Refocusing Generalised Normalisation

Abstract

When defined with general elimination/application rules, natural deduction and λ-calculus become closer to sequent calculus. In order to get real isomorphism, normalisation has to be defined in a “multiary” variant, in which reduction rules are necessarily non-local (reason: nomalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). Non-local rules are bad, for instance, for the mechanization of the system. A solution is to extend natural deduction even further to a unified calculus based on the unification of cut and general elimination. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.