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

TL;DR

This paper explores various types of non-determinism in abstract computation models over first-order structures, analyzing how properties like decidability influence computational power and efficiency.

Contribution

It compares different non-deterministic approaches, including Moschovakis' operator, and examines how decidability properties affect computational capabilities.

Findings

Semi-decidability of sets implies their decidability.

Identity relation's decidability impacts computational power.

Different non-determinisms have varying effects on efficiency.

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.