Space-Bounded OTMs and REG$^{\infty}$

TL;DR

The paper explores a transfinite analogue of a classical complexity theorem, introducing deterministic ordinal automata and relating them to languages decidable by ordinal Turing machines with limited space.

Contribution

It introduces deterministic ordinal automata and establishes their connection to languages decidable by space-bounded ordinal Turing machines.

Findings

Deterministic ordinal automata satisfy properties similar to finite automata.

Languages decidable by space-limited OTMs are also decidable by DOAs.

The paper extends classical complexity results to transfinite automata and Turing machines.

Abstract

An important theorem in classical complexity theory is that LOGLOGSPACE=REG, i.e. that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce deterministic ordinal automata (DOAs), show that they satisfy many of the basic statements of the theory of deterministic finite automata and regular languages. We then consider languages decidable by an ordinal Turing machine (OTM), introduced by P. Koepke in 2005 and show that if the working space of an OTM is of strictly smaller cardinality than the input length for all sufficiently long inputs, the language so decided is also decidable by a DOA.