Abstract
A normalization procedure is presented for a classical natural deduction (ND) proof system. This proof system, called N-Graphs, has a multiple conclusion proof structure where cycles are allowed. With this, we have developed a thorough treatment of cycles, including cycles normalization via an algorithm. We also demonstrate the usefulness of the graphical framework of N-Graphs, where derivations are seen as digraphs. We use geometric perspective techniques to establish the normalization mechanism, thus giving a direct normalization proof.
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References
de Oliveira, A.G.: Proofs from a Geometric Perspective. PhD thesis, Centro de Informática - Universidade Federal de Pernambuco (April 2001)
de Oliveira, A.G., de Queiroz, R.J.G.B.: 1 of Logic for Concurrency and Synchronisation. Trends in Logic - Studia Logic Library. In: Geometry of Deductions via Graphs of Proofs, pp. 1–86. Kluwer Academic Publishers, Dordrecht (2003)
Statman, R.: Structural Complexity of Proofs. PhD thesis, Stanford University (May 1974)
Carbone, A.: Duplication of directed graphs and exponential blow up of proofs. Ann. Pure Appl. Logic 100(1-3), 1–67 (1999)
Guglielmi, A., Gundersen, T.: Normalisation control in deep inference via atomic flows. Logical Methods in Computer Science 4, 1–36 (2008)
Blute, R., Cockett, J.R.B., Seely, R.A.G., Trimble, T.H.: Natural deduction and coherence for weakly distributive categories. Journal of Pure and Applied Algebra 13(3), 229–296 (1996)
Prawitz, D.: Ideas and results in proof theory. In: Fenstad, J.E., ed.: Proceedings 2nd Scandinavian Logic Symp., Oslo, Norway, 18–20 June (1970); Studies in Logic and the Foundations of Mathematics, vol. 63, pp. 235–307. North-Holland, Amsterdam (1971)
Pereira, L.C.P.D., Massi, C.D.B.: Normalização para a lógica clássica. III Encontro Nacional de Filosofia, 5 Gramado, RS - Brasil (1988)
Stålmarck, G.: Normalization theorems for full first order classical natural deduction. Journal of Symbolic Logic 56(1), 129–149 (1991)
Ungar, A.M.: Normalization, Cut-Elimination and the Theory of Proofs. CSLI Lecture Notes, vol. 28. CSLI Publications, Stanford (1992)
Cellucci, C.: Existential instantiation and normalization in sequent natural deduction. Annals of Pure and Applied Logic 58(2), 111–148 (1992)
Abramsky, S.: Proofs as processes. Theoretical Computer Science 135, 5–9 (1994)
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Alves, G.V., de Oliveira, A.G., de Queiroz, R. (2009). Transformations via Geometric Perspective Techniques Augmented with Cycles Normalization. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2009. Lecture Notes in Computer Science(), vol 5514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02261-6_8
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DOI: https://doi.org/10.1007/978-3-642-02261-6_8
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