 24]-=-, player 1 has to optimize his long-run average gain per edge taken whereas, in TP games [20, 19], player 1 has to optimize his long-run sum of weights. Energy games [10, 5, 23] model safety-like prop... |

9 | The complexity of multi-mean-payoff and multi-energy games
- Velner, Chatterjee, et al.
(Show Context)
Citation Context ...ary to go beyond them to accurately model practical applications. For example, multi-dimensional games and conjunctions with a parity objective model trade-offs between different quantitative aspects =-=[11, 14, 32]-=-. Similarly, window objectives address the need for strategies ensuring good quantitative behaviors within reasonable time frames [12]. Average-energy games. We study the average-energy (AE) payoff fu... |

5 | 2009): Games through Nested Fixpoints
- Gawlitza, Seidl
(Show Context)
Citation Context ... quantitative games include mean-payoff (MP), total-payoff (TP) and energy games (EG). In MP games [15, 33, 24], player 1 has to optimize his long-run average gain per edge taken whereas, in TP games =-=[20, 19]-=-, player 1 has to optimize his long-run sum of weights. Energy games [10, 5, 23] model safety-like properties: the goal is to ensure that the running sum of weights never drops below zero and/or that ... |

4 | Markov decision processes and stochastic games with total effective payoff, RUTCOR Research report
- Boros, Elbassioni, et al.
(Show Context)
Citation Context ...ntly. Nothing was known about memoryless determinacy and complexity of deciding the winner. Independently to our work, Boros et al. recently studied the same payoff (under the name total-payoff ). In =-=[4]-=-, they study Markov decision processes and stochastic games with the payoff of Eq. (1) and solve both questions. Their results overlap with ours for AE games (Table 1). Let us first mention that our r... |

4 | Optimal bounds for multiweighted and parametrised energy games
- Juhl, Larsen, et al.
- 2013
(Show Context)
Citation Context ...mes (EG). In MP games [15, 33, 24], player 1 has to optimize his long-run average gain per edge taken whereas, in TP games [20, 19], player 1 has to optimize his long-run sum of weights. Energy games =-=[10, 5, 23]-=- model safety-like properties: the goal is to ensure that the running sum of weights never drops below zero and/or that it never exceeds a given upper bound U ∈ N. All three classes share common prope... |

3 |
Vorobyov (2004): Memoryless Determinacy of Parity and Mean Payoff Games: A Simple Proof. Theoretical Computer Science 310(1-3
- Björklund, Sandberg, et al.
(Show Context)
Citation Context ...E equal to +∞. 3.2 Useful properties of the average-energy Classical sufficient criteria. Various sufficient criteria—or connected approaches—to deduce memoryless determinacy appear in the literature =-=[15, 2, 1, 20, 27]-=-. Unfortunately, they cannot be applied straight out of the box to the AE payoff. Intuitively, a common requirement is for winning objectives to be closed under cyclic permutation and under concatenat... |

2 |
Srba (2008): Infinite Runs in Weighted Timed Automata with Energy Constraints
- Bouyer, Fahrenberg, et al.
(Show Context)
Citation Context ...mes (EG). In MP games [15, 33, 24], player 1 has to optimize his long-run average gain per edge taken whereas, in TP games [20, 19], player 1 has to optimize his long-run sum of weights. Energy games =-=[10, 5, 23]-=- model safety-like properties: the goal is to ensure that the running sum of weights never drops below zero and/or that it never exceeds a given upper bound U ∈ N. All three classes share common prope... |

2 |
Gentilini & J.-F. Raskin (2011): Faster algorithms for mean-payoff games
- Brim, Chaloupka, et al.
(Show Context)
Citation Context ...win, or player 2 has one, and in both cases no memory is required to win). Second, deciding the winner for those games is in NP ∩ coNP and no polynomial algorithm is known despite many efforts (e.g., =-=[8, 12]-=-). Energy games with both lower and upper bounds (EGLU) are more complex: they are EXPTIME-complete and winning requires memory in general [5]. While those classes are well-known, it is sometimes nece... |

2 |
Reynier (2009): Automatic Synthesis of Robust and
- Cassez, Jessen, et al.
(Show Context)
Citation Context ...on: in AE games, the goal of player 1 is to optimize the long-run average accumulated energy over a play. We introduce this objective to formalize the specification desired in a practical application =-=[9]-=-, which we detail in the following as a motivating example. Interestingly, it turns out that this payoff first appeared long ago [31], but it was not subject to a systematic study until very recently:... |

2 |
Laroussinie (2008): Model Checking Probabilistic Timed Automata with One or Two Clocks
- Jurdziński, Sproston, et al.
(Show Context)
Citation Context ...er no legitimate move is available. If such a configuration is of the form (v,0),P1 wins the play, otherwise P2 wins. Deciding the winner given an initial configuration (vinit,c0) is EXPTIME-complete =-=[25]-=-. Our reduction is depicted in Fig. 6. The EL is initialized to c0, then it is decreasing along any play. Consider the AEL objective for AE threshold t := 0. To ensure that the energy always stays non... |

2 | 2013): Automated Synthesis of Reliable and Efficient Systems Through Game Theory: A Case Study
- Randour
(Show Context)
Citation Context ...play, whatever the strategy played by his adversary. To reason about resource constraints and the performance of strategies, quantitative games have been considered in the literature. See for example =-=[10, 3, 29]-=-, or [30] for an overview. Those games are played on weighted graphs, where edges are fitted with integer weights modeling rewards or costs. The performance of a play is evaluated via a payoff functio... |

