Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T08:48:47.749Z Has data issue: false hasContentIssue false
  • English
  • Français

An introduction to differential linear logic: proof-nets, models and antiderivatives

Published online by Cambridge University Press:  09 February 2017

THOMAS EHRHARD
THOMAS EHRHARD*
Affiliation:
CNRS, IRIF, UMR 8243, Univ Paris Diderot, Sorbonne Paris Cité F-75205 Paris, France Email: thomas.ehrhard@pps.univ-paris-diderot.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Differential linear logic enriches linear logic with additional logical rules for the exponential connectives, dual to the usual rules of dereliction, weakening and contraction. We present a proof-net syntax for differential linear logic and a categorical axiomatization of its denotational models. We also introduce a simple categorical condition on these models under which a general antiderivative operation becomes available. Last, we briefly describe the model of sets and relations and give a more detailed account of the model of finiteness spaces and linear and continuous functions.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Special Issue 7: Differential Linear Logic, Nets and Other Quantitative and Parallel Approaches to Proof-Theory , August 2018 , pp. 995 - 1060
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. (1993). Computational interpretations of linear logic. Theoretical Computer Science 111 3–57.CrossRefGoogle Scholar
Bierman, G. (1995). What is a categorical model of intuitionistic linear logic? In: Dezani-Ciancaglini, M. and Plotkin, G.D. (eds.) Proceedings of the second Typed Lambda-Calculi and Applications Conference, Lecture Notes in Computer Science, vol. 902, Springer-Verlag, 73–93.Google Scholar
Blute, R., Cockett, R. and Seely, R. (2006). Differential categories. Mathematical Structures in Computer Science 16 (6) 1049–1083.CrossRefGoogle Scholar
Blute, R., Ehrhard, T. and Tasson, C. (2012). A convenient differential category. Cahiers de Topologie et Géométrie Différentielle Catégoriques 53 (3) 211–232.Google Scholar
Boudol, G., Curien, P.-L. and Lavatelli, C. (1999). A semantics for lambda calculi with resource. Mathematical Structures in Computer Science 9 (4) 437–482.Google Scholar
Bucciarelli, A., Carraro, A., Ehrhard, T. and Manzonetto, G. (2011). Full abstraction for resource calculus with tests. In: Bezem, M. (ed.) CSL, LIPIcs, vol. 12, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik.Google Scholar
Danos, V. and Regnier, L. (1999). Reversible, irreversible and optimal lambda-machines. Theoretical Computer Science 227 (1–2) 273–291.Google Scholar
De Bruijn, N.G. (1987). Generalizing Automath by means of a lambda-typed lambda calculus. In: Kueker, D.W., Lopez-Escobar, E.G.K. and Smith, C.H. (eds.) Mathematical Logic and Theoretical Computer Science, Lecture Notes in Pure and Applied Mathematics, vol. 106, Marcel Dekker, 71–92. Reprinted in: Selected papers on Automath, Studies in Logic, vol. 133, North-Holland, 313–337.Google Scholar
Ehrhard, T. (2002). On Köthe sequence spaces and linear logic. Mathematical Structures in Computer Science 12 579–623.Google Scholar
Ehrhard, T. (2005). Finiteness spaces. Mathematical Structures in Computer Science 15 (4) 615–646.Google Scholar
Ehrhard, T. (2010). A finiteness structure on resource terms. In: LICS, IEEE Computer Society, 402–410.Google Scholar
Ehrhard, T. (2012). The Scott model of linear logic is the extensional collapse of its relational model. Theoretical Computer Science 424 20–45.Google Scholar
Ehrhard, T. (2014). A new correctness criterion for MLL proof nets. In: Henzinger, T.A. and Miller, D. (eds.) Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS '14, Vienna, Austria, July 14–18, 2014, ACM, 8.Google Scholar
Ehrhard, T. and Regnier, L. (2003). The differential lambda-calculus. Theoretical Computer Science 309 (1–3) 1–41.Google Scholar
Ehrhard, T. and Regnier, L. (2006). Böhm trees, Krivine machine and the Taylor expansion of ordinary lambda-terms. In: Beckmann, A., Berger, U., Löwe, B. and Tucker, J.V. (eds.) Logical Approaches to Computational Barriers, Lecture Notes in Computer Science, vol. 3988, Springer-Verlag, 186–197. Long version available on http://www.pps.univ-paris-diderot.fr/~ehrhard/.Google Scholar
