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Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

Authors: A. Carbone and S. Semmes
Journal: Bull. Amer. Math. Soc. 34 (1997), 131-159
MSC (1991): Primary 03F05; Secondary 68Q15
MathSciNet review: 1423203
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Abstract: Modus Ponens says that if you know $A$ and you know that $A$ implies $B$, then you know $B$. This is a basic rule that we take for granted and use repeatedly, but there is a gem of a theorem in logic by Gentzen to the effect that it is not needed in some logical systems. It is fun to say, ``You can make proofs without lemmas'' to mathematicians and watch how they react, but our true intention here is to let go of logic as a reflection of reasoning and move towards combinatorial aspects. Proofs contain basic problems of algorithmic complexity within their framework, and there is strong geometric and dynamical flavor inside them.

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A. Carbone
Affiliation: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
Address at time of publication: Mathématiques/Informatique, Université de Paris XII, 94010 Creteil Cedex, France

S. Semmes
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251

Received by editor(s): July 3, 1996
Received by editor(s) in revised form: December 1, 1996
Additional Notes: The first author was supported by the Lise-Meitner Stipendium # M00187-MAT (Austrian FWF) and the second author was supported by the U.S. National Science Foundation. Both authors are grateful to IHES for its hospitality.
Article copyright: © Copyright 1997 American Mathematical Society