A Proof Theory of ( \(\omega \) -)Context-Free Languages, via Non-wellfounded Proofs

We investigate the proof theory of regular expressions with fixed points, construed as a notation for ( \(\omega \) -)context-free grammars. Starting with a hypersequential system for regular expressions due to Das and Pous [15], we define its extension by least fixed points and prove the soundness and completeness of its non-wellfounded proofs for the standard language model. From here we apply proof-theoretic techniques to recover an infinitary axiomatisation of the resulting equational theory, complete for inclusions of context-free languages. Finally, we extend our syntax by greatest fixed points, now computing \(\omega \) -context-free languages. We show the soundness and completeness of the corresponding system using a mixture of proof-theoretic and game-theoretic techniques.