Herbrand's theorem revisited

Abstract

We present recent results on the deepening connection between proof theory and formal language theory. To each first-order proof with prenex cuts of complexity at most Πn/Σn, we associate a typed (non-deterministic) tree grammar of order n (equivalently, an order n recursion scheme) that abstracts the computation of Herbrand sets obtained through Gentzen-style cut elimination. Apart from offering a means to compute Herbrand expansions directly from proofs with cuts, these grammars provide a structural counterpart to Herbrand's theorem that opens the door to tackling a number of questions in proof theory such as proof equivalence, proof compression and proof complexity.