37 research outputs found
Succinct progress measures for solving parity games
The recent breakthrough paper by Calude et al. has given the first algorithm
for solving parity games in quasi-polynomial time, where previously the best
algorithms were mildly subexponential. We devise an alternative
quasi-polynomial time algorithm based on progress measures, which allows us to
reduce the space required from quasi-polynomial to nearly linear. Our key
technical tools are a novel concept of ordered tree coding, and a succinct tree
coding result that we prove using bounded adaptive multi-counters, both of
which are interesting in their own right
Hereditary History Preserving Bisimilarity is Undecidable
We show undecidability of hereditary history preserving bisimilarityfor finite asynchronous transition systems by a reduction from the haltingproblem of deterministic 2-counter machines. To make the proof moretransparent we introduce an intermediate problem of checking dominobisimilarity for origin constrained tiling systems. First we reduce thehalting problem of deterministic 2-counter machines to origin constraineddomino bisimilarity. Then we show how to model domino bisimulations ashereditary history preserving bisimulations for finite asynchronous transitionssystems. We also argue that the undecidability result holds forfinite 1-safe Petri nets, which can be seen as a proper subclass of finiteasynchronous transition systems
Alternating weak automata from universal trees
An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and it is polynomial if the asymptotic number of priorities is at most logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if - like all presently known such translations - it is efficiently constructive) lead to algorithms for solving parity games that are asymptotically faster in the worst case than the current state of the art (Calude, Jain, Khoussainov, Li, and Stephan, 2017; Jurdzinski and Lazic, 2017; and Fearnley, Jain, Schewe, Stephan, and Wojtczak, 2017), and hence it would yield a significant breakthrough
Perfect half space games
We introduce perfect half space games, in which the goal of Player 2 is to make the sums of encountered multidimensional 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 Jurdzinski 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 Niwinski, 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
Mean-Payoff Games on Timed Automata
Mean-payoff games on timed automata are played on the infinite weighted graph of configurations of priced timed automata between two players - Player Min and Player Max - by moving a token along the states of the graph to form an infinite run. The goal of Player Min is to minimize the limit average weight of the run, while the goal of the Player Max is the opposite. Brenguier, Cassez, and Raskin recently studied a variation of these games and showed that mean-payoff games are undecidable for timed automata with five or more clocks. We refine this result by proving the undecidability of mean-payoff games with three clocks. On a positive side, we show the decidability of mean-payoff games on one-clock timed automata with binary price-rates. A key contribution of this paper is the application of dynamic programming based proof techniques applied in the context of average reward optimization on an uncountable state and action space
Sets Which Contain a Quadratic Residue Modulo p for Almost All p
In a mean-payoff parity game, one of the two players aims both to achieve a qualitative parity objective and to minimize a quantitative long-term average of payoffs (aka. mean payoff). The game is zero-sum and hence the aim of the other player is to either foil the parity objective or to maximize the mean payoff. Our main technical result is a pseudo-quasi-polynomial algorithm for solving mean-payoff parity games. All algorithms for the problem that have been developed for over a decade have a pseudo-polynomial and an exponential factors in their running times; in the running time of our algorithm the latter is replaced with a quasi-polynomial one. By the results of Chatterjee and Doyen (2012) and of Schewe, Weinert, and Zimmermann (2018), our main technical result implies that there are pseudo-quasi-polynomial algorithms for solving parity energy games and for solving parity games with weights. Our main conceptual contributions are the definitions of strategy decompositions for both players, and a notion of progress measures for mean-payoff parity games that generalizes both parity and energy progress measures. The former provides normal forms for and succinct representations of winning strategies, and the latter enables the application to mean-payoff parity games of the order-theoretic machinery that underpins a recent quasi-polynomial algorithm for solving parity games
A technique to speed up symmetric attractor-based algorithms for parity games
The classic McNaughton-Zielonka algorithm for solving parity games has excellent performance in practice, but its worst-case asymptotic complexity is worse than that of the state-of-the-art algorithms. This work pinpoints the mechanism that is responsible for this relative underperformance and proposes a new technique that eliminates it. The culprit is the wasteful manner in which the results obtained from recursive calls are indiscriminately discarded by the algorithm whenever subgames on which the algorithm is run change. Our new technique is based on firstly enhancing the algorithm to compute attractor decompositions of subgames instead of just winning strategies on them, and then on making it carefully use attractor decompositions computed in prior recursive calls to reduce the size of subgames on which further recursive calls are made. We illustrate the new technique on the classic example of the recursive McNaughton-Zielonka algorithm, but it can be applied to other symmetric attractor-based algorithms that were inspired by it, such as the quasi-polynomial versions of the McNaughton-Zielonka algorithm based on universal trees
07471 Abstracts Collection -- Equilibrium Computation
From 18 to 23 November 2007, the Dagstuhl Seminar 07471 ``Equilibrium Computation\u27\u27 was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Stochastic Timed Automata
A stochastic timed automaton is a purely stochastic process defined on a
timed automaton, in which both delays and discrete choices are made randomly.
We study the almost-sure model-checking problem for this model, that is, given
a stochastic timed automaton A and a property , we want to decide whether
A satisfies with probability 1. In this paper, we identify several
classes of automata and of properties for which this can be decided. The proof
relies on the construction of a finite abstraction, called the thick graph,
that we interpret as a finite Markov chain, and for which we can decide the
almost-sure model-checking problem. Correctness of the abstraction holds when
automata are almost-surely fair, which we show, is the case for two large
classes of systems, single- clock automata and so-called weak-reactive
automata. Techniques employed in this article gather tools from real-time
verification and probabilistic verification, as well as topological games
played on timed automata.Comment: 40 pages + appendi