649 research outputs found
Reachability for Branching Concurrent Stochastic Games
We give polynomial time algorithms for deciding almost-sure and limit-sure reachability in Branching Concurrent Stochastic Games (BCSGs). These are a class of infinite-state imperfect-information stochastic games that generalize both finite-state concurrent stochastic reachability games ([L. de Alfaro et al., 2007]) and branching simple stochastic reachability games ([K. Etessami et al., 2018])
Reachability analysis of branching probabilistic processes
We study a fundamental class of infinite-state stochastic processes and stochastic
games, namely Branching Processes, under the properties of (single-target) reachability
and multi-objective reachability.
In particular, we study Branching Concurrent Stochastic Games (BCSGs), which
are an imperfect-information game extension to the classical Branching Processes, and
show that these games are determined, i.e., have a value, under the fundamental objective
of reachability, building on and generalizing prior work on Branching Simple
Stochastic Games and finite-state Concurrent Stochastic Games. We show that, unlike
in the turn-based branching games, in the concurrent setting the almost-sure and limitsure
reachability problems do not coincide and we give polynomial time algorithms
for deciding both almost-sure and limit-sure reachability. We also provide a discussion
on the complexity of quantitative reachability questions for BCSGs.
Furthermore, we introduce a new model, namely Ordered Branching Processes
(OBPs), which is a hybrid model between classical Branching Processes and Stochastic
Context-Free Grammars. Under the reachability objective, this model is equivalent
to the classical Branching Processes. We study qualitative multi-objective reachability
questions for Ordered Branching Markov Decision Processes (OBMDPs), or equivalently
context-free MDPs with simultaneous derivation. We provide algorithmic results
for efficiently checking certain Boolean combinations of qualitative reachability
and non-reachability queries with respect to different given target non-terminals.
Among the more interesting multi-objective reachability results, we provide two
separate algorithms for almost-sure and limit-sure multi-target reachability for OBMDPs.
Specifically, given an OBMDP, given a starting non-terminal, and given a set
of target non-terminals, our first algorithm decides whether the supremum probability,
of generating a tree that contains every target non-terminal in the set, is 1. Our second
algorithm decides whether there is a strategy for the player to almost-surely (with
probability 1) generate a tree that contains every target non-terminal in the set. The
two separate algorithms are needed: we show that indeed, in this context, almost-sure
and limit-sure multi-target reachability do not coincide. Both algorithms run in time
polynomial in the size of the OBMDP and exponential in the number of targets. Hence,
they run in polynomial time when the number of targets is fixed. The algorithms are
fixed-parameter tractable with respect to this number. Moreover, we show that the
qualitative almost-sure (and limit-sure) multi-target reachability decision problem is in
general NP-hard, when the size of the set of target non-terminals is not fixed
Recursive Concurrent Stochastic Games
We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent
analysis of recursive simple stochastic games to a concurrent setting where the
two players choose moves simultaneously and independently at each state. For
multi-exit games, our earlier work already showed undecidability for basic
questions like termination, thus we focus on the important case of single-exit
RCSGs (1-RCSGs).
We first characterize the value of a 1-RCSG termination game as the least
fixed point solution of a system of nonlinear minimax functional equations, and
use it to show PSPACE decidability for the quantitative termination problem. We
then give a strategy improvement technique, which we use to show that player 1
(maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM)
strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM
strategies. Thus, such games are r-SM-determined. These results mirror and
generalize in a strong sense the randomized memoryless determinacy results for
finite stochastic games, and extend the classic Hoffman-Karp strategy
improvement approach from the finite to an infinite state setting. The proofs
in our infinite-state setting are very different however, relying on subtle
analytic properties of certain power series that arise from studying 1-RCSGs.
We show that our upper bounds, even for qualitative (probability 1)
termination, can not be improved, even to NP, without a major breakthrough, by
giving two reductions: first a P-time reduction from the long-standing
square-root sum problem to the quantitative termination decision problem for
finite concurrent stochastic games, and then a P-time reduction from the latter
problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure
Optimal Strategies in Infinite-state Stochastic Reachability Games
We consider perfect-information reachability stochastic games for 2 players
on infinite graphs. We identify a subclass of such games, and prove two
interesting properties of it: first, Player Max always has optimal strategies
in games from this subclass, and second, these games are strongly determined.
The subclass is defined by the property that the set of all values can only
have one accumulation point -- 0. Our results nicely mirror recent results for
finitely-branching games, where, on the contrary, Player Min always has optimal
strategies. However, our proof methods are substantially different, because the
roles of the players are not symmetric. We also do not restrict the branching
of the games. Finally, we apply our results in the context of recently studied
One-Counter stochastic games
Strategy Complexity of Reachability in Countable Stochastic 2-Player Games
We study countably infinite stochastic 2-player games with reachability
objectives. Our results provide a complete picture of the memory requirements
of -optimal (resp. optimal) strategies. These results depend on
the size of the players' action sets and on whether one requires strategies
that are uniform (i.e., independent of the start state).
