577 research outputs found
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
Borel Ranks and Wadge Degrees of Context Free Omega Languages
We show that, from a topological point of view, considering the Borel and the
Wadge hierarchies, 1-counter B\"uchi automata have the same accepting power
than Turing machines equipped with a B\"uchi acceptance condition. In
particular, for every non null recursive ordinal alpha, there exist some
Sigma^0_alpha-complete and some Pi^0_alpha-complete omega context free
languages accepted by 1-counter B\"uchi automata, and the supremum of the set
of Borel ranks of context free omega languages is the ordinal gamma^1_2 which
is strictly greater than the first non recursive ordinal. This very surprising
result gives answers to questions of H. Lescow and W. Thomas [Logical
Specifications of Infinite Computations, In:"A Decade of Concurrency", LNCS
803, Springer, 1994, p. 583-621]
Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra
Let \g be any simple Lie algebra over . Recall that there
exists an embedding of into \g, called a principal TDS,
passing through a principal nilpotent element of \g and uniquely determined
up to conjugation. Moreover, \wedge (\g^*)^\g is freely generated (in the
super-graded sense) by primitive elements , where
is the rank of \g. N. Hitchin conjectured that for any primitive
element \omega \in \wedge^d (\g^*)^\g, there exists an irreducible
-submodule V_\omega \subset \g of dimension such that
is non-zero on the line . We prove that the
validity of this conjecture for simple simply-laced Lie algebras implies its
validity for any simple Lie algebra.
Let G be a connected, simply-connected, simple, simply-laced algebraic group
and let be a diagram automorphism of G with fixed subgroup K. Then, we
show that the restriction map R(G) \to R(K) is surjective, where R denotes the
representation ring over . As a corollary, we show that the
restriction map in the singular cohomology H^*(G)\to H^*(K) is surjective. Our
proof of the reduction of Hitchin's conjecture to the simply-laced case relies
on this cohomological surjectivity.Comment: 14 page
Disjoint Infinity-Borel Functions
This is a followup to a paper by the author where the disjointness relation
for definable functions from to is
analyzed. In that paper, for each we defined a Baire
class one function which encoded
in a certain sense. Given , let
be the statement that is disjoint from at most countably many of
the functions . We show the consistency strength of is that of an inaccessible cardinal. We show that
implies . Finally, we show that assuming large
cardinals, holds in models of the form
where is a selective ultrafilter on
.Comment: 16 page
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