Categoricity in Quasiminimal Pregeometry Classes

Quasiminimal pregeometry classes were introduces by Zilber [2005a] to isolate the model theoretical core of several interesting examples. He proves that a quasiminimal pregeometry class satisfying an additional axiom, called excellence, is categorical in all uncountable cardinalities. Recently Bays et al. [2014] showed that excellence follows from the rest of axioms. In this paper we present a direct proof of the categoricity result without using excellence

Decidability of the Clark's Completion Semantics for Monadic Programs and Queries

There are many different semantics for general logic programs (i.e. programs that use negation in the bodies of clauses). Most of these semantics are Turing complete (in a sense that can be made precise), implying that they are undecidable. To obtain decidability one needs to put additional restrictions on programs and queries. In logic programming it is natural to put restrictions on the underlying first-order language. In this note we show the decidability of the Clark's completion semantics for monadic general programs and queries. To appear in Theory and Practice of Logic Programming (TPLP

Existentially closed exponential fields

We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. We extend some notions from classification theory to positive logic and position the category of existentially closed exponential fields in the stability hierarchy as NSOP1 but TP2

Aspects of nonelementary stability theory

This thesis aims to contribute to neostability and in particular to the stability theory of nonelementary classes. The central themes of the thesis are quasiminimality and excellence. We explore both classical and geometric aspects of stability theory for such classes. In particular our methods aim to utilise the topology on the space of types in these settings. Chapter 1 is introductory and sets the necessary background on nonelementary classes. In Chapter 2 we use infinitary methods to study quasiminimal excellent classes. We give a simplified proof of the Categoricity Theorem. In Chapter 3 we study quasiminimality from the first order perspective. We look for conditions on a first order theory that allow us to build quasiminimal models with various additional properties. In Chapter 4 we look at excellent groups. We aim to generalise various results from (geometric) stability theory to this nonelementary setting.</p