On the Executability of Interactive Computation

TL;DR

This paper compares interactive Turing machines and reactive Turing machines, showing their behaviors are interrelated and that advice enhances their expressive power, unifying theories of reactive and interactive computability.

Contribution

It demonstrates that reactive executability encompasses interactive computability and introduces RTMs with advice, significantly expanding their behavioral expressiveness.

Findings

ITMs' behavior is reactively executable

RTMs' stream translations are interactively computable

RTMs with advice can simulate any countable transition system

Abstract

The model of interactive Turing machines (ITMs) has been proposed to characterise which stream translations are interactively computable; the model of reactive Turing machines (RTMs) has been proposed to characterise which behaviours are reactively executable. In this article we provide a comparison of the two models. We show, on the one hand, that the behaviour exhibited by ITMs is reactively executable, and, on the other hand, that the stream translations naturally associated with RTMs are interactively computable. We conclude from these results that the theory of reactive executability subsumes the theory of interactive computability. Inspired by the existing model of ITMs with advice, which provides a model of evolving computation, we also consider RTMs with advice and we establish that a facility of advice considerably upgrades the behavioural expressiveness of RTMs: every countable transition system can be simulated by some RTM with advice up to a fine notion of behavioural equivalence.