Deletion Operations on Deterministic Families of Automata

TL;DR

This paper explores how various deletion operations affect deterministic automata languages, revealing closure properties and limitations, especially involving reversal-bounded multicounter machines and nondeterministic language classes.

Contribution

It demonstrates closure of one-way deterministic reversal-bounded multicounter languages under certain quotients and shows how increased reversals expand the language class.

Findings

One-way deterministic reversal-bounded multicounter languages are closed under right quotient with many language families.

Left quotient with one reversal-bounded counters yields increased counters but remains within deterministic reversal-bounded multicounter languages.

Higher reversals or additional counters can produce languages outside deterministic and one-way nondeterministic automata classes.

Abstract

Many different deletion operations are investigated applied to languages accepted by one-way and two-way deterministic reversal-bounded multicounter machines, deterministic pushdown automata, and finite automata. Operations studied include the prefix, suffix, infix and outfix operations, as well as left and right quotient with languages from different families. It is often expected that language families defined from deterministic machines will not be closed under deletion operations. However, here, it is shown that one-way deterministic reversal-bounded multicounter languages are closed under right quotient with languages from many different language families; even those defined by nondeterministic machines such as the context-free languages. Also, it is shown that when starting with one-way deterministic machines with one counter that makes only one reversal, taking the left quotient with languages from many different language families -- again including those defined by nondeterministic machines such as the context-free languages -- yields only one-way deterministic reversal-bounded multicounter languages (by increasing the number of counters). However, if there are two more reversals on the counter, or a second 1-reversal-bounded counter, taking the left quotient (or even just the suffix operation) yields languages that can neither be accepted by deterministic reversal-bounded multicounter machines, nor by 2-way nondeterministic machines with one reversal-bounded counter.