On the Herbrand content of LK
Bahareh Afshari, Stefan Hetzl, Graham E. Leigh
TL;DR
This paper introduces a structural, tree grammar-based representation of the Herbrand content in LK-proofs with cuts of certain complexity, analyzing how this content evolves under proof reductions.
Contribution
It provides a novel typed non-deterministic tree grammar framework to represent Herbrand content and studies its behavior during cut-elimination in LK proofs.
Findings
Grammar accurately captures Herbrand expansions.
Language equality and inclusion are characterized during reductions.
Finite language generation for Herbrand content.
Abstract
We present a structural representation of the Herbrand content of LK-proofs with cuts of complexity prenex Sigma-2/Pi-2. The representation takes the form of a typed non-deterministic tree grammar of order 2 which generates a finite language of first-order terms that appear in the Herbrand expansions obtained through cut-elimination. In particular, for every Gentzen-style reduction between LK-proofs we study the induced grammars and classify the cases in which language equality and inclusion hold.
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