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.
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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
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DOI: https://doi.org/10.1007/0-306-48088-3_3
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