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On categorical models of classical logic and the Geometry of Interaction

Published online by Cambridge University Press:  01 October 2007

CARSTEN FÜHRMANN  and
DAVID PYM
CARSTEN FÜHRMANN
Affiliation:
University of Bath, England, U.K.
DAVID PYM
Affiliation:
University of Bath and HP Labs, Bristol, England, U.K.
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Abstract

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models called classical categories that is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models called Dummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.

Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 17 , Issue 5 , October 2007 , pp. 957 - 1027
Copyright
Copyright © Cambridge University Press 2007

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References

Abramsky, S. (1996) Retracing some paths in process algebra. In: CONCUR 96. Springer-Verlag Lecture Notes in Computer Science 1119 1–17.CrossRefGoogle Scholar
Abramsky, S., Haghverdi, E. and Scott, P. (2002) Geometry of interaction and linear combinatory algebras. Mathematical Structures in Computer Science 12 (5)625–665.CrossRefGoogle Scholar
Abramsky, S. and Jagadeesan, R. (1994) New foundations for the geometry of interaction. J. Pure Applied Algebra 111 (1)53–119.Google Scholar
Barbanera, F. and Berardi, S. (1996) A symmetric lambda calculus for classical program extraction. In: Special issue: Symposium on Theoretical Aspects of Computer Software TACS'94. Information and Computation 125 103–117.CrossRefGoogle Scholar
Bellin, G. (2003) Two paradigms of logic computation in affine logic? In: deQueiroz, R. J. Queiroz, R. J. (ed.) Logic for Concurrency and Synchronisation, Trends in Logic 18, Kluwer Academic Publishers.Google Scholar
Bellin, G., Hyland, J., Robinson, E. and Urban, C. (2004) Proof theory of classical propositional calculus (submitted).Google Scholar
Blute, R., Cockett, J. and Seely, R. (2000) Feedback for linearly distributive categories: traces and fixpoints. J. Pure Applied Algebra 154 27–69.CrossRefGoogle Scholar
Blute, R., Cockett, J., Seely, R. and Trimble, T. (1996) Natural deduction and coherence for weakly distributive categories. J. Pure Applied Algebra 113 (3)229–296.CrossRefGoogle Scholar
Cockett, J., Koslowski, J. and Seely, R. (2003) Morphisms and modules for poly-bicategories. Theory and Applications of Categories 11 (2)15–74.Google Scholar
Cockett, J. and Seely, R. (1997a) Proof Theory for Full Intuitionistic Linear Logic, and MIX Categories. Theory and Applications of Categories 3 (5)85–131.Google Scholar
Cockett, J. and Seely, R. (1997b) Weakly distributive categories. J. Pure Applied Algebra 114 (2)133–173. (Updated version available at http://www.math.mcgill.ca/~rags)CrossRefGoogle Scholar
Cockett, J. and Seely, R. (1999) Linearly distributive functors. J. Pure Applied Algebra 143 155–203.CrossRefGoogle Scholar
Curien, P.-L. and Herbelin, H. (2000) The duality of computation. In: Proc. International Conference on Functional Programming, Montreal, IEEE.CrossRefGoogle Scholar
Danos, V. and Regnier, L. (1989) The structure of multiplicatives. Arch. Math. Logic 28 181–203.CrossRefGoogle Scholar
Dosen, K. (1999) Cut Elimination in Categories, Trends in Logic 6, Kluwer Academic Publishers.CrossRefGoogle Scholar
Dosen, K. and Petric, Z. (2004) Proof-theoretical coherence. Preprint, Mathematical Institute, Belgrade.Google Scholar
Dummett, M. (1977) Elements of Intuitionism, Oxford University Press.Google Scholar
Filinski, A. (1989) Declarative continuations and categorical duality, Master's thesis, Computer Science Department, University of Copenhagen, DIKU Report 89/11.Google Scholar
Führmann, C. and Pym, D. (2004) On the Geometry of Interaction for Classical Logic. In: Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science (LICS 2004), Turku (Finland) 211–220.CrossRefGoogle Scholar
