Basic interactive algorithms: Preview

TL;DR

This paper previews an upcoming work on axiomatizing basic interactive algorithms, emphasizing their foundational role and relation to the Church-Turing thesis, including probabilistic and quantum algorithms.

Contribution

It introduces the concept of basic algorithms and discusses their axiomatization, extending classical notions to probabilistic and quantum algorithms within a unified framework.

Findings

Basic algorithms can be axiomatized and are behaviorally equivalent to abstract state machines.

Probabilistic and quantum algorithms can be viewed as basic algorithms with oracles.

The distinction between the classical and physical Church-Turing theses is clarified.

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.