9,025 research outputs found
Infinite time decidable equivalence relation theory
We introduce an analog of the theory of Borel equivalence relations in which
we study equivalence relations that are decidable by an infinite time Turing
machine. The Borel reductions are replaced by the more general class of
infinite time computable functions. Many basic aspects of the classical theory
remain intact, with the added bonus that it becomes sensible to study some
special equivalence relations whose complexity is beyond Borel or even
analytic. We also introduce an infinite time generalization of the countable
Borel equivalence relations, a key subclass of the Borel equivalence relations,
and again show that several key properties carry over to the larger class.
Lastly, we collect together several results from the literature regarding Borel
reducibility which apply also to absolutely Delta_1^2 reductions, and hence to
the infinite time computable reductions.Comment: 30 pages, 3 figure
Ramsey-type graph coloring and diagonal non-computability
A function is diagonally non-computable (d.n.c.) if it diagonalizes against
the universal partial computable function. D.n.c. functions play a central role
in algorithmic randomness and reverse mathematics. Flood and Towsner asked for
which functions h, the principle stating the existence of an h-bounded d.n.c.
function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper,
we prove that for every computable order h, there exists an~-model of
DNR_h which is not a not model of the Ramsey-type graph coloring principle for
two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines
bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to
transform a computable non-reducibility into a separation over omega-models.Comment: 18 page
Iterative forcing and hyperimmunity in reverse mathematics
The separation between two theorems in reverse mathematics is usually done by
constructing a Turing ideal satisfying a theorem P and avoiding the solutions
to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a
forcing technique for iterating a computable non-reducibility in order to
separate theorems over omega-models. In this paper, we present a modularized
version of their framework in terms of preservation of hyperimmunity and show
that it is powerful enough to obtain the same separations results as Wang did
with his notion of preservation of definitions.Comment: 15 page
- …