57,197 research outputs found
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
Dynamics in atomic signaling games
We study an atomic signaling game under stochastic evolutionary dynamics.
There is a finite number of players who repeatedly update from a finite number
of available languages/signaling strategies. Players imitate the most fit
agents with high probability or mutate with low probability. We analyze the
long-run distribution of states and show that, for sufficiently small mutation
probability, its support is limited to efficient communication systems. We find
that this behavior is insensitive to the particular choice of evolutionary
dynamic, a property that is due to the game having a potential structure with a
potential function corresponding to average fitness. Consequently, the model
supports conclusions similar to those found in the literature on language
competition. That is, we show that efficient languages eventually predominate
the society while reproducing the empirical phenomenon of linguistic drift. The
emergence of efficiency in the atomic case can be contrasted with results for
non-atomic signaling games that establish the non-negligible possibility of
convergence, under replicator dynamics, to states of unbounded efficiency loss
Innovation and Nested Preferential Growth in Chess Playing Behavior
Complexity develops via the incorporation of innovative properties. Chess is
one of the most complex strategy games, where expert contenders exercise
decision making by imitating old games or introducing innovations. In this
work, we study innovation in chess by analyzing how different move sequences
are played at the population level. It is found that the probability of
exploring a new or innovative move decreases as a power law with the frequency
of the preceding move sequence. Chess players also exploit already known move
sequences according to their frequencies, following a preferential growth
mechanism. Furthermore, innovation in chess exhibits Heaps' law suggesting
similarities with the process of vocabulary growth. We propose a robust
generative mechanism based on nested Yule-Simon preferential growth processes
that reproduces the empirical observations. These results, supporting the
self-similar nature of innovations in chess are important in the context of
decision making in a competitive scenario, and extend the scope of relevant
findings recently discovered regarding the emergence of Zipf's law in chess.Comment: 8 pages, 4 figures, accepted for publication in Europhysics Letters
(EPL
Deciding the Borel complexity of regular tree languages
We show that it is decidable whether a given a regular tree language belongs
to the class of the Borel hierarchy, or equivalently whether
the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure
An Upper Bound on the Complexity of Recognizable Tree Languages
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class
for some natural number , where is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
into the class , and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
Fair Simulation for Nondeterministic and Probabilistic Buechi Automata: a Coalgebraic Perspective
Notions of simulation, among other uses, provide a computationally tractable
and sound (but not necessarily complete) proof method for language inclusion.
They have been comprehensively studied by Lynch and Vaandrager for
nondeterministic and timed systems; for B\"{u}chi automata the notion of fair
simulation has been introduced by Henzinger, Kupferman and Rajamani. We
contribute to a generalization of fair simulation in two different directions:
one for nondeterministic tree automata previously studied by Bomhard; and the
other for probabilistic word automata with finite state spaces, both under the
B\"{u}chi acceptance condition. The former nondeterministic definition is
formulated in terms of systems of fixed-point equations, hence is readily
translated to parity games and is then amenable to Jurdzi\'{n}ski's algorithm;
the latter probabilistic definition bears a strong ranking-function flavor.
These two different-looking definitions are derived from one source, namely our
coalgebraic modeling of B\"{u}chi automata. Based on these coalgebraic
observations, we also prove their soundness: a simulation indeed witnesses
language inclusion
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