Our main result is that -optimal (resp. optimal) Maximizer
strategies require infinite memory if Minimizer is allowed infinite action
sets. This lower bound holds even under very strong restrictions. Even in the
special case of infinitely branching turn-based reachability games, even if all
states allow an almost surely winning Maximizer strategy, strategies with a
step counter plus finite private memory are still useless.
Regarding uniformity, we show that for Maximizer there need not exist
positional (i.e., memoryless) uniformly -optimal strategies even
in the special case of finite action sets or in finitely branching turn-based
games. On the other hand, in games with finite action sets, there always exists
a uniformly -optimal Maximizer strategy that uses just one bit of
public memory
Equilibria-based Probabilistic Model Checking for Concurrent Stochastic Games
Probabilistic model checking for stochastic games enables formal verification
of systems that comprise competing or collaborating entities operating in a
stochastic environment. Despite good progress in the area, existing approaches
focus on zero-sum goals and cannot reason about scenarios where entities are
endowed with different objectives. In this paper, we propose probabilistic
model checking techniques for concurrent stochastic games based on Nash
equilibria. We extend the temporal logic rPATL (probabilistic alternating-time
temporal logic with rewards) to allow reasoning about players with distinct
quantitative goals, which capture either the probability of an event occurring
or a reward measure. We present algorithms to synthesise strategies that are
subgame perfect social welfare optimal Nash equilibria, i.e., where there is no
incentive for any players to unilaterally change their strategy in any state of
the game, whilst the combined probabilities or rewards are maximised. We
implement our techniques in the PRISM-games tool and apply them to several case
studies, including network protocols and robot navigation, showing the benefits
compared to existing approaches
Memoryless Strategies in Stochastic Reachability Games
We study concurrent stochastic reachability games played on finite graphs.
Two players, Max and Min, seek respectively to maximize and minimize the
probability of reaching a set of target states. We prove that Max has a
memoryless strategy that is optimal from all states that have an optimal
strategy. Our construction provides an alternative proof of this result by
Bordais, Bouyer and Le Roux, and strengthens it, as we allow Max's action sets
to be countably infinite
Memoryless strategies in stochastic reachability games
We study concurrent stochastic reachability games played on finite graphs. Two players, Max and Min, seek respectively to maximize and minimize the probability of reaching a set of target states. We prove that Max has a memoryless strategy that is optimal from all states that have an optimal strategy. Our construction provides an alternative proof of this result by Bordais, Bouyer and Le Roux [4], and strengthens it, as we allow Maxâs action sets to be countably infinite.<br/
Qualitative Reachability in Stochastic BPA Games
We consider a class of infinite-state stochastic games generated by stateless
pushdown automata (or, equivalently, 1-exit recursive state machines), where
the winning objective is specified by a regular set of target configurations
and a qualitative probability constraint `>0' or `=1'. The goal of one player
is to maximize the probability of reaching the target set so that the
constraint is satisfied, while the other player aims at the opposite. We show
that the winner in such games can be determined in PTIME for the `>0'
constraint, and both in NP and coNP for the `=1' constraint. Further, we prove
that the winning regions for both players are regular, and we design algorithms
which compute the associated finite-state automata. Finally, we show that
winning strategies can be synthesized effectively.Comment: Submitted to Information and Computation. 48 pages, 3 figure
Liveness of Randomised Parameterised Systems under Arbitrary Schedulers (Technical Report)
We consider the problem of verifying liveness for systems with a finite, but
unbounded, number of processes, commonly known as parameterised systems.
Typical examples of such systems include distributed protocols (e.g. for the
dining philosopher problem). Unlike the case of verifying safety, proving
liveness is still considered extremely challenging, especially in the presence
of randomness in the system. In this paper we consider liveness under arbitrary
(including unfair) schedulers, which is often considered a desirable property
in the literature of self-stabilising systems. We introduce an automatic method
of proving liveness for randomised parameterised systems under arbitrary
schedulers. Viewing liveness as a two-player reachability game (between
Scheduler and Process), our method is a CEGAR approach that synthesises a
progress relation for Process that can be symbolically represented as a
finite-state automaton. The method is incremental and exploits both
Angluin-style L*-learning and SAT-solvers. Our experiments show that our
algorithm is able to prove liveness automatically for well-known randomised
distributed protocols, including Lehmann-Rabin Randomised Dining Philosopher
Protocol and randomised self-stabilising protocols (such as the Israeli-Jalfon
Protocol). To the best of our knowledge, this is the first fully-automatic
method that can prove liveness for randomised protocols.Comment: Full version of CAV'16 pape
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