Herbrand Disjunctions, Cut Elimination and Context-Free Tree Grammars
Bahareh Afshari, Stefan Hetzl, Graham E. Leigh · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2015
Recently a new connection between proof theory and formal language theory was introduced. It was shown that the operation of cut elimination for proofs in first-order predicate logic involving Pi_1-cuts corresponds to computing the language of a particular class of regular tree grammars. The present paper expands this connection to the level of Pi_2-cuts. Given a proof pi of a Sigma_1 formula with cuts only on Pi_2 formulæ, we show there is associated to pi a natural context-free tree grammar whose language is finite and yields a Herbrand disjunction for pi.