Führmann, C. and Pym, D. (2006) Order-enriched categorical models of the classical sequent calculus. Journal of Pure and Applied Algebra 204 (2006) 21–78. (Manuscript available at http://www.cs.bath.ac.uk/~pym/oecm.pdf)CrossRefGoogle Scholar
Gentzen, G. (1934) Untersuchungen über das logische Schliesen. Mathematische Zeitschrift 39 176–210, 405–431.CrossRefGoogle Scholar
Girard, J.-Y. (1987) Linear logic. Theoret. Comp. Sci. 50 1–102.CrossRefGoogle Scholar
Girard, J.-Y. (1989) Geometry of interaction I: Interpretation of system F. In: Logic Colloqium (Padova, 1988). Stud. Logic Found. Math. 127 221–260.CrossRefGoogle Scholar
Girard, J.-Y. (1990) Geometry of interaction II: Deadlock-free algorithms. In: Proceedings COLOG (Tallinn, 1988). Springer-Verlag Lecture Notes in Computer Science 417 76–93.CrossRefGoogle Scholar
Girard, J.-Y. (1995) Geometry of interaction III: Accommodating the additives. In: Advances in Linear Logic (Ithaca, NY, 1993). London Math. Soc. Lecture Note Ser. 222 329–389.Google Scholar
Girard, J.-Y., Lafont, Y. and Taylor, P. (1989) Proofs and Types, Cambridge University Press.Google Scholar
Haghverdi, E. and Scott, P. (2006) A categorical model for the geometry of interaction. Theoretical Computer Science 350 (2)252–274.CrossRefGoogle Scholar
Hasegawa, M. (2002) Classical linear logic of implications. In: Proc. 11th Annual Conference of the European Association for Computer Science Logic (CSL'02), Edinburgh. Springer-Verlag Lecture Notes in Computer Science 2471.CrossRefGoogle Scholar
Hyland, J. (2002) Proof theory in the abstract. Ann. of Pure Applied Logic 114 (1-3)43–78.CrossRefGoogle Scholar
Hyland, J. (2004) Abstract interpretation of proofs: Classical propositional calculus. In: Proceedings CSL 2004. Springer-Verlag Lecture Notes in Computer Science 3210 6–21.CrossRefGoogle Scholar
Hyland, J. and Schalk, A. (2003) Glueing and orthogonality for models of linear logic. Theoretical Computer Science 294 183–231.CrossRefGoogle Scholar
Joyal, A., Street, R. and Verity, D. (1996) Traced monoidal categories. Math. Proc. Camb. Phil. Soc. 119 447–468.CrossRefGoogle Scholar
Kelly, G. and Laplaza, M. (1980) Coherence for compact closed categories. J. Pure Applied Algebra 19 193–213.CrossRefGoogle Scholar
Lamarche, F. and Straßburger, L. (2004) Naming proofs in propositional classical logic (submitted).CrossRefGoogle Scholar
Loader, R. (1994) Models of Lambda Calculi and Linear Logic: Structural, Equational and Proof-theoretic Characterisations, Ph.D. thesis, St. Hugh's College, Oxford.Google Scholar
McKinley, R. (2006) Categorical models of first-order classical proofs, Ph.D. thesis, University of Bath.Google Scholar
Ong, C.-H. L. (1996) A semantic view of classical proofs. In: Proc. LICS 96, IEEE Computer Society Press, 230–241.Google Scholar
Parigot, M. (1992) λμ-calculus: an algorithmic interpretation of classical natural deduction. In: Proceedings of the International Conference on Logic Programming and Automated Reasoning LPAR'92. Springer-Verlag Lecture Notes in Computer Science 624 190–201.CrossRefGoogle Scholar
Plotkin, G. (1975) Call-by-name, call-by-value, and the λ-calculus. Theoretical Computer Science 1 125–159.CrossRefGoogle Scholar
Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study, Almquist and Wiksell, Stockholm.Google Scholar
Pym, D. and Ritter, E. (2001) On the semantics of classical disjunction. J. Pure Applied Algebra 159 315–338.CrossRefGoogle Scholar
Robinson, E. (2003) Proof Nets for Classical Logic. J. Logic Computat. 13 (5)777–797.CrossRefGoogle Scholar
Selinger, P. (2001) Control categories and duality: on the categorical semantics of the lambda-mu calculus. Mathematical Structures in Computer Science 11 207–260.CrossRefGoogle Scholar
Tan, A. (1997) Full completeness for models of linear logic, Ph.D. thesis, University of Cambridge.Google Scholar
Troelstra, A. and Schwichtenberg, H. (1996) Basic Proof Theory, Cambridge University Press.Google Scholar
Wadler, P. (2003) Call-by-value is dual to call-by-name. In: Proc. International Conference on Functional Programming. ACM SIGPLAN Notices 38 (9) 189–201.CrossRefGoogle Scholar