Once-Marking and Always-Marking 1-Limited Automata

TL;DR

This paper introduces restricted variants of 1-limited automata, analyzing their size complexity and showing how these restrictions affect their succinctness compared to deterministic finite automata.

Contribution

It defines once-marking and always-marking 1-limited automata and studies their descriptional complexity, revealing new size relationships with finite automata.

Findings

Once-marking 1-limited automata have a double exponential size gap to one-way deterministic finite automata.

Deterministic restriction of once-marking automata is polynomially related to two-way deterministic automata.

Always-marking 1-limited automata have a single exponential size gap to one-way deterministic finite automata.

Abstract

Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case the size gap from 1-limited automata to one-way deterministic finite automata is double exponential. Here we introduce two restricted versions of 1-limited automata, once-marking 1-limited automata and always-marking 1-limited automata, and study their descriptional complexity. We prove that once-marking 1-limited automata still exhibit a double exponential size gap to one-way deterministic finite automata. However, their deterministic restriction is polynomially related in size to two-way deterministic finite automata, in contrast to deterministic 1-limited automata, whose equivalent two-way deterministic finite automata in the worst case are exponentially larger. For always-marking 1-limited automata, we prove that the size gap to one-way deterministic finite automata is only a single exponential. The gap remains exponential even in the case the given machine is deterministic. We obtain other size relationships between different variants of these machines and finite automata and we present some problems that deserve investigation.