370 research outputs found
How unprovable is Rabin's decidability theorem?
We study the strength of set-theoretic axioms needed to prove Rabin's theorem
on the decidability of the MSO theory of the infinite binary tree. We first
show that the complementation theorem for tree automata, which forms the
technical core of typical proofs of Rabin's theorem, is equivalent over the
moderately strong second-order arithmetic theory to a
determinacy principle implied by the positional determinacy of all parity games
and implying the determinacy of all Gale-Stewart games given by boolean
combinations of sets. It follows that complementation for
tree automata is provable from - but not -comprehension.
We then use results due to MedSalem-Tanaka, M\"ollerfeld and
Heinatsch-M\"ollerfeld to prove that over -comprehension, the
complementation theorem for tree automata, decidability of the MSO theory of
the infinite binary tree, positional determinacy of parity games and
determinacy of Gale-Stewart games are all
equivalent. Moreover, these statements are equivalent to the
-reflection principle for -comprehension. It follows in
particular that Rabin's decidability theorem is not provable in
-comprehension.Comment: 21 page
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
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
Permutation Games for the Weakly Aconjunctive -Calculus
We introduce a natural notion of limit-deterministic parity automata and
present a method that uses such automata to construct satisfiability games for
the weakly aconjunctive fragment of the -calculus. To this end we devise a
method that determinizes limit-deterministic parity automata of size with
priorities through limit-deterministic B\"uchi automata to deterministic
parity automata of size and with
priorities. The construction relies on limit-determinism to avoid the full
complexity of the Safra/Piterman-construction by using partial permutations of
states in place of Safra-Trees. By showing that limit-deterministic parity
automata can be used to recognize unsuccessful branches in pre-tableaux for the
weakly aconjunctive -calculus, we obtain satisfiability games of size
with priorities for weakly aconjunctive
input formulas of size and alternation-depth . A prototypical
implementation that employs a tableau-based global caching algorithm to solve
these games on-the-fly shows promising initial results
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