169 research outputs found
The Determinacy of Context-Free Games
We prove that the determinacy of Gale-Stewart games whose winning sets are
accepted by real-time 1-counter B\"uchi automata is equivalent to the
determinacy of (effective) analytic Gale-Stewart games which is known to be a
large cardinal assumption. We show also that the determinacy of Wadge games
between two players in charge of omega-languages accepted by 1-counter B\"uchi
automata is equivalent to the (effective) analytic Wadge determinacy. Using
some results of set theory we prove that one can effectively construct a
1-counter B\"uchi automaton A and a B\"uchi automaton B such that: (1) There
exists a model of ZFC in which Player 2 has a winning strategy in the Wadge
game W(L(A), L(B)); (2) There exists a model of ZFC in which the Wadge game
W(L(A), L(B)) is not determined. Moreover these are the only two possibilities,
i.e. there are no models of ZFC in which Player 1 has a winning strategy in the
Wadge game W(L(A), L(B)).Comment: To appear in the Proceedings of the 29 th International Symposium on
Theoretical Aspects of Computer Science, STACS 201
Wadge Degrees of -Languages of Petri Nets
We prove that -languages of (non-deterministic) Petri nets and
-languages of (non-deterministic) Turing machines have the same
topological complexity: the Borel and Wadge hierarchies of the class of
-languages of (non-deterministic) Petri nets are equal to the Borel and
Wadge hierarchies of the class of -languages of (non-deterministic)
Turing machines which also form the class of effective analytic sets. In
particular, for each non-null recursive ordinal there exist some -complete and some -complete -languages of Petri nets, and the supremum of
the set of Borel ranks of -languages of Petri nets is the ordinal
, which is strictly greater than the first non-recursive ordinal
. We also prove that there are some -complete, hence non-Borel, -languages of Petri nets, and
that it is consistent with ZFC that there exist some -languages of
Petri nets which are neither Borel nor -complete. This
answers the question of the topological complexity of -languages of
(non-deterministic) Petri nets which was left open in [DFR14,FS14].Comment: arXiv admin note: text overlap with arXiv:0712.1359, arXiv:0804.326
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
Baire reductions and good Borel reducibilities
In reference [8] we have considered a wide class of "well-behaved"
reducibilities for sets of reals. In this paper we continue with the study of
Borel reducibilities by proving a dichotomy theorem for the degree-structures
induced by good Borel reducibilities. This extends and improves the results of
[8] allowing to deal with a larger class of notions of reduction (including,
among others, the Baire class functions).Comment: 21 page
Lipschitz and uniformly continuous reducibilities on ultrametric Polish spaces
We analyze the reducibilities induced by, respectively, uniformly continuous,
Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces,
and determine whether under suitable set-theoretical assumptions the induced
degree-structures are well-behaved.Comment: 37 pages, 2 figures, revised version, accepted for publication in the
Festschrift that will be published on the occasion of Victor Selivanov's 60th
birthday by Ontos-Verlag. A mistake has been corrected in Section
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