What is an Algorithm?: a Modern View

TL;DR

This paper proposes a modern, unified definition of an algorithm based on computability logic, viewing it as a sequence of moves transforming inputs into outputs, integrating previous perspectives.

Contribution

It introduces a new definition of algorithms combining abstract state machines and recursors within a computability logic framework.

Findings

Defines algorithms as sequences of legal moves in a game-like setting.

Unifies different views of algorithms into a single formal framework.

Provides a formal language-based representation of algorithms.

Abstract

Although algorithm is one of the central subjects, there have been little common understandings of what an algorithm is. For example, Gurevich view algorithms as abstract state machines, while others view algorithms as recursors. We promote a third view: it is a combination to these two disparate views. This approach -- based on computability logic -- describes an algorithm as $A(I,O)$ where $I$ is a set of input services and $O$ an output service. It leads to the following modern definition: {\it An algorithm $A$ is a (tree of) sequence of legal moves for providing $O$ using $I$. } In the above, $A$ is written in an imperative language/abstract state machine and $I,O$ are written in recursors/logical specifications.