372 research outputs found
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem
Average-energy games
Two-player quantitative zero-sum games provide a natural framework to
synthesize controllers with performance guarantees for reactive systems within
an uncontrollable environment. Classical settings include mean-payoff games,
where the objective is to optimize the long-run average gain per action, and
energy games, where the system has to avoid running out of energy.
We study average-energy games, where the goal is to optimize the long-run
average of the accumulated energy. We show that this objective arises naturally
in several applications, and that it yields interesting connections with
previous concepts in the literature. We prove that deciding the winner in such
games is in NP inter coNP and at least as hard as solving mean-payoff games,
and we establish that memoryless strategies suffice to win. We also consider
the case where the system has to minimize the average-energy while maintaining
the accumulated energy within predefined bounds at all times: this corresponds
to operating with a finite-capacity storage for energy. We give results for
one-player and two-player games, and establish complexity bounds and memory
requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
Obligation Blackwell Games and p-Automata
We recently introduced p-automata, automata that read discrete-time Markov
chains. We used turn-based stochastic parity games to define acceptance of
Markov chains by a subclass of p-automata. Definition of acceptance required a
cumbersome and complicated reduction to a series of turn-based stochastic
parity games. The reduction could not support acceptance by general p-automata,
which was left undefined as there was no notion of games that supported it.
Here we generalize two-player games by adding a structural acceptance
condition called obligations. Obligations are orthogonal to the linear winning
conditions that define winning. Obligations are a declaration that player 0 can
achieve a certain value from a configuration. If the obligation is met, the
value of that configuration for player 0 is 1.
One cannot define value in obligation games by the standard mechanism of
considering the measure of winning paths on a Markov chain and taking the
supremum of the infimum of all strategies. Mainly because obligations need
definition even for Markov chains and the nature of obligations has the flavor
of an infinite nesting of supremum and infimum operators. We define value via a
reduction to turn-based games similar to Martin's proof of determinacy of
Blackwell games with Borel objectives. Based on this definition, we show that
games are determined. We show that for Markov chains with Borel objectives and
obligations, and finite turn-based stochastic parity games with obligations
there exists an alternative and simpler characterization of the value function.
Based on this simpler definition we give an exponential time algorithm to
analyze finite turn-based stochastic parity games with obligations. Finally, we
show that obligation games provide the necessary framework for reasoning about
p-automata and that they generalize the previous definition
Energy Parity Games
Energy parity games are infinite two-player turn-based games played on
weighted graphs. The objective of the game combines a (qualitative) parity
condition with the (quantitative) requirement that the sum of the weights
(i.e., the level of energy in the game) must remain positive. Beside their own
interest in the design and synthesis of resource-constrained omega-regular
specifications, energy parity games provide one of the simplest model of games
with combined qualitative and quantitative objective. Our main results are as
follows: (a) exponential memory is necessary and sufficient for winning
strategies in energy parity games; (b) the problem of deciding the winner in
energy parity games can be solved in NP \cap coNP; and (c) we give an algorithm
to solve energy parity by reduction to energy games. We also show that the
problem of deciding the winner in energy parity games is polynomially
equivalent to the problem of deciding the winner in mean-payoff parity games,
while optimal strategies may require infinite memory in mean-payoff parity
games. As a consequence we obtain a conceptually simple algorithm to solve
mean-payoff parity games
Blackwell-Optimal Strategies in Priority Mean-Payoff Games
We examine perfect information stochastic mean-payoff games - a class of
games containing as special sub-classes the usual mean-payoff games and parity
games. We show that deterministic memoryless strategies that are optimal for
discounted games with state-dependent discount factors close to 1 are optimal
for priority mean-payoff games establishing a strong link between these two
classes
EPTCS
First cycle games (FCG) are played on a finite graph by two players who push a token along the edges until a vertex is repeated, and a simple cycle is formed. The winner is determined by some fixed property Y of the sequence of labels of the edges (or nodes) forming this cycle. These games are traditionally of interest because of their connection with infinite-duration games such as parity and mean-payoff games. We study the memory requirements for winning strategies of FCGs and certain associated infinite duration games. We exhibit a simple FCG that is not memoryless determined (this corrects a mistake in Memoryless determinacy of parity and mean payoff games: a simple proof by Bj⋯orklund, Sandberg, Vorobyov (2004) that claims that FCGs for which Y is closed under cyclic permutations are memoryless determined). We show that θ (n)! memory (where n is the number of nodes in the graph), which is always sufficient, may be necessary to win some FCGs. On the other hand, we identify easy to check conditions on Y (i.e., Y is closed under cyclic permutations, and both Y and its complement are closed under concatenation) that are sufficient to ensure that the corresponding FCGs and their associated infinite duration games are memoryless determined. We demonstrate that many games considered in the literature, such as mean-payoff, parity, energy, etc., satisfy these conditions. On the complexity side, we show (for efficiently computable Y) that while solving FCGs is in PSPACE, solving some families of FCGs is PSPACE-hard
Hyperplane Separation Technique for Multidimensional Mean-Payoff Games
We consider both finite-state game graphs and recursive game graphs (or
pushdown game graphs), that can model the control flow of sequential programs
with recursion, with multi-dimensional mean-payoff objectives. In pushdown
games two types of strategies are relevant: global strategies, that depend on
the entire global history; and modular strategies, that have only local memory
and thus do not depend on the context of invocation. We present solutions to
several fundamental algorithmic questions and our main contributions are as
follows: (1) We show that finite-state multi-dimensional mean-payoff games can
be solved in polynomial time if the number of dimensions and the maximal
absolute value of the weight is fixed; whereas if the number of dimensions is
arbitrary, then problem is already known to be coNP-complete. (2) We show that
pushdown graphs with multi-dimensional mean-payoff objectives can be solved in
polynomial time. (3) For pushdown games under global strategies both single and
multi-dimensional mean-payoff objectives problems are known to be undecidable,
and we show that under modular strategies the multi-dimensional problem is also
undecidable (whereas under modular strategies the single dimensional problem is
NP-complete). We show that if the number of modules, the number of exits, and
the maximal absolute value of the weight is fixed, then pushdown games under
modular strategies with single dimensional mean-payoff objectives can be solved
in polynomial time, and if either of the number of exits or the number of
modules is not bounded, then the problem is NP-hard. (4) Finally we show that a
fixed parameter tractable algorithm for finite-state multi-dimensional
mean-payoff games or pushdown games under modular strategies with
single-dimensional mean-payoff objectives would imply the solution of the
long-standing open problem of fixed parameter tractability of parity games.Comment: arXiv admin note: text overlap with arXiv:1201.282
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