Cut Elimination inside a Deep Inference System for Classical Predicate Logic
- 55 Downloads
Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic. As a consequence we derive Herbrand's Theorem, which we express as a factorisation of derivations.
Keywordscut elimination deep inference first-order predicate logic
Unable to display preview. Download preview PDF.
- Avron, A., ‘The method of hypersequents in the proof theory of Propositional non-classical logics’, in Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, (eds.), Logic: from foundations to applications. Proc. Logic Colloquium, Keele, UK, 1993. Oxford University Press, New York, 1996, pp. 1–32.Google Scholar
- Basin, D., M. D'Agostino, D. M. Gabbay, S. Matthews, and L. Vigano (eds.), Labelled Deduction, volume 17 of Applied Logic Series. Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
- Belnap, N. D., Jr., ‘Display logic’, Journal of Philosophical Logic, 11: 375–417, 1982.Google Scholar
- Brünnler, K., ‘Atomic cut elimination for classical logic’, in M. Baaz and J. A. Makowsky, (eds.), CSL 8003, volume 2803 of Lecture Notes in Computer Science. Springer-Verlag, 2003, pp. 86–97.Google Scholar
- Brünnler, K., Deep Inference and Symmetry in Classical Proofs. PhD thesis, Tech-nische Universität Dresden, September 2003.Google Scholar
- Buss, S. R., ‘On Herbrand's theorem’, in Logic and Computational Complexity, volume 960 of Lecture Notes in Computer Science. Springer-Verlag, 1995, pp. 195–209.Google Scholar
- Dl Gianantonio, P., ‘Structures for multiplicative cyclic linear logic: Deepness vs cyclicity’, in J. Marcinkowski and A. Tarlecki, (eds.), CSL 2004, volume 3210 of Lecture Notes in Computer Science. Springer-Verlag, 2004, pp. 130–144.Google Scholar
- Guglielmi, A., ‘A system of interaction and structure’, Technical Report WV-02–10, Technische Universität Dresden, 2002. To appear in ACM Transactions on Computational Logic.Google Scholar
- Guglielmi, A., ‘Polynomial size deep-inference proofs instead of exponential size shallow-inference proofs’, Manuscript, 2003. http://www.ki.inf.tu-dresden.de/~guglielm/res/notes/AG12.pdf.
- Guglielmi, A., ‘Resolution in the calculus of structures’, Manuscript, 2003. http://www.ki.inf.tu-dresden.de/~guglielm/res/notes/AG10.pdf.
- Guglielmi, A., and L. Strassburger, ‘Non-commutativity and MELL in the calculus of structures’, in L. Fribourg, (ed.), CSL 2001, volume 2142 of Lecture Notes in Computer Science. Springer-Verlag, 2001, pp. 54–68.Google Scholar
- Stewart, Ch., and Ph. Stouppa, ‘A systematic proof theory for several modal logics’, Technical Report WV-03–08, Technische Universität Dresden, 2003. Accepted at Advances in Modal Logic 2004, to appear in proceedings published by King's College Publications.Google Scholar
- Stouppa, P., The design of modal proof theories: The case of S5, Master's thesis. Technische Universität Dresden, 2004.Google Scholar
- Strassburger, L., Linear Logic and Noncommutativity in the Calculus of Structures, PhD thesis. Technische Universität Dresden, 2003.Google Scholar
- Tait, W. W., ‘Normal derivability in classical logic’, in The Syntax and Semantics of Infinitary Languages, volume 72 of Lecture Notes in Mathematics. Springer, 1968, pp. 204–236.Google Scholar
- Tiu, A.F., ‘Properties of a Logical System in the Calculus of Structures’, Master's thesis. Technische Universität Dresden, 2001.Google Scholar