Deep Inference and Symmetry in Classical Proofs

• Main Author: Brünnler, Kai
• Other Authors: Technische Universität Dresden, Informatik
• Format: Doctoral Thesis
• Language: English
• Published: Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden 2003
• Subjects: Beweistheorie; Schnittelimination; calculus of structures; classical logic; cut elimination; deep inference; proof theory; symmetry; ddc:28; rvk:SK 130; Klassische Logik
• Online Access: http://nbn-resolving.de/urn:nbn:de:swb:14-1064911987703-38192; http://nbn-resolving.de/urn:nbn:de:swb:14-1064911987703-38192; http://www.qucosa.de/fileadmin/data/qucosa/documents/1073/1064911987703-3819.pdf

In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems.