# On the Density of Context-Free and Counter Languages

## TL;DR

This paper investigates the decidability of density properties in various formal languages, showing that certain density questions are decidable for some automata classes but undecidable for others, and introduces new language classes.

## Contribution

It provides new decidability results for density in languages accepted by specific automata and introduces the concept of bounded-dense languages.

## Key findings

- Decidability of infix-density for nondeterministic reversal-bounded pushdown automata.
- Undecidability of density for deterministic pushdown automata and nondeterministic one-counter automata.
- Decidability of suffix-density with fewer reversals in deterministic one-counter automata.

## Abstract

A language $L$ is said to be dense if every word in the universe is an infix of some word in $L$. This notion has been generalized from the infix operation to arbitrary word operations $\varrho$ in place of the infix operation ($\varrho$-dense, with infix-dense being the standard notion of dense). It is shown here that it is decidable, for a language $L$ accepted by a one-way nondeterministic reversal-bounded pushdown automaton, whether $L$ is infix-dense. However, it becomes undecidable for both deterministic pushdown automata (with no reversal-bound), and for nondeterministic one-counter automata. When examining suffix-density, it is undecidable for more restricted families such as deterministic one-counter automata that make three reversals on the counter, but it is decidable with less reversals. Other decidability results are also presented on dense languages, and contrasted with a marked version called $\varrho$-marked-density. Also, new languages are demonstrated to be outside various deterministic language families after applying different deletion operations from smaller families. Lastly, bounded-dense languages are defined and examined.