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A resource aware semantics for a focused intuitionistic calculus

Published online by Cambridge University Press:  22 May 2017

DELIA KESNER  and
DANIEL VENTURA
DELIA KESNER
Affiliation:
IRIF, CNRS, Univ. Paris-Diderot, Paris, France Email: [email protected]
DANIEL VENTURA
Affiliation:
INF, Univ. Federal de Goiás, Goiânia, Brazil Email: [email protected]
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Abstract

We investigate a new computational interpretation for an intuitionistic focused sequent calculus which is compatible with a resource aware semantics. For that, we associate to Herbelin's syntax a type system based on non-idempotent intersection types, together with a set of reduction rules – inspired from the substitution at a distance paradigm – that preserves (and decreases the size of) typing derivations. The non-idempotent approach allows us to use very simple combinatorial arguments, only based on this measure decreasingness, to characterize linear-head and strongly normalizing terms by means of typability. For the sake of completeness, we also study typability (and the corresponding strong normalization characterization) in the calculus obtained from the former one by projecting the explicit cuts.

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Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Special Issue 1: Special Issue: Best Papers Presented at ICTAC 2015 , January 2019 , pp. 93 - 126
Copyright
Copyright © Cambridge University Press 2017 

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References

Accattoli, B. (2012). An abstract factorization theorem for explicit substitutions. In: LIPIcs, vol. 15, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 6–21.Google Scholar
Accattoli, B., Bonelli, E., Kesner, D. and Lombardi, C. (2014). A nonstandard standardization theorem. In: Sewell, P. (ed.) Proceedings of the 41st Annual ACM Symposium on Principles of Programming Languages (POPL), ACM, 659–670.Google Scholar
Accattoli, B. and Kesner, D. (2010). The structural lambda-calculus. In: Dawar, A. and Veith, H. (eds.) Proceedings of 24th EACSL Conference on Computer Science Logic (CSL), Lecture Notes in Computer Science, vol. 6247, Springer-Verlag, 381–395.Google Scholar
Andreoli, J. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2 (3) 297–347.Google Scholar
Baader, F. and Nipkow, T. (1998). Term Rewriting and All That, Cambridge University Press.Google Scholar
Barendregt, H. (1984). The Lambda Calculus: Its Syntax and Semantics (revised edition), Studies in Logic and the Foundations of Mathematics, vol. 103, Elsevier Science, Amsterdan, The Netherlands.Google Scholar
Barendregt, H., Coppo, M. and Dezani-Ciancaglini, M. (1983). A filter lambda model and the completeness of type assignment. Bulletin of Symbolic Logic 48 (4) 931–940.Google Scholar
Bernadet, A. and Lengrand, S. (2011). Complexity of strongly normalising λ-terms via non-idempotent intersection types. In: Hofmann, M. (ed.) Proceedings of the 14th International Conference on Foundations of Software Science and Computation Structures (FOSSACS), Lecture Notes in Computer Science, vol. 6604, Springer-Verlag.Google Scholar
Bernadet, A. and Lengrand, S. (2013). Non-idempotent intersection types and strong normalisation. Logical Methods in Computer Science 9 (4). doi: 10.2168/LMCS-9(4:3)2013Google Scholar
Boudol, G., Curien, P.-L. and Lavatelli, C. (1999). A semantics for lambda calculi with resources. Mathematical Structures in Computer Science 9 (4) 437–482.Google Scholar
Bucciarelli, A., Kesner, D. and Ronchi Della Rocca, S. (2014). The inhabitation problem for non-idempotent intersection types. In: IFIP Theoretical Computer Science (TCS), LNCS, Springer-Verlag, 341–354.Google Scholar
Coppo, M. and Dezani-Ciancaglini, M. (1978). A new type-assignment for lambda terms. Archiv für Mathematische Logik und Grundlagenforschung 19 (1) 139–156.Google Scholar
Coppo, M. and Dezani-Ciancaglini, M. (1980). An extension of the basic functionality theory for the λ-calculus. Notre Dame, Journal of Formal Logic 21 (4) 685–693.Google Scholar
Coppo, M., Dezani-Ciancaglini, M. and Venneri, B. (1981). Functional characters of solvable terms. Mathematical Logic Quarterly 27 (2–6) 45–58.Google Scholar
Damiani, F. and Giannini, P. (1994). A decidable intersection type system based on relevance. In: International Symposium on Theoretical Computer Science (TACS), Lecture Notes in Computer Science, vol. 789, Springer-Verlag, 707–725.Google Scholar
Danos, V. and Regnier, L. (2003). Head Linear Reduction. Available at http://iml.univ-mrs.fr/~regnier/articles/pam.ps.gz.Google Scholar
De Benedetti, E. and Ronchi Della Rocca, S. (2013). Bounding normalization time through intersection types. In: Paolini, L. (ed.) Proceedings of 6th Workshop on Intersection Types and Related Systems (ITRS), EPTCS, Cornell University Library, 48–57.Google Scholar
de Carvalho, D. (2007). Sémantiques de la logique linéaire et temps de calcul. These de doctorat, Université Aix-Marseille II.Google Scholar
Díaz, J., Lanese, I. and Sangiorgi, D. (eds.) (2014). IFIP Theoretical Computer Science (TCS), LNCS, Springer-Verlag.Google Scholar
Dyckhoff, R. and Urban, C. (2003). Strong normalization of herbelin's explicit substitution calculus with substitution propagation. Journal of Logic and Computation 13 (5) 689–706.Google Scholar
Espírito Santo, J. (2000). Revisiting the correspondence between cut elimination and normalisation. In: Montanari, U., Rolim, J. D. P. and Welzl, E. (eds.) Proceedings of the Automata, Languages and Programming, 27th International Colloquium, ICALP 2000, Lecture Notes in Computer Science, vol. 1853, Springer-Verlag, 600–611.Google Scholar
Espírito Santo, J. (2009). The lambda-calculus and the unity of structural proof theory. Theory of Computing Systems 45 (4) 963–994.Google Scholar
Espírito Santo, J., Ivetic, J. and Likavec, S. (2012). Characterising strongly normalising intuitionistic terms. Fundamenta Informaticae 121 (1–4) 83–120.Google Scholar
Gardner, P. (1994). Discovering needed reductions using type theory. In: Hagiya, M. and Mitchell, J. C. (eds.) Proceedings of the Theoretical Aspects of Computer Software, International Conference TACS '94, Lecture Notