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Abstract computation over first-order structures. Part IIa: Moschovakis' operator and other non-determinisms

Christine Gaßner

TL;DR

This paper extends abstract computation over first-order structures by integrating Moschovakis' non-deterministic ν-operator into BSS RAMs, enabling structured non-deterministic guessing and inverses relative to fixed oracle sets. It defines ν formally, provides illustrative examples, and analyzes the expanded computational power, particularly for n-ary and decidable oracles. A central contribution is showing that ν-oracle computations can be simulated by non-deterministic machines with fixed resource bounds, and it develops a detailed simulation framework using 3-tape ND machines and a rich set of subprograms and pseudo-instructions. These results establish a bridge between Moschovakis' operator-based theories and machine-oriented models, setting the stage for deeper comparisons of non-determinisms across first-order structures in Part IIb, and clarifying how non-deterministic search interacts with structural computability to affect semi-decidability and inverses.

Abstract

BSS RAMs over first-order structures help to characterize algorithms for processing objects by means of useful operations and relations. They are the result of a generalization of several types of abstract machines. We want to discuss whether this concept that allows a machine-oriented characterization of algorithms is sufficiently general for describing also other models of computation. Yiannis N. Moschovakis introduced a concept of abstract computability of functions on the basis of recursive definability over first-order structures. Moschovakis' search operator is the counterpart to the operator introduced by Stephen C. Kleene and suitable for structures without computable minima. To compare our concept with Moschovakis' generalization of the theory of recursive functions, we extend the abilities of BSS RAMs by an operator that makes it possible to provide information about computable functions and their inverses in a non-deterministic way. In Part IIb, we compare several non-determinisms, summarize effects resulting from the restriction of guesses to constants, and take into account properties such as the semi-decidability of oracle sets, the semi-decidability of the identity relation, and the recognizability of constants.

Abstract computation over first-order structures. Part IIa: Moschovakis' operator and other non-determinisms

TL;DR

This paper extends abstract computation over first-order structures by integrating Moschovakis' non-deterministic ν-operator into BSS RAMs, enabling structured non-deterministic guessing and inverses relative to fixed oracle sets. It defines ν formally, provides illustrative examples, and analyzes the expanded computational power, particularly for n-ary and decidable oracles. A central contribution is showing that ν-oracle computations can be simulated by non-deterministic machines with fixed resource bounds, and it develops a detailed simulation framework using 3-tape ND machines and a rich set of subprograms and pseudo-instructions. These results establish a bridge between Moschovakis' operator-based theories and machine-oriented models, setting the stage for deeper comparisons of non-determinisms across first-order structures in Part IIb, and clarifying how non-deterministic search interacts with structural computability to affect semi-decidability and inverses.

Abstract

BSS RAMs over first-order structures help to characterize algorithms for processing objects by means of useful operations and relations. They are the result of a generalization of several types of abstract machines. We want to discuss whether this concept that allows a machine-oriented characterization of algorithms is sufficiently general for describing also other models of computation. Yiannis N. Moschovakis introduced a concept of abstract computability of functions on the basis of recursive definability over first-order structures. Moschovakis' search operator is the counterpart to the operator introduced by Stephen C. Kleene and suitable for structures without computable minima. To compare our concept with Moschovakis' generalization of the theory of recursive functions, we extend the abilities of BSS RAMs by an operator that makes it possible to provide information about computable functions and their inverses in a non-deterministic way. In Part IIb, we compare several non-determinisms, summarize effects resulting from the restriction of guesses to constants, and take into account properties such as the semi-decidability of oracle sets, the semi-decidability of the identity relation, and the recognizability of constants.
Paper Structure (10 sections, 4 theorems, 4 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 4 equations, 1 figure.

Table of Contents

  1. BSS RAMs with Moschovakis' operator
  2. Moschovakis' non-deterministic operator $\nu$
  3. Examples
  4. The power of non-deterministic BSS RAMs
  5. Evaluation of $\nu$-instructions for $n$-ary or decidable $Q$'s
  6. Simulation of $\nu$-oracle machines
  7. Useful subprograms and more
  8. The non-deterministic simulation of $\nu$-oracle machines
  9. Replacing $\nu$-instructions by trying to semi-decide $Q$
  10. More details: The simulation of ${\cal N}_Q$ semi-deciding $Q$

Key Result

Theorem 4.4

Let ${\cal A}\in {\sf Struc}$, $Q\subseteq U_{\cal A}^\infty$, and $Q\in {\rm SDEC}_{\cal A}$. Then, there holds

Figures (1)

  • Figure 1: The execution of ${\sf init}(Z_{I_{k_{\cal M}+1}})$

Theorems & Definitions (8)

  • Example 4.1: Computable functions and decidable relations
  • Example 4.2: Non-deterministically computable inverses
  • Example 4.3: Non-deterministically $\nu$-computable inverses
  • Theorem 4.4: $\nu$-queries and the ${\rm ND}$-semi-decidability
  • Remark 4.5: Another algorithm with identity test
  • Theorem 4.6: Guessing and the $\nu$-semi-decidability
  • Theorem 4.7: Non-deterministic computability
  • Corollary 4.8: Special oracles