Abstract
We propose a notion of symmetric reduction for a system of proof-nets for Multiplicative Affine Logic with Mix (MAL + Mix) (namely, multiplicative linear logic with the mix-rule the unrestricted weakening-rule). We prove that such a reduction has the strong normalization and Church-Rosser properties. A notion of irrelevance in a proof-net is defined and the possibility of cancelling the irrelevant parts of a proof-net without erasing the entire net is taken as one of the correctness conditions; therefore purely local cut-reductions are given, minimizing cancellation and suggesting a paradigm of “computation without garbage collection”. Reconsidering Ketonen and Weyhrauch’s decision procedure for affine logic [15, 4], the use of the mix-rule is related to the non-determinism of classical proof-theory. The question arises, whether these features of classical cut-elimination are really irreducible to the familiar paradigm of cut-elimination for intuitionistic and linear logic.
Paper submitted in 1999, revised in 2001. Research supported by EPSRC senior research fellowship on grant GL/L 33382. This research started during a visit to the University of Leeds in 1997: thanks to Stan Wainer, John Derrick and Michael Rathjen and Diane McMagnus for their hospitality. Thanks to Martin Hyland, Edmund Robinson and especially Arnaud Fleury for extremely useful discussions during the final revision.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Asperti. Causal Dependencies in Multiplicative Linear Logic with MIX. Mathematical Structures in Computer Science 5:351–380, 1995.
G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh.
G. Bellin. Subnets of proof-nets in multiplicative linear logic with Mix, Mathematical Structures in Computer Science 7:663–699, 1997.
G. Bellin and J. Ketonen. A Decision Procedure Revisited: Notes on Direct Logic, Linear Logic and its Implementation, Theoretical Computer Science 95(1): 115–142, 1992.
G. Bellin and P. J. Scott. Selected papers of the meeting on Mathematical Foundations of Programming Semantics (MFPS 92), Part 1 (Oxford 1992), Theoretical Computer Science 135(1):11–65, 1994.
G. Bellin and J. van de Wiele. Subnets of Proof-nets in MLL−, in Advances in Linear Logic, J-Y. Girard, Y Lafont and L. Regnier editors, London Mathematical Society Lecture Note Series 222, Cambridge University Press, 1995, pp. 249–270.
C. Urban and G. M. Bierman. Strong Normalization of Cut-Elimination in Classical Logic, Fundamenta Informaticae XX: 1–32, 2000.
V. Danos, J-B. Joinet and H. Schellinx. LKQ and LKT: Sequent calculi for second order logic based upon linear decomposition of classical implication, in Advances in Linear Logic, J.-Y. Girard, Y. Lafont and L. Regnier editors, London Mathematical Society Lecture Note Series 222, Cambridge University Press, 1995, pp. 211–224.
V. Danos, J-B. Joinet and H. Schellinx. A new deconstructive logic: linear logic. Preprint n.52, Équipe de Logique Mathématique, Université Paris VII, Octobre 1994.
A. Fleury and C. Retoré. The Mix Rule. Mathematical Structures in Computer Science 4:273–85, 1994.
J-Y. Girard. Linear Logic. Theoretical Computer Science 50:1–102, 1987.
J-Y. Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science 1:255–296, 1991.
J-Y. Girard. On the unity of logic. Annals of Pure and Applied Logic 59:201–217, 1993.
J-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7, Cambridge University Press, 1989.
J. Ketonen and R. Weyhrauch. A Decidable Fragment of Predicate Calculus, Theoretical Computer Science 32:297–307, 1984.
P. Lincoln and T. Winkler. Constant-only multiplicative linear logic is NP-complete. Selected papers of the meeting on Mathematical Foundations of Programming Semantics (MFPS 92), Part 1 (Oxford 1992), Theoretical Computer Science 135(1):155–169, 1994.
M. Parigot. Free Deduction: an analysis of computation in classical logic. In Voronkov, A., editor, Russian Conference on Logic Programming, pp. 361–380. Sprinver Verlag. LNAI 592, 1991.
M. Parigot. Strong normalization for second order classical natural deduction, in Logic in Computer Science, pp. 39–46. IEEE Computer Society Press. Proceedings of the Eight Annual Symposium LICS, Montreal, June 19–23, 1992.
D. Prawitz. Natural deduction. A proof-theoretic study. Almquist and Wiksell, Stockholm, 1965.
D. Prawitz. Ideas and Results in Proof Theory, in Proceedings of the Second Scandinavian Logic Symposium, ed. Fenstad, North-Holland, 1971.
Edmund Robinson. Proof Nets for Classical Logic, preprint, Dagstuhl workshop 1141/1 on Semantic Foundations of Proof-search, April 2001. http://www.dcs.qmw.ac.uk/~edmundr.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Kluwer Academic Publishers
About this chapter
Cite this chapter
Bellin, G. (2003). Two Paradigms of Logical Computation in Affine Logic?. In: de Queiroz, R.J.G.B. (eds) Logic for Concurrency and Synchronisation. Trends in Logic, vol 15. Springer, Dordrecht. https://doi.org/10.1007/0-306-48088-3_3
Download citation
DOI: https://doi.org/10.1007/0-306-48088-3_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-1270-9
Online ISBN: 978-0-306-48088-1
eBook Packages: Springer Book Archive