Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

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 application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

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

On effective sigma-boundedness and sigma-compactness

We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a countable union of compact lightface \Delta^1_1 sets), or contains a superperfect subset (and then the set is not sigma-bounded, of course). We add different generalizations of this result, in particular, 1) such that the boundedness property involved includes covering by compact sets and equivalence classes of a given finite collection of lightface \Delta^1_1 equivalence relations, 2) generalizations to lightface \Sigma^1_2 sets, 3) generalizations true in the Solovay model. As for effective sigma-compactness, we start with a theorem by Louveau, saying that any lightface \Delta^1_1 set of the Baire space either is effectively sigma-compact (that is, is equal to a countable union of compact lightface \Delta^1_1 sets), or it contains a relatively closed superperfect subset. Then we prove a generalization of this result to lightface \Sigma^1_1 sets.Comment: arXiv admin note: substantial text overlap with arXiv:1103.106

On the complexity of the relations of isomorphism and bi-embeddability

Given an L_{\omega_1 \omega}-elementary class C, that is the collection of the countable models of some L_{\omega_1 \omega}-sentence, denote by \cong_C and \equiv_C the analytic equivalence relations of, respectively, isomorphism and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal [LR05], in [FMR09] it was proposed the problem of determining which pairs of analytic equivalence relations (E,F) can be realized (up to Borel bireducibility) as pairs of the form (\cong_C,\equiv_C), C some L_{\omega_1 \omega}-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such problem: under very mild conditions on E and F, it is always possible to find such an L_{\omega_1 \omega}-elementary class C.Comment: 15 page

The universality of polynomial time Turing equivalence

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees

Classes of structures with no intermediate isomorphism problems

We say that a theory $T$ is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence $T$ is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle on a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle on a cone