61 research outputs found
A survey of stochastic ω regular games
We summarize classical and recent results about two-player games played on graphs with ω-regular objectives. These games have applications in the verification and synthesis of reactive systems. Important distinctions are whether a graph game is turn-based or concurrent; deterministic or stochastic; zero-sum or not. We cluster known results and open problems according to these classifications
The Complexity of Nash Equilibria in Stochastic Multiplayer Games
We analyse the computational complexity of finding Nash equilibria in
turn-based stochastic multiplayer games with omega-regular objectives. We show
that restricting the search space to equilibria whose payoffs fall into a
certain interval may lead to undecidability. In particular, we prove that the
following problem is undecidable: Given a game G, does there exist a Nash
equilibrium of G where Player 0 wins with probability 1? Moreover, this problem
remains undecidable when restricted to pure strategies or (pure) strategies
with finite memory. One way to obtain a decidable variant of the problem is to
restrict the strategies to be positional or stationary. For the complexity of
these two problems, we obtain a common lower bound of NP and upper bounds of NP
and PSPACE respectively. Finally, we single out a special case of the general
problem that, in many cases, admits an efficient solution. In particular, we
prove that deciding the existence of an equilibrium in which each player either
wins or loses with probability 1 can be done in polynomial time for games where
the objective of each player is given by a parity condition with a bounded
number of priorities
The Complexity of Nash Equilibria in Stochastic Multiplayer Games
We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with -regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game~, does there exist a pure-strategy Nash equilibrium of~ where player 0 wins with probability~. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if randomised strategies are allowed, decidability remains an open problem; we can only prove NP-hardness in this case. One way to obtain a provably decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability~ can be done in polynomial time for games where, for instance, the objective of each player is given by a parity condition with a bounded number of priorities
Pure Nash Equilibria in Concurrent Deterministic Games
We study pure-strategy Nash equilibria in multi-player concurrent
deterministic games, for a variety of preference relations. We provide a novel
construction, called the suspect game, which transforms a multi-player
concurrent game into a two-player turn-based game which turns Nash equilibria
into winning strategies (for some objective that depends on the preference
relations of the players in the original game). We use that transformation to
design algorithms for computing Nash equilibria in finite games, which in most
cases have optimal worst-case complexity, for large classes of preference
relations. This includes the purely qualitative framework, where each player
has a single omega-regular objective that she wants to satisfy, but also the
larger class of semi-quantitative objectives, where each player has several
omega-regular objectives equipped with a preorder (for instance, a player may
want to satisfy all her objectives, or to maximise the number of objectives
that she achieves.)Comment: 72 page
Fast Symbolic Algorithms for Omega-Regular Games under Strong Transition Fairness
We consider fixpoint algorithms for two-player games on graphs with -regular winning conditions, where the environment is constrained by a strong transition fairness assumption. Strong transition fairness is a widely occurring special case of strong fairness, which requires that any execution is strongly fair with respect to a specified set of live edges: whenever the source vertex of a live edge is visited infinitely often along a play, the edge itself is traversed infinitely often along the play as well. We show that, surprisingly, strong transition fairness retains the algorithmic characteristics of the fixpoint algorithms for -regular games -- the new algorithms can be obtained simply by replacing certain occurrences of the controllable predecessor by a new almost sure predecessor operator. For Rabin games with pairs, the complexity of the new algorithm is symbolic steps, which is independent of the number of live edges in the strong transition fairness assumption. Further, we show that GR(1) specifications with strong transition fairness assumptions can be solved with a 3-nested fixpoint algorithm, same as the usual algorithm. In contrast, strong fairness necessarily requires increasing the alternation depth depending on the number of fairness assumptions. We get symbolic algorithms for (generalized) Rabin, parity and GR(1) objectives under strong transition fairness assumptions as well as a direct symbolic algorithm for qualitative winning in stochastic -regular games that runs in symbolic steps, improving the state of the art. Finally, we have implemented a BDD-based synthesis engine based on our algorithm. We show on a set of synthetic and real benchmarks that our algorithm is scalable, parallelizable, and outperforms previous algorithms by orders of magnitude
Symbolic Algorithms for Graphs and Markov Decision Processes with Fairness Objectives
Given a model and a specification, the fundamental model-checking problem
asks for algorithmic verification of whether the model satisfies the
specification. We consider graphs and Markov decision processes (MDPs), which
are fundamental models for reactive systems. One of the very basic
specifications that arise in verification of reactive systems is the strong
fairness (aka Streett) objective. Given different types of requests and
corresponding grants, the objective requires that for each type, if the request
event happens infinitely often, then the corresponding grant event must also
happen infinitely often. All -regular objectives can be expressed as
Streett objectives and hence they are canonical in verification. To handle the
state-space explosion, symbolic algorithms are required that operate on a
succinct implicit representation of the system rather than explicitly accessing
the system. While explicit algorithms for graphs and MDPs with Streett
objectives have been widely studied, there has been no improvement of the basic
symbolic algorithms. The worst-case numbers of symbolic steps required for the
basic symbolic algorithms are as follows: quadratic for graphs and cubic for
MDPs. In this work we present the first sub-quadratic symbolic algorithm for
graphs with Streett objectives, and our algorithm is sub-quadratic even for
MDPs. Based on our algorithmic insights we present an implementation of the new
symbolic approach and show that it improves the existing approach on several
academic benchmark examples.Comment: Full version of the paper. To appear in CAV 201
Simple and tight complexity lower bounds for solving Rabin games
We give a simple proof that assuming the Exponential Time Hypothesis (ETH),
determining the winner of a Rabin game cannot be done in time , where is the number of pairs of vertex subsets involved in
the winning condition and is the vertex count of the game graph. While this
result follows from the lower bounds provided by Calude et al [SIAM J. Comp.
2022], our reduction is simpler and arguably provides more insight into the
complexity of the problem. In fact, the analogous lower bounds discussed by
Calude et al, for solving Muller games and multidimensional parity games,
follow as simple corollaries of our approach. Our reduction also highlights the
usefulness of a certain pivot problem -- Permutation SAT -- which may be of
independent interest.Comment: 10 pages, 5 figures. To appear in SOSA 202
Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games
2.5 player parity games combine the challenges posed by 2.5 player
reachability games and the qualitative analysis of parity games. These two
types of problems are best approached with different types of algorithms:
strategy improvement algorithms for 2.5 player reachability games and recursive
algorithms for the qualitative analysis of parity games. We present a method
that - in contrast to existing techniques - tackles both aspects with the best
suited approach and works exclusively on the 2.5 player game itself. The
resulting technique is powerful enough to handle games with several million
states
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