Pushdown Conjecture for ID Languages

Determine whether every infinite meta-state DAG language (ID)—that is, every language generated by a minimal deterministic DAG grammar whose minimal set of meta-states 2min is infinite—is recognizable by a pushdown automaton.

Background

The paper introduces meta-states to adapt classical finite-state techniques to directed acyclic graph (DAG) languages. It defines classes FID (finite induced meta-state DAG languages) and FD (finite meta-state DAG languages), for which the authors construct deterministic finite automata (DFAs) that recognize the corresponding DAG languages.

In contrast, the class ID (infinite meta-state DAG languages) consists of languages generated by minimal deterministic DAG grammars whose minimal meta-state set 2min is infinite. The paper proves that notable families (e.g., fully balanced trees of unbounded size and certain DAGs with cycles and chord paths) fall into ID and provides a characterization (Theorem 6.2) of when 2min is infinite.

Motivated by placing these classes within a hierarchy analogous to the Chomsky hierarchy, the authors explicitly conjecture that ID languages have PDA recognizers, paralleling the DFA recognizability of FD/FID languages.