Discrete Transfinite Computation

TL;DR

This paper explores models of computation extended to transfinite steps, including machines with ordinal-based tapes, connecting them to higher-type recursion theories and demonstrating their relevance to polynomial time on infinite strings.

Contribution

It introduces and analyzes various transfinite computational models, linking them to existing theories and illustrating their applicability to infinite string computation.

Findings

Transfinite models generalize Turing machines to ordinal steps.

Connections established between these models and higher-type recursion theories.

Polynomial time on ω-strings is effectively modeled by these transfinite frameworks.

Abstract

We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as $\alpha$-recursion. We conclude that, in particular, polynomial time computation on $\omega$-strings is well modelled by several convergent conceptions.