Abstract
This article examines the computational content of the classical Gentzen sequent calculus. There are a number of well-known methods that extract computational content from first-order logic but applying these to the sequent calculus involves first translating proofs into other formalisms, Hilbert calculi or Natural Deduction for example. A direct approach which mirrors the symmetry inherent in sequent calculus has potential merits in relation to proof-theoretic considerations such as the (non-)confluence of cut elimination, the problem of cut introduction, proof compression and proof equivalence. Motivated by such applications, we provide a representation of sequent calculus proofs as higher order recursion schemes. Our approach associates to an LK proof π of ⇒∃vF, where F is quantifier free, an acyclic higher order recursion scheme H with a finite language yielding a Herbrand disjunction for ∃vF. More generally, we show that the language of H contains all Herbrand disjunctions computable from π via a broad range of cut elimination strategies.
Highlights
The property of being a valid first-order formula is intimately tied to the consideration of the ground, i.e., variable-free, instances of that formula
Our representation of Herbrand’s theorem is tailored for the classical sequent calculus in the sense that it remains faithful to the non-deterministic process of computing Herbrand expansions via cut elimination
An example of the latter is an upper bound on the size of Herbrand expansions which can be obtained via a broad array of cut elimination strategies
Summary
The property of being a valid first-order formula is intimately tied to the consideration of the ground, i.e., variable-free, instances of that formula. Our representation of Herbrand’s theorem is tailored for the classical sequent calculus in the sense that it remains faithful to the non-deterministic process of computing Herbrand expansions via (reductive) cut elimination. In this respect, we believe the present work marks the first method of Herbrand extraction that operates directly on sequent calculus proofs. The framework of higher order recursion schemes opens the door to applying techniques and results from formal language theory directly to structural proof theory An example of the latter is an upper bound on the size (and, number) of Herbrand expansions which can be obtained via a broad array of cut elimination strategies. The article concludes with a discussion of the results and potential extensions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
