New Results on the Minimum Amount of Useful Space

TL;DR

This paper explores minimal space requirements for recognizing nonregular languages across various automata models, establishing tight bounds and demonstrating the power of quantum and classical states in space-efficient recognition.

Contribution

It provides new tight bounds on space complexity for recognizing nonregular languages with different automata models, including quantum and classical automata.

Findings

Realtime nondeterministic Turing machines recognize nonregular unary languages within weak log log n space.

Log log n is a tight lower bound for nonregular language acceptance on weak realtime pushdown automata.

Unary nonregular languages can be accepted by automata with weak log n space, including quantum and classical models.

Abstract

We present several new results on minimal space requirements to recognize a nonregular language: (i) realtime nondeterministic Turing machines can recognize a nonregular unary language within weak $\log\log n$ space, (ii) $\log\log n$ is a tight space lower bound for accepting general nonregular languages on weak realtime pushdown automata, (iii) there exist unary nonregular languages accepted by realtime alternating one-counter automata within weak $\log n$ space, (iv) there exist nonregular languages accepted by two-way deterministic pushdown automata within strong $\log\log n$ space, and, (v) there exist unary nonregular languages accepted by two-way one-counter automata using quantum and classical states with middle $\log n$ space and bounded error.