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?

  • $\begingroup$ 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/… $\endgroup$ – Hendrik Jan Oct 18 '16 at 21:02
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    $\begingroup$ Chomsky-Schützenberger comes to mind as well. $\endgroup$ – Raphael Oct 18 '16 at 22:29
  • $\begingroup$ See also this very closely related question. $\endgroup$ – Raphael Oct 18 '16 at 22:34
  • $\begingroup$ Would it be OK to narrow this question down as Can CFLs be characterized in terms of their closure properties?? $\endgroup$ – reinierpost Sep 15 '17 at 9:00
  • $\begingroup$ @reinierpost Sorry for the late reply. Yes, that would work. $\endgroup$ – user76284 Feb 1 at 22:20

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