1 |
2014): First Cycle Games
- Aminof, Rubin
(Show Context)
Citation Context ...E equal to +∞. 3.2 Useful properties of the average-energy Classical sufficient criteria. Various sufficient criteria—or connected approaches—to deduce memoryless determinacy appear in the literature =-=[15, 2, 1, 20, 27]-=-. Unfortunately, they cannot be applied straight out of the box to the AE payoff. Intuitively, a common requirement is for winning objectives to be closed under cyclic permutation and under concatenat... |

1 |
Jobstmann (2009): Better Quality in Synthesis through Quantitative Objectives
- Bloem, Chatterjee, et al.
(Show Context)
Citation Context ...play, whatever the strategy played by his adversary. To reason about resource constraints and the performance of strategies, quantitative games have been considered in the literature. See for example =-=[10, 3, 29]-=-, or [30] for an overview. Those games are played on weighted graphs, where edges are fitted with integer weights modeling rewards or costs. The performance of a play is evaluated via a payoff functio... |

1 |
2015): Average-energy games. Research Report, Laboratoire Spécification et Vérification
- Bouyer, Markey, et al.
(Show Context)
Citation Context ...is strategy is trivially winning for the AEL problem as well. This observation leads to an incremental algorithm that is correct (no false positives) but incomplete (it is not guaranteed to stop). In =-=[6]-=-, we draw the outline of a potential approach to obtain completeness hence decidability. Lemma 11. There is an algorithm that takes as input an AEL problem and iteratively solves corresponding AELU pr... |

1 |
Novotný (2014): Minimizing Running Costs in Consumption Systems
- Brázdil, Klaška, et al.
(Show Context)
Citation Context ...sions are completely new, exhibit theoretical interest and are adequate for practical applications in constrained energy systems, as witnessed by the case study of [9]. Recent work of Brázdil et al. =-=[7]-=- considers the optimization of a payoff under energy constraint. They study mean-payoff in consumption systems, i.e., simplified one-player energy games where all edges consume energy but some states ... |

1 |
Stoelinga (2003): Resource Interfaces
- Chakrabarti, Alfaro, et al.
(Show Context)
Citation Context ...play, whatever the strategy played by his adversary. To reason about resource constraints and the performance of strategies, quantitative games have been considered in the literature. See for example =-=[10, 3, 29]-=-, or [30] for an overview. Those games are played on weighted graphs, where edges are fitted with integer weights modeling rewards or costs. The performance of a play is evaluated via a payoff functio... |

1 |
Randour & J.-F. Raskin (2015): Looking at mean-payoff and total-payoff through windows
- Chatterjee, Doyen, et al.
(Show Context)
Citation Context ...win, or player 2 has one, and in both cases no memory is required to win). Second, deciding the winner for those games is in NP ∩ coNP and no polynomial algorithm is known despite many efforts (e.g., =-=[8, 12]-=-). Energy games with both lower and upper bounds (EGLU) are more complex: they are EXPTIME-complete and winning requires memory in general [5]. While those classes are well-known, it is sometimes nece... |

1 |
Prabhu (2013): Quantitative timed simulation functions and refinement metrics for real-time systems
- Chatterjee, S
(Show Context)
Citation Context ...not subject to a systematic study until very recently: see related work for more discussion. In addition to being meaningful w.r.t. practical applications, AE games also have theoretical interest. In =-=[13]-=-, Chatterjee and Prabhu define the average debit-sum level objective, which can be seen as a variation of the average-energy where the accumulated energy is taken to be zero in any point where it is a... |

1 |
Randour & J.-F. Raskin (2014): Strategy synthesis for multi-dimensional quantitative objectives
- Chatterjee, M
(Show Context)
Citation Context ...ary to go beyond them to accurately model practical applications. For example, multi-dimensional games and conjunctions with a parity objective model trade-offs between different quantitative aspects =-=[11, 14, 32]-=-. Similarly, window objectives address the need for strategies ensuring good quantitative behaviors within reasonable time frames [12]. Average-energy games. We study the average-energy (AE) payoff fu... |

1 |
Jurdziński (2013): Reachability in two-clock timed automata is PSPACE-complete
- Fearnley, M
(Show Context)
Citation Context ...objective 1-player 2-player memory MP in P [26] in NP ∩ coNP [33] memoryless [15] TP in P [17] in NP ∩ coNP [19] memoryless [20] EGL in P [5] in NP ∩ coNP [10, 5] memoryless [10] EGLU PSPACE-complete =-=[16]-=- EXPTIME-complete [5] pseudo-polynomial AE in P in NP ∩ coNP memoryless AELU , polynomial U in P in NP ∩ coNP polynomial AELU , arbitrary U in EXPTIME / PSPACE-hard EXPTIME-complete pseudo-polynomial ... |

1 |
2004): When Can You Play Positionnaly
- Gimbert, Zielonka
(Show Context)
Citation Context |

1 |
2014): Synthesis in Multi-Criteria Quantitative Games
- Randour
(Show Context)
Citation Context ...he strategy played by his adversary. To reason about resource constraints and the performance of strategies, quantitative games have been considered in the literature. See for example [10, 3, 29], or =-=[30]-=- for an overview. Those games are played on weighted graphs, where edges are fitted with integer weights modeling rewards or costs. The performance of a play is evaluated via a payoff function that ma... |

1 |
The Complexity of
- Zwick, Paterson
- 1996
(Show Context)
Citation Context |