1,189 research outputs found
REGISTER GAMES
The complexity of parity games is a long standing open problem that saw a
major breakthrough in 2017 when two quasi-polynomial algorithms were published.
This article presents a third, independent approach to solving parity games in
quasi-polynomial time, based on the notion of register game, a parameterised
variant of a parity game. The analysis of register games leads to a
quasi-polynomial algorithm for parity games, a polynomial algorithm for
restricted classes of parity games and a novel measure of complexity, the
register index, which aims to capture the combined complexity of the priority
assignement and the underlying game graph.
We further present a translation of alternating parity word automata into
alternating weak automata with only a quasi-polynomial increase in size, based
on register games; this improves on the previous exponential translation.
We also use register games to investigate the parity index hierarchy: while
for words the index hierarchy of alternating parity automata collapses to the
weak level, and for trees it is strict, for structures between trees and words,
it collapses logarithmically, in the sense that any parity tree automaton of
size n is equivalent, on these particular classes of structures, to an
automaton with a number of priorities logarithmic in n
On the Borel Inseparability of Game Tree Languages
The game tree languages can be viewed as an automata-theoretic counterpart of
parity games on graphs. They witness the strictness of the index hierarchy of
alternating tree automata, as well as the fixed-point hierarchy over binary
trees. We consider a game tree language of the first non-trivial level, where
Eve can force that 0 repeats from some moment on, and its dual, where Adam can
force that 1 repeats from some moment on. Both these sets (which amount to one
up to an obvious renaming) are complete in the class of co-analytic sets. We
show that they cannot be separated by any Borel set, hence {\em a fortiori} by
any weakly definable set of trees. This settles a case left open by
L.Santocanale and A.Arnold, who have thoroughly investigated the separation
property within the -calculus and the automata index hierarchies. They
showed that separability fails in general for non-deterministic automata of
type , starting from level , while our result settles
the missing case
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
On the Problem of Computing the Probability of Regular Sets of Trees
We consider the problem of computing the probability of regular languages of
infinite trees with respect to the natural coin-flipping measure. We propose an
algorithm which computes the probability of languages recognizable by
\emph{game automata}. In particular this algorithm is applicable to all
deterministic automata. We then use the algorithm to prove through examples
three properties of measure: (1) there exist regular sets having irrational
probability, (2) there exist comeager regular sets having probability and
(3) the probability of \emph{game languages} , from automata theory,
is if is odd and is otherwise
Formats of Winning Strategies for Six Types of Pushdown Games
The solution of parity games over pushdown graphs (Walukiewicz '96) was the
first step towards an effective theory of infinite-state games. It was shown
that winning strategies for pushdown games can be implemented again as pushdown
automata. We continue this study and investigate the connection between game
presentations and winning strategies in altogether six cases of game arenas,
among them realtime pushdown systems, visibly pushdown systems, and counter
systems. In four cases we show by a uniform proof method that we obtain
strategies implementable by the same type of pushdown machine as given in the
game arena. We prove that for the two remaining cases this correspondence
fails. In the conclusion we address the question of an abstract criterion that
explains the results
Emptiness Of Alternating Tree Automata Using Games With Imperfect Information
We consider the emptiness problem for alternating tree automata,
with two acceptance semantics: classical (all branches are accepted)
and qualitative (almost all branches are accepted). For the classical semantics, the usual technique to tackle this problem relies on a Simulation Theorem which constructs an equivalent non-deterministic automaton from the original alternating one, and then checks emptiness by a reduction to a two-player perfect information game.
However, for the qualitative semantics, no simulation of alternation by means of non-determinism is known.
We give an alternative technique to decide the emptiness problem of alternating tree automata, that does not rely on a Simulation Theorem.
Indeed, we directly reduce the emptiness problem to solving an imperfect information two-player parity game. Our new approach can successfully be applied to both semantics, and yields decidability results with optimal complexity; for the qualitative semantics, the key ingredient in the proof is a positionality result for stochastic games played over infinite graphs
Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games
The McNaughton-Zielonka divide et impera algorithm is the simplest and most
flexible approach available in the literature for determining the winner in a
parity game. Despite its theoretical worst-case complexity and the negative
reputation as a poorly effective algorithm in practice, it has been shown to
rank among the best techniques for the solution of such games. Also, it proved
to be resistant to a lower bound attack, even more than the strategy
improvements approaches, and only recently a family of games on which the
algorithm requires exponential time has been provided by Friedmann. An easy
analysis of this family shows that a simple memoization technique can help the
algorithm solve the family in polynomial time. The same result can also be
achieved by exploiting an approach based on the dominion-decomposition
techniques proposed in the literature. These observations raise the question
whether a suitable combination of dynamic programming and game-decomposition
techniques can improve on the exponential worst case of the original algorithm.
In this paper we answer this question negatively, by providing a robustly
exponential worst case, showing that no intertwining of the above mentioned
techniques can help mitigating the exponential nature of the divide et impera
approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176
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