Uri Andrews

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

We study computably enumerable equivalence relations (ceers), under the reducibility R ≤ S if there exists a computable function f such that x R y if and only if f(x) S f(y), for every x, y. We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if R′ ≤ R, where R′ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are Σ3-complete (the former answering an open question of Gao and Gerdes).

Spectra of theories and structures

We introduce the notion of a degree spectrum of a complete theory to be the set of Turing degrees that contain a copy of some model of the theory. We generate examples showing that not all degree spectra of theories are degree spectra of structures and vice-versa. To this end, we give a new necessary condition on the degree spectrum of a structure, specifically showing that the set of PA degrees and the upward closure of the set of 1-random degrees are not degree spectra of structures but are degree spectra of theories.

New Spectra of Strongly Minimal Theories in Finite Languages

Abstract We describe strongly minimal theories T n with finite languages such that in the chain of countable models of T n , only the first n models have recursive presentations. Also, we describe a strongly minimal theory with a finite language such that every non-saturated model has a recursive presentation.

Separable models of randomizations

Every complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of T is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T . This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.

A Survey on Universal Computably Enumerable Equivalence Relations

We review the literature on universal computably enumerable equivalence relations, i.e. the computably enumerable equivalence relations (ceers) which are \(\Sigma ^0_1\)-complete with respect to computable reducibility on equivalence relations. Special attention will be given to the so-called uniformly effectively inseparable (u.e.i.) ceers, i.e. the nontrivial ceers yielding partitions of the natural numbers in which each pair of distinct equivalence classes is effectively inseparable (uniformly in their representatives). The u.e.i. ceers comprise infinitely many isomorphism types. The relation of provable equivalence in Peano Arithmetic plays an important role in the study and classification of the u.e.i. ceers.

The complexity of index sets of classes of computably enumerable equivalence relations

Let ďc be computable reducibility on ceers. We show that for every computably enumerable equivalence relation (or ceer) R with infinitely many equivalence classes, the index sets ti : Ri ďc Ru (with R non-universal), ti : Ri ěc Ru, and ti : Ri ”c Ru are Σ3 complete, whereas in case R has only finitely many equivalence classes, we have that ti : Ri ďc Ru is Π2 complete, and ti : Ri ěc Ru (with R having at least two distinct equivalence classes) is Σ2 complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is Π4 complete. Finally, we prove that the 1-reducibility pre-ordering on c.e. sets is a Σ 0 3 complete pre-ordering relation, a fact that is used to show that the pre-ordering relation ďc on ceers is a Σ3 complete pre-ordering relation.

Jumps of computably enumerable equivalence relations

Abstract We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let E ( a ) denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then X ( 2 ) computes A ( ω ) , we show that there is a ceer R such that R ≥ Id ( n ) , for every finite ordinal n, but, for all k, R ( k ) ≱ Id ( ω ) (here Id is the identity equivalence relation). We show that if a , b are notations of the same ordinal less than ω 2 , then E ( a ) ≡ E ( b ) , but there are notations a , b of ω 2 such that Id ( a ) and Id ( b ) are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form Id ( a ) where a is a notation for ω 2 .

THE COMPLEMENTS OF LOWER CONES OF DEGREES AND THE DEGREE SPECTRA OF STRUCTURES

We study Turing degrees a for which there is a countable structure A whose degree spectrum is the collection {x : x 6≤ a}. In particular, for degrees a from the interval [0′,0′′], such a structure exists if a′ = 0′′, and there are no such structures if a′′ > 0′′′.

DECIDABLE MODELS OF ω -STABLE THEORIES

We characterize the ω-stable theories all of whose countable models admit decidable presentations. In particular, we show that for a countable ω-stable T , every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable models. We further characterize the decidable models of ω-stable theories with countably many countable models as those which realize only recursive types. §

The degrees of categorical theories with recursive models

We show that even for categorical theories, recursiveness of the models guarantees no information regarding the complexity of the theory. In particular, we show that every tt-degree reducible to 0(ω) contains both א1categorical and א0-categorical theories in finite languages all of whose countable models have recursive presentations.