Proof Search Tree and Cut Elimination

Abstract

A new cut elimination method is obtained here by “proof mining” (unwinding) from the following non-effective proof that begins with extracting an infinite branch $\mathcal{B}$ when the canonical search tree $\mathcal{T}$ for a given formula E of first order logic is not finite. The branch $\mathcal{B}$ determines a semivaluation so that $\mathcal{B}\models \bar{E}$ and (*) every semivaluation can be extended to a total valuation. Since for every derivation d of E and every model $\mathcal{M}$, ${\mathcal M}\models E$, this provides a contradiction showing that $\mathcal{T}$ is finite, $\exists l(\mathcal{T}<l)$. A primitive recursive function L(d) such that $\mathcal{T}< L(d)$ is obtained using instead of (*) the statement: For every r, if the canonical search tree $\mathcal{T}^{r+1}$ with cuts of complexity r + 1 is finite, then $\mathcal{T}^r$ is finite.

In our proof the reduction of (r + 1)-cuts does not introduce new r-cuts but preserves only one of the branches.