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

Christine Gaßner

TL;DR

This work develops a unified framework for abstract computation over arbitrary first-order structures by generalizing BSS RAMs to ${\cal A}$-machines with programs ${\sf P}_{\sigma}$ and a reduct ${\cal B}$ of ${\cal A}$. It defines finite and infinite memory models, an extensive instruction set, and precise semantics via input/output procedures and the result function ${\rm Res}_{\cal M}$, enabling notions of semi-decidability and decidability ${\rm SDEC}_{\cal A}$ and ${\rm DEC}_{\cal A}$. The paper further expands to non-deterministic and oracle variants, introduces Moschovakis style operators, and provides concrete examples for computing basic functions and evaluating formulas within this abstract computation framework. By isolating algorithmic reasoning from concrete execution environments, the work lays groundwork for comparing complexity and decidability across diverse structures and sets the stage for hierarchies and completeness results via non-deterministic and oracle machines in later parts.

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.

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

TL;DR

This work develops a unified framework for abstract computation over arbitrary first-order structures by generalizing BSS RAMs to -machines with programs and a reduct of . It defines finite and infinite memory models, an extensive instruction set, and precise semantics via input/output procedures and the result function , enabling notions of semi-decidability and decidability and . The paper further expands to non-deterministic and oracle variants, introduces Moschovakis style operators, and provides concrete examples for computing basic functions and evaluating formulas within this abstract computation framework. By isolating algorithmic reasoning from concrete execution environments, the work lays groundwork for comparing complexity and decidability across diverse structures and sets the stage for hierarchies and completeness results via non-deterministic and oracle machines in later parts.

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.

Paper Structure

This paper contains 14 sections, 4 theorems, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. BSS-RAMs over first-order structures
  3. First-order structures and their signatures
  4. The $\sigma$-programs
  5. Finite machines and more
  6. Infinite machines and their configurations
  7. Computation and semi-decidability by BSS RAMs
  8. Non-deterministic BSS RAMs
  9. Oracle machines
  10. Other non-deterministic BSS RAMs
  11. Examples
  12. Input and output functions
  13. Computing functions and deciding sets
  14. Evaluating formulas

Key Result

Theorem 2.6

Let ${\cal A}$ be a first-order structure of any signature. If $Q\in {\rm SDEC}_{\cal A}$, then ${\rm SDEC}_{\cal A}\subseteq({\rm SDEC}_{\cal A}^\nu)^Q\subseteq {\rm SDEC}_{\cal A}^{\rm ND}$.

Figures (3)

  • Figure 1: Examples for pairs $(con_s,con_{s+1})$ of configurations in $\to_{\cal M}$
  • Figure 2: A further pair of configurations $(con_s,con_{s+1})$ in $\to_{\cal M}$
  • Figure 3: Two pairs of configurations $(con_s,con_{s+1})$ in $\to_{\cal M}$

Theorems & Definitions (15)

  • Definition 2.3: Computable function
  • Definition 2.4: Non-deterministically computable function
  • Definition 2.5: $\nu$-computable functions
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9: Computability by non-deterministic branching
  • Theorem 2.10
  • Example 3.1: Input and output functions for machines over ${\cal A}_0$
  • Example 3.2: Computing the sum of three numbers
  • ...and 5 more