62,189 research outputs found
Concurrent Games and Semi-Random Determinacy
Consider concurrent, infinite duration, two-player win/lose games played on graphs. If the winning condition satisfies some simple requirement, the existence of Player 1 winning (finite-memory) strategies is equivalent to the existence of winning (finite-memory) strategies in finitely many derived one-player games. Several classical winning conditions satisfy this simple requirement.
Under an additional requirement on the winning condition, the non-existence of Player 1 winning strategies from all vertices is equivalent to the existence of Player 2 stochastic strategies almost-sure winning from all vertices. Only few classical winning conditions satisfy this additional requirement, but a fairness variant of omega-regular languages does
Additional Winning Strategies in Two-Player Games
Abstract. We study the problem of checking whether a two-player reachability game admits more than a winning strategy. We investigate this in case of perfect and imperfect information, and, by means of an automata approach we provide a linear-time procedure and an exponential-time procedure, respectively. In both cases, the results are tight
Synthesizing Permissive Winning Strategy Templates for Parity Games
We present a novel method to compute \emph{permissive winning strategies} in
two-player games over finite graphs with -regular winning conditions.
Given a game graph and a parity winning condition , we compute a
\emph{winning strategy template} that collects an infinite number of
winning strategies for objective in a concise data structure. We use
this new representation of sets of winning strategies to tackle two problems
arising from applications of two-player games in the context of cyber-physical
system design -- (i) \emph{incremental synthesis}, i.e., adapting strategies to
newly arriving, \emph{additional} -regular objectives , and (ii)
\emph{fault-tolerant control}, i.e., adapting strategies to the occasional or
persistent unavailability of actuators. The main features of our strategy
templates -- which we utilize for solving these challenges -- are their easy
computability, adaptability, and compositionality. For \emph{incremental
synthesis}, we empirically show on a large set of benchmarks that our technique
vastly outperforms existing approaches if the number of added specifications
increases. While our method is not complete, our prototype implementation
returns the full winning region in all 1400 benchmark instances, i.e., handling
a large problem class efficiently in practice.Comment: CAV'2
The Fixpoint-Iteration Algorithm for Parity Games
It is known that the model checking problem for the modal mu-calculus reduces
to the problem of solving a parity game and vice-versa. The latter is realised
by the Walukiewicz formulas which are satisfied by a node in a parity game iff
player 0 wins the game from this node. Thus, they define her winning region,
and any model checking algorithm for the modal mu-calculus, suitably
specialised to the Walukiewicz formulas, yields an algorithm for solving parity
games. In this paper we study the effect of employing the most straight-forward
mu-calculus model checking algorithm: fixpoint iteration. This is also one of
the few algorithms, if not the only one, that were not originally devised for
parity game solving already. While an empirical study quickly shows that this
does not yield an algorithm that works well in practice, it is interesting from
a theoretical point for two reasons: first, it is exponential on virtually all
families of games that were designed as lower bounds for very particular
algorithms suggesting that fixpoint iteration is connected to all those.
Second, fixpoint iteration does not compute positional winning strategies. Note
that the Walukiewicz formulas only define winning regions; some additional work
is needed in order to make this algorithm compute winning strategies. We show
that these are particular exponential-space strategies which we call
eventually-positional, and we show how positional ones can be extracted from
them.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Solving Odd-Fair Parity Games
This paper discusses the problem of efficiently solving parity games where
player Odd has to obey an additional 'strong transition fairness constraint' on
its vertices -- given that a player Odd vertex is visited infinitely often,
a particular subset of the outgoing edges (called live edges) of has to be
taken infinitely often. Such games, which we call 'Odd-fair parity games',
naturally arise from abstractions of cyber-physical systems for planning and
control.
In this paper, we present a new Zielonka-type algorithm for solving Odd-fair
parity games. This algorithm not only shares 'the same worst-case time
complexity' as Zielonka's algorithm for (normal) parity games but also
preserves the algorithmic advantage Zielonka's algorithm possesses over other
parity solvers with exponential time complexity.
We additionally introduce a formalization of Odd player winning strategies in
such games, which were unexplored previous to this work. This formalization
serves dual purposes: firstly, it enables us to prove our Zielonka-type
algorithm; secondly, it stands as a noteworthy contribution in its own right,
augmenting our understanding of additional fairness assumptions in two-player
games.Comment: To be published in FSTTCS 202
Repairing Multi-Player Games
Synthesis is the automated construction of systems from their specifications. Modern systems often consist of interacting components, each having its own objective. The interaction among the components is modeled by a multi-player game. Strategies of the components induce a trace in the game, and the objective of each component is to force the game into a trace that satisfies its specification. This is modeled by augmenting the game with omega-regular winning conditions. Unlike traditional synthesis games, which are zero-sum, here the objectives of the components do not necessarily contradict each other. Accordingly, typical questions about these games concern their stability - whether the players reach an equilibrium, and their social welfare - maximizing the set of (possibly weighted) specifications that are satisfied.
We introduce and study repair of multi-player games. Given a game, we study the possibility of modifying the objectives of the players in order to obtain stability or to improve the social welfare. Specifically, we solve the problem of modifying the winning conditions in a given concurrent multi-player game in a way that guarantees the existence of a Nash equilibrium. Each modification has a value, reflecting both the cost of strengthening or weakening the underlying specifications, as well as the benefit of satisfying specifications in the obtained equilibrium. We seek optimal modifications, and we study the problem for various omega-regular objectives and various cost and benefit functions. We analyze the complexity of the problem in the general setting as well as in one with a fixed number of players. We also study two additional types of repair, namely redirection of transitions and control of a subset of the players
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
Learning-Based Synthesis of Safety Controllers
We propose a machine learning framework to synthesize reactive controllers
for systems whose interactions with their adversarial environment are modeled
by infinite-duration, two-player games over (potentially) infinite graphs. Our
framework targets safety games with infinitely many vertices, but it is also
applicable to safety games over finite graphs whose size is too prohibitive for
conventional synthesis techniques. The learning takes place in a feedback loop
between a teacher component, which can reason symbolically about the safety
game, and a learning algorithm, which successively learns an overapproximation
of the winning region from various kinds of examples provided by the teacher.
We develop a novel decision tree learning algorithm for this setting and show
that our algorithm is guaranteed to converge to a reactive safety controller if
a suitable overapproximation of the winning region can be expressed as a
decision tree. Finally, we empirically compare the performance of a prototype
implementation to existing approaches, which are based on constraint solving
and automata learning, respectively
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