Fast Cut-Elimination by Projection Matthias Baaz∗ Alexander Leitsch Technische Universit¨at Wien, Austria
1
Introduction
The transformation of arbitrary proofs in Gentzen’s calculus LK into cut-free proofs is of central importance not only to proof theory itself but also to computer science. Particularly the extraction of programs from proofs requires cutelimination as a preparatory step; the size of the resulting program then depends on that of the cut-free proof. This illustrates the relevance 1. of the design of efficient cut-elimination procedures and 2. of a characterization of problem classes admitting fast cut-elimination. The famous results of Statman [Statman 79] and Orevkov [Orevkov 82] show that ”fast” cut-elimination is generally impossible; indeed the complexity of cut-elimination (independent of the particular procedure) is even nonelementary. However this result does not imply that the choice of specific elimination procedures is irrelevant at all: We prove for a wide class of endsequents which contain all equational theories – among them Statman’s worst-case examples – that the elimination of monotone cuts is only of at most exponential expense if, as additional elimination technique, direct projections of cutformulas to subformulas are admitted. Moreover we show that, for this class of endsequents and for monotone cuts, the worst-case complexity of usual Gentzen-type elimination procedures is nonelementary. The method of incorporating more mathematical information into cut-elimination procedures is in the spirit of Goad [Goad 80]; Goad’s method uses additional information based on the evaluation of subformulas, and thus is of semantic nature, whereas the projection method is purely syntactic.
2
Notation and Definitions
We use LK in an usual formulation (cf. [Takeuti 87]) with atomic axiom sequents and ⊥ ` added; in binary rules no implicit contractions are performed. The length l(ω) of a derivation ω is the number of sequents in the derivation. ∗ Corresponding author. Current address: TU-Wien, 1040 Wien/Austria; phone: +43(1)58801-4088; email:
[email protected] Karlsplatz 13/E185-2,
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Definition 2.1 We write B