Formalizing the notions of non-interactive and interactive algorithms

TL;DR

This paper extends formal definitions of algorithms to include non-interactive and interactive, deterministic and non-deterministic types, introducing proto-algorithms and equivalence relations to unify these concepts.

Contribution

It generalizes previous formalizations by defining proto-algorithms for both non-interactive and interactive cases, incorporating equivalence relations to characterize algorithm classes.

Findings

Introduces non-interactive and interactive proto-algorithms.

Defines three equivalence relations for proto-algorithms.

Proposes a framework for classifying algorithms based on these relations.

Abstract

An earlier paper gives an account of a quest for a satisfactory formalization of the classical informal notion of an algorithm. That notion only covers algorithms that are deterministic and non-interactive. In this paper, an attempt is made to generalize the results of that quest first to a notion of an algorithm that covers both deterministic and non-deterministic algorithms that are non-interactive and then further to a notion of an algorithm that covers both deterministic and non-deterministic algorithms that are interactive. The notions of an non-interactive proto-algorithm and an interactive proto-algorithm are introduced. Non-interactive algorithms and interactive algorithms are expected to be equivalence classes of non-interactive proto-algorithms and interactive proto-algorithms, respectively, under an appropriate equivalence relation. On both non-interactive proto-algorithms and interactive proto-algorithms, three equivalence relations are defined. Two of them are deemed to be bounds for an appropriate equivalence relation and the third is likely an appropriate one.