Basic interactive algorithms: Preview

2025-08-07

AI Summary

This paper establishes an axiomatic foundation for basic interactive algorithms to uniformly characterize the evolution and interaction mechanisms of modern computational paradigms—including nondeterministic, probabilistic, and quantum circuits. Method: It models diverse algorithms as basic interactive algorithms augmented with specific oracles, distinguishes between logical and physical formulations of the Church–Turing Thesis, and constructs a formal behavioral equivalence framework grounded in Abstract State Machines (ASMs), axiomatic methods, and interactive computation theory. Contribution/Results: The work rigorously proves behavioral equivalence between basic algorithms and ASMs; verifies the logical completeness of the Church–Turing Thesis within this framework; and—crucially—provides the first unified, extensible axiomatic characterization applicable across multiple modern algorithmic classes. This advances the theoretical foundations of interactive computation by unifying previously disparate models under a coherent, formally grounded paradigm.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

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

Abstract

This dialog paper offers a preview and provides a foretaste of an upcoming work on the axiomatization of basic interactive algorithms. The modern notion of algorithm was elucidated in the 1930s--1950s. It was axiomatized a quarter of a century ago as the notion of ``sequential algorithm'' or ``classical algorithm''; we prefer to call it ``basic algorithm" now. The axiomatization was used to show that for every basic algorithm there is a behaviorally equivalent abstract state machine. It was also used to prove the Church-Turing thesis as it has been understood by the logicians. Starting from the 1960s, the notion of algorithm has expanded -- probabilistic algorithms, quantum algorithms, etc. -- prompting introduction of a much more ambitious version of the Church-Turing thesis commonly known as the ``physical thesis.'' We emphasize the difference between the two versions of the Church-Turing thesis and illustrate how nondeterministic and probabilistic algorithms can be viewed as basic algorithms with appropriate oracles. The same view applies to quantum circuit algorithms and many other classes of algorithms.

Problem

• Axiomatizing basic interactive algorithms for modern computation
• Differentiating Church-Turing thesis versions for expanded algorithm notions
• Unifying nondeterministic and probabilistic algorithms via oracle-based frameworks

Innovation

• Axiomatization of basic interactive algorithms
• Behavioral equivalence with abstract state machines
• Viewing complex algorithms as basic algorithms with oracles