Deletion Operations on Deterministic Families of Automata

Many different deletion operations are investigated applied to languages accepted by one-way and twoway 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 oneway 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 contextfree 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). These results are surprising given the nondeterministic nature of the deletion operation. 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.