Afshari, B., Hetzl, S., & Leigh, G. (2015). Herbrand Disjunctions, Cut Elimination and Context-Free Tree Grammars. In T. Altenkirch (Ed.), 13th International Conference on Typed Lambda Calculi and Applications - TLCA 2015 (pp. 1–16). Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.TLCA.2015.1
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.
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