Related papers: The universality of polynomial time Turing equival…
We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of…
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…
We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to…
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the…
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in…
There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem…
Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We…
We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel…
A set of valuable universal similarity factorization equalities are established over complex Clifford algebras $\Cn.$ Through them matrix representations of complex Clifford algebras $\Cn$ can directly be derived, and their properties can…
We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting.…
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and…
Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.
We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second…
We consider countable Borel equivalence relations on quotient Borel spaces. We prove a generalization of the Feldman-Moore representation theorem, but provide some examples showing that other very simple properties of countable equivalence…
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of…
Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…
It is shown that the isomorphism relation between continuous t-norms is Borel bireducible with the relation of order isomorphism between linear orders on the set of natural numbers, and therefore, it is a Borel complete equivalence…
We introduce infinite time computable model theory, the computable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time…
We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel…
We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite then the complexity of hyperfinite countable Borel equivalence relations is as high as possible, namely,…