Effective Reducibilities on Structures and Degrees of Presentability

In this survey paper we consider presentations of structures in admissible sets and some effective reducibilities between structures and their degrees of presentability. Degrees of Σ-definability of structures, as well as degrees of presentability with respect to different effective reducibilities, are natural measures of complexity which are total, i.e. defined for any structure. We also consider properties of structures invariant under various effective reducibilities, and study how a degree of presentability of a structure depends from a domain for presentations ( i.e. from the choice of an admissible set). In this survey paper we consider presentations of structures in admissible sets and some effective reducibilities between structures and their degrees of presentability. The main object of study are semilattices of degrees of Σ-definability, which can be considered as a theoretical model of object-oriented programming, based on a generalization of oracle computability regarding oracles, as well as the results of computation, as abstract structures. On the other hand, the notion of Σ-definability of a structure in an admissible set is an effectivization of one of central notions of model theory, the notion of interpretability of one structure in another, and, at the same time, a generalization of the notion of constructivizability of a structure on natural numbers. We show that the semilattices of Turing and enumeration degrees of subsets of natural numbers are embeddable in a natural way into the semilattices of degrees of Σ-definability. The notion of a structure having a degree, known in computable model theory, gives only a partial measure of complexity, since there are a lot of structures which do not have a degree. Degrees of Σ-definability, as well as degrees of presentability with respect to different effective reducibilities, are natural measures of complexity which are total, i.e. defined for any structure. We can also consider properties of structures invariant under various effective reducibilities, and study how a degree of presentability of a structure depends from a domain for presentations ( i.e. from the choice of an admissible set). Most of notations and terminology we use here are standard and corresponds to [4, 3, 11]. We denote the domains of a structure M by M , and its signature by ? This work was supported by the INTAS YSF (grant 04-83-3310) and the Russian Foundation for Basic Research (grants 05-01-00481 and 06-01-04002).