European Summer Meeting of the Association for Symbolic Logic: Manchester, England, 1984

s of contributed talks given (in person or by title) by members of the Association for Symbolic Logic follow. For the Program Committee Wilfrid Hodges, Program Chair Abstracts of contributed paperss of contributed papers MARAT M. ARSLANOV, An interplay between d-r.e. and r.e. sets. Department of Mathematics, Kazan State University, Kazan, Russia, Tatarstan 420008. E-mail: [email protected]. A set A ⊆ ù is called d-r.e. if there are r.e. sets A1 and A2 such that A = A1\A2. A Turing degree is called a d-r.e. degree if it contains a d-r.e. set; it is called properly d-r.e. if it is d-r.e. but not r.e. Recently it was shown (S. B. Cooper and others) that the partial order of the d-r.e. degrees is not dense. Earlier the author proved the following statements. Theorem 1. For each r.e. degree a < 0 there exists a properly d-r.e. degree b such that a < b < 0. Theorem 2. For any high r.e. degree a 6= 0, 0 there exists d-r.e. degrees b and c such that b < a < c and a is the only r.e. degree between b and c. Studying related questions, the author proved the following two theorems. Theorem 3. There is a low properly d-r.e. degree h such that for any r.e. degree a, if a < h then a < b < h for some r.e. b. Theorem 4. There is a properly d-r.e. degree d such that for any r.e. degree a, if d < a then d < b < a for some r.e. degree b. JOHN T. BALDWIN ANDMASANORI ITAI, Properties of K -generic models. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60680, USA. E-mail: [email protected], [email protected]. Let (K,≤) be a pair whereK is a class of finite relational structures and≤, strong submodel, is a relation on the members of K such that A ≤ B implies A ⊆ B . If (K,≤) satisfies the amalgamation property and the joint embedding property, there exists a structure M called (K,≤)-generic with the following properties: i) every finite member of K can be strongly embedded inM , ii) every finite substructure of M is in K , iii) if A and B are finite strong submodels ofM that are isomorphic, say by f, then there is an automorphism of M extending f. Hrushovski constructed, as a (K,≤)-generic model, a stable א0-categorical pseudoplane refuting a conjecture of Lachlan. He also defined, as yet another (K,≤)-generic model, a strongly minimal set of a new type giving a counter example to a Zil’ber’s conjecture. By generalizing the idea, Baldwin was able to define a family of almost strongly minimal, non-Desarguesian projective planes. We study some model theoretic properties of various (K,≤)-generic models, e.g., determining their Morley ranks, I (T,א0) where T is the theory of a (K,≤)-generic model. EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC 87