# Can we define CFL without grammars or automata?

The set of regular languages $R$ over an alphabet $\Sigma$ can be defined as the smallest set satisfying these 5 axioms:

• Empty language: $\{\} \in R$
• Singleton languages: $\forall a \in \Sigma : \{a\} \in R$
• Closure under union: $\forall A, B \in R : A \cup B \in R$
• Closure under concatenation: $\forall A, B \in R : A B \in R$
• Closure under Kleene star: $\forall A \in R: A^* = \bigcup_{i \in \mathbb{N}} A^i = \{\varepsilon\} \cup A \cup A^2 \cup \ldots \in R$

Is there a similar way to define the set of context-free languages in terms of set operations, without invoking context-free grammars or pushdown automata?

• Does this old question on expressions for cf-languages answer your own question, or are you interested in another generalization of regularity? cs.stackexchange.com/questions/59923/… – Hendrik Jan Oct 18 '16 at 21:02
• Chomsky-Schützenberger comes to mind as well. – Raphael Oct 18 '16 at 22:29
• Please ask only one question per post. I'm removing the last two as they are very independent of the first and make this way too broad. (You may want/have to do your own research, you know.) – Raphael Oct 18 '16 at 22:29
• See also this very closely related question. – Raphael Oct 18 '16 at 22:34
• Would it be OK to narrow this question down as Can CFLs be characterized in terms of their closure properties?? – reinierpost Sep 15 '17 at 9:00