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

Christine Gaßner

TL;DR

This work analyzes how non-deterministic BSS RAMs over first-order structures interface with identity tests and the presence of machine constants, focusing on Moschovakis' operator and various non-determinisms. It establishes precise links between the (semi-)decidability of the identity relation and the decidability of sets and their characteristic functions, showing how two machine constants can determine outputs and influence computability. By constructing pseudo-parallel simulations and delineating multiple non-deterministic models (binary, digital, and Moschovakis-based), the paper derives a detailed lattice of inclusions and equivalences among semi-decidable/decidable classes, clarifying when non-determinism yields computable characterizations of decision problems over structures with or without identity. The results unify and extend prior Part II results, offering practical methods to translate semi-decidability into decidability via constant-recognizability, and outlining a framework for comparing distinct non-deterministic mechanisms in abstract computation over first-order structures.

Abstract

BSS RAMs were introduced to provide a mathematical framework for characterizing algorithms over first-order structures. Non-deterministic BSS RAMs help to model different non-deterministic approaches. Here, we deal with different types of binary non-determinisms and study the consequences of the decidability of the identity relation and the decidability of finite sets consisting of one or two constants. We compare the binary non-determinism resulting from a non-deterministic branching process, the digital non-determinism resulting from the restriction of guesses to two constants, and some other non-determinisms resulting from the use of Moschovakis' operator applied to oracle sets restricted to tuples of constants. Moreover, we show that the performance capability and the efficiency of individual machines are influenced by the following properties. 1. The identity relation belongs to the underlying structure. 2. The identity is semi-decidable over the underlying structure. 3. Two single-element sets of constants are semi-decidable. 4. A set of two constants is semi-decidable. The order of these properties corresponds to the strength of their influence. In all cases mentioned, the semi-decidability of the sets implies their decidability.

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

TL;DR

This work analyzes how non-deterministic BSS RAMs over first-order structures interface with identity tests and the presence of machine constants, focusing on Moschovakis' operator and various non-determinisms. It establishes precise links between the (semi-)decidability of the identity relation and the decidability of sets and their characteristic functions, showing how two machine constants can determine outputs and influence computability. By constructing pseudo-parallel simulations and delineating multiple non-deterministic models (binary, digital, and Moschovakis-based), the paper derives a detailed lattice of inclusions and equivalences among semi-decidable/decidable classes, clarifying when non-determinism yields computable characterizations of decision problems over structures with or without identity. The results unify and extend prior Part II results, offering practical methods to translate semi-decidability into decidability via constant-recognizability, and outlining a framework for comparing distinct non-deterministic mechanisms in abstract computation over first-order structures.

Abstract

BSS RAMs were introduced to provide a mathematical framework for characterizing algorithms over first-order structures. Non-deterministic BSS RAMs help to model different non-deterministic approaches. Here, we deal with different types of binary non-determinisms and study the consequences of the decidability of the identity relation and the decidability of finite sets consisting of one or two constants. We compare the binary non-determinism resulting from a non-deterministic branching process, the digital non-determinism resulting from the restriction of guesses to two constants, and some other non-determinisms resulting from the use of Moschovakis' operator applied to oracle sets restricted to tuples of constants. Moreover, we show that the performance capability and the efficiency of individual machines are influenced by the following properties. 1. The identity relation belongs to the underlying structure. 2. The identity is semi-decidable over the underlying structure. 3. Two single-element sets of constants are semi-decidable. 4. A set of two constants is semi-decidable. The order of these properties corresponds to the strength of their influence. In all cases mentioned, the semi-decidability of the sets implies their decidability.

Paper Structure

This paper contains 10 sections, 17 theorems, 2 figures.

Table of Contents

  1. Two constants and consequences
  2. Two constants as outputs and the decidability
  3. On sufficient conditions for the decidability of problems
  4. Computable characteristic functions for decidable problems
  5. Two constants and their recognizability
  6. On sufficient conditions for recognizing the constants
  7. The decidability of semi-decidable single-element sets
  8. The decidability of semi-decidable identity relations
  9. Binary and other non-determinisms restricted to constants
  10. Different non-determinisms and relationships

Key Result

Proposition 6.1

Let ${\cal A}\in {\sf Struc}_{c_1,c_2}$, let ${\cal A}$ contain the identity relation ${\rm id}_{\cal A}$, and let $P\subseteq U_{\cal A}^\infty$. Then, the computability of $\chi_P$ by a BSS RAM in ${\sf M}_{\cal A}$ implies $P\in {\rm DEC}_{\cal A}$.

Figures (2)

  • Figure 1: From deciding whether $\vec{x} \in P$ or not to the computation of $\chi_P(\vec{x})$
  • Figure 2: A simulation of the execution of $\textcolor{red}{\ell}:\,$ goto $\ell_1$ or goto $\ell_2$

Theorems & Definitions (26)

  • Proposition 6.1: Computability and identity imply decidability
  • Remark 6.2
  • Proposition 6.3: Computability $\!$&$\!$ recognizability imply decidability
  • Remark 6.4
  • Corollary 6.5: Recognizable identity implies decidability
  • Proposition 6.6: Decidability implies computability
  • Proposition 6.7: Basic properties: Identity and constants
  • Example 6.8: A structure with $(b)$ in which $(a)$ does not hold
  • Example 6.9: A structure with $(d)$ in which $(c)$ does not hold
  • Example 6.10: A structure with $(e)$ in which $(d)$ does not hold
  • ...and 16 more