Formalizing the notions of non-interactive and interactive algorithms

2024-05-29

AI Summary

Formal theories of algorithms have long been confined to non-interactive settings, leaving interactive and nondeterministic algorithms without rigorous foundational treatment. Method: This work introduces a unified formal framework encompassing both non-interactive and interactive, deterministic and nondeterministic algorithms. It proposes the “prototype algorithm” as an abstract computational model and rigorously defines its behavioral semantics. Three equivalence relations—behavioral, implementation, and specification equivalence—are formally introduced; their relationships are established, and specification equivalence is proven to be the appropriate criterion for capturing essential algorithmic identity. Contribution: The framework breaks the traditional boundaries of algorithm definitions, providing the first formal foundation for interactive algorithms. It establishes a layered, extensible meta-theory of algorithms and delivers a rigorous logical basis for reasoning about algorithmic essence, correctness verification, and cross-model comparison—thereby unifying previously fragmented formal approaches under a coherent theoretical umbrella.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Humans and AI: Other Foundations of Human Computation & AI
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Algorithmic accountability and transparency on the web

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.

Problem

• Formalizing non-interactive and interactive algorithms
• Generalizing deterministic and non-deterministic algorithm notions
• Defining equivalence relations for proto-algorithms

Innovation

• Formalizing non-interactive and interactive algorithms
• Introducing proto-algorithms for generalization
• Defining equivalence relations on proto-algorithms