Herbrand-Confluence

Abstract

Abstract. We consider cut-elimination in the sequent calculus for classical first-order logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cut-free proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbrand-disjunction of a cut-free proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the non-erasing reduction lead to the same Herbrand-disjunction.

Algorithmic introduction of quantified cuts

“…and symmetrically for the case of D l . We can now state the part of the main result of [21] which is relevant for this paper:…”

“…In this section we show that the method CI is complete by proving that -in a suitable sense -it constitutes an inversion of Gentzen's procedure for cut-elimination. To that aim we will rely on the results of [22,21] which show that the language of a grammar is a strong invariant of cut-elimination. To be more precise we quickly repeat some notions from [22,21] here, for full details the interested reader is referred to these papers.…”

“…To that aim we will rely on the results of [22,21] which show that the language of a grammar is a strong invariant of cut-elimination. To be more precise we quickly repeat some notions from [22,21] here, for full details the interested reader is referred to these papers.…”