Mathematical Informatics: Algorithms

TL;DR

This paper develops an intensional approach to computability, modeling programs as dynamical systems constrained by monoid actions, and formalizes algorithms as labeled graphs separating control and data.

Contribution

It introduces a formal definition of algorithms as finite labeled graphs and establishes a framework for their implementation within models of computation.

Findings

Defines algorithms as finite directed graphs with partial maps as labels

Separates control flow from data operations in the model

Provides a formal notion of algorithm implementation in computational models

Abstract

This work continues the development of an intensional approach to computability initiated in previous work, in which programs and computations, rather than functions, constitute the primary objects of study. In this setting, models of computation are described as monoid actions on a configuration space, and programs as dynamical systems constrained by this action. Within this framework, we introduce a formal notion of algorithm as a finite directed graph whose edges are labelled by partial maps over an abstract data structure. This definition separates control from data, representing the former as a graph and the latter as an algebra of operations. We then define what it means for a program, in a given model of computation, to implement such an algorithm, by requiring a correspondence between computational steps and labelled transitions that preserves the induced transformations on representations of data. This yields a precise notion of implementation and situates algorithms as abstract partial specifications of computational behaviour.