Abstract computation over first-order structures. Part I: Deterministic and non-deterministic BSS RAMs

Abstract: Most ideas about what an algorithm is are very similar. Basic operations are used for transforming objects. The evaluation of internal and external states by relations has impact on the further process. A more precise definition can lead to a model of abstract computation over an arbitrary first-order structure. Formally, the algorithms can be determined by strings. Their meaning can be described purely mathematically by functions and relations derived from the operations and relations of a first-order structure. Our model includes models of computability and derivation systems from different areas of mathematics, logic, and computer science. To define the algorithms, we use so-called programs. Since we do this independently of their executability by computers, the so-called execution of our programs can be viewed as a form of abstract computation. This concept helps to highlight common features of algorithms that are independent of the underlying structures. Here, in Part I, we define BSS RAMs step by step. In Part II, we study Moschovakis' operator which is known from a general recursion theory over first-order structures. Later, we study hierarchies defined analogously to the arithmetical hierarchy by means of quantified formulas of an infinitary logic in this framework.