Workshop in Computability Theory Paris - July 2010

A pair of sets of natural numbers A and B forms a K-pair if there exists a c.e. set W , such that A × B ⊆ W and A × B ⊆ W . Kpairs are introduced by Kalimullin and used by him to prove the first order definability of the enumeration jump operator in the global structure of the enumeration degrees. He shows that the property of being a K-pair is degree theoretic and first order definable in the global structure. In this talk we examine the properties of K-pairs in the local structure of the enumeration degrees. We show that the class of K-pairs of Σ2 enumeration degrees is also locally definable. This is the first definability results in the local structure of the enumeration degrees. It unlocks the definability of further classes of Σ2 degrees and allows us to characterize the complexity of the first order theory of the local structure. This is joint work with Hristo Ganchev. Ekaterina Fokina. Title: Isomorphism and Bi-Embeddability among Σ1 Equivalence Relations Abstract: We discuss possible approaches to measure and compare the complexity of equivalence relations on classes of computable structures. In particular, we will consider such natural equivalence relations as isomorphism and bi-embeddability. If a class of structures is closed under isomorphism and the set of its computable members is definable by a computable infinitary sentence, then the relations of isomorphism or bi-embeddability are Σ1. We consider several reducibilities (in the context of sets and equivalence relations) that allows us to compare the complexity of such relations. In particular, we show that the isomorphism may be complete among Σ1 equivalence relations on ω. For the bi-embeddability we can show more. In fact, for every Σ1 equivalence relation E on ω there is a computable infinitary sentence φ such that the bi-embeddability relation on computable models of φ is equivalent (with respect to the chosen reducibility) to E. The main results are joint work with S. Friedman, V. Harizanov, J. Knight, C. McCoy and A. Montalbán. We also discuss some general facts about the structure of degrees of Σ1 equivalence relations on ω under the reducibilities we have studied. This part is based on joint results with S. Friedman. Barbara Csima. Title: Introducing the Bounded Jump Abstract: In recent work with Rod Downey and Selwyn Ng, we show that, for the usual Turing Jump, analogs of Sacks’ and Shoenfield’s jump inversion fail for both the truth table and bounded Turing (more commonly known as weak truth table) reducibilities. This gives further evidence that since the Turing jump is defined with respect to Turing reducibility, perhaps it is not always the “right” jump to consider on stronger reducibilities. Together with Bernie Anderson, we have defined what we call the bounded jump, a jump operator on the bounded Turing degrees. We have found that the bounded jump has many desirable properties, which I will present.