Pushdown automata for context-free languages over plain categories

Investigate and formalize how pushdown automata emerge for context-free languages defined over plain categories (i.e., categories without additional monoidal structure); develop an appropriate categorical notion of pushdown automaton and establish its relationship to context-free grammars within the parsing-as-lifting framework.

Background

The paper proposes a structural approach to context-free languages in categorical and monoidal settings, proving a monoidal representation theorem. While regular and context-free monoidal languages are treated, a corresponding automaton model for context-freeness is not yet developed.

In the conclusion, the authors explicitly state that the emergence of pushdown automata in the categorical framework (even for plain categories) remains to be elaborated, highlighting a key open direction to bridge grammars and automata in this generalized setting.