Extension without cut

Annals of Pure and Applied Logic 163 (12):1995-2007 (2012)
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Abstract

In proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proof complexity one distinguishes Frege systems and extended Frege systems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut-free systems with extension, which is neither possible with Frege systems, nor with the sequent calculus. We show that the propositional pigeonhole principle admits polynomial-size proofs in a cut-free system with extension. We also define cut-free systems with substitution and show that the cut-free system with extension p-simulates the cut-free system with substitution

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Grundzüge der theoretischen Logik.D. Hilbert & W. Ackermann - 1928 - Annalen der Philosophie Und Philosophischen Kritik 7:157-157.
Grundzüge der theoretischen logik.David Hilbert - 1928 - Berlin,: G. Springer. Edited by Wilhelm Ackermann.

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