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

TL;DR

This paper introduces a formal model of abstract computation over first-order structures, defining algorithms via programs and exploring their properties in deterministic and non-deterministic BSS RAM models.

Contribution

It provides a rigorous mathematical framework for abstract algorithms over first-order structures, including definitions of BSS RAMs and their properties, independent of physical implementation.

Findings

Defined BSS RAM models step by step.

Connected abstract computation to first-order logic and structures.

Laid groundwork for hierarchy analysis using infinitary logic.

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.