Ehrhard, T. and Regnier, L. (2008). Uniformity and the Taylor expansion of ordinary lambda-terms. Theoretical Computer Science 403 (2–3) 347–372.Google Scholar
Fernández, M. and Mackie, I. (1999). A calculus for interaction nets. In: Nadathur, G. (ed.) PPDP, Lecture Notes in Computer Science, vol. 1702, Springer-Verlag, 170–187.Google Scholar
Fiore, M.P. (2007) Differential structure in models of multiplicative biadditive intuitionistic linear logic. In: DellaRocca, S.R. Rocca, S.R. (ed.) TLCA, Lecture Notes in Computer Science, vol. 4583, Springer, 163–177.Google Scholar
Gimenez, S. (2011). Realizability proof for normalization of full differential linear logic. In: Luke Ong, C.-H. (ed.) TLCA, Lecture Notes in Computer Science, vol. 6690 Springer-Verlag, 107–122.Google Scholar
Girard, J.-Y. (1986). The system F of variable types, fifteen years later. Theoretical Computer Science 45 159–192.CrossRefGoogle Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1–102.CrossRefGoogle Scholar
Girard, J.-Y. (1988). Normal functors, power series and the λ-calculus. Annals of Pure and Applied Logic 37 129–177.Google Scholar
Huth, M. (1993). Linear domains and linear maps. In: Brookes, S.D., Main, M.G., Melton, A., Mislove, M.W. and Schmidt, D.A. (eds.) MFPS, Lecture Notes in Computer Science, vol. 802, Springer-Verlag, 438–453.Google Scholar
Krivine, J.-L. (1985). Un interpréteur du lambda-calcul. Unpublished note. Available at: https://www.irif.fr/~krivine/articles/interprt.pdfGoogle Scholar
Krivine, J.-L. (2007). A call-by-name lambda-calculus machine. Higher-Order and Symbolic Computation 20 (3) 199–207.Google Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, Springer-Verlag.CrossRefGoogle Scholar
Mackie, I. and Sato, S. (2008). A calculus for interaction nets based on the linear chemical abstract machine. Electronic Notes in Theoretical Computer Science 192 (3) 59–70.Google Scholar
Melliès, P.-A. (2009). Categorical semantics of linear logic. In: Interactive Models of Computation and Program Behavior. Panoramas et Synthèses, vol. 27, Soc. Math. France, Paris, 1–196.Google Scholar
Pagani, M. (2009). The cut-elimination theorem for differential nets with promotion. In: Curien, P.-L. (ed.) Proceedings of the 9th International Conference, TLCA 2009, Lecture Notes in Computer Science, vol. 5608, Springer, 219–233.Google Scholar
Pagani, M. and Tranquilli, P. (2009). Parallel reduction in resource lambda-calculus. In: Hu, Z. (ed.) APLAS, Lecture Notes in Computer Science, vol. 5904, Springer, 226–242.Google Scholar
Pagani, M. and Tranquilli, P. (2011). The conservation theorem for differential nets. Mathematical Structures in Computer Science. Forthcoming.Google Scholar
Retoré, C. (2003). Handsome proof-nets: Perfect matchings and cographs. Theoretical Computer Science 294 (3) 473–488.CrossRefGoogle Scholar
Tasson, C. (2009a). Algebraic totality, towards completeness. In: Curien (2009), 325–340.Google Scholar
Tasson, C. (2009b). Sémantiques et syntaxes vectorielles de la logique linéaire. Thèse de doctorat, Université Paris Diderot – Paris 7.Google Scholar
Tasson, C. and Vaux, L. (2010). Transport of finiteness structures and applications. Mathematical Structures in Computer Science. Forthcoming.Google Scholar
Tranquilli, P. (2009). Confluence of pure differential nets with promotion. In: Grädel, E. and Kahle, R. (eds.) CSL, Lecture Notes in Computer Science, vol. 5771, Springer-Verlag, 500–514.Google Scholar
Vaux, L. (2005). The differential lambda-mu calculus. Theoretical Computer Science 379 (1–2) 166–209.CrossRefGoogle Scholar
Vaux, L. (2009). The algebraic lambda-calculus. Mathematical Structures in Computer Science 19 (5) 1029–1059.Google Scholar
Winskel, G. (2004). Linearity and non linearity in distributed computation. In: Ehrhard, T., Girard, J.-Y., Ruet, P. and Scott, P. (eds.) Linear Logic in Computer Science, London Mathematical Society Lecture Notes Series, vol. 316, Cambridge University Press.Google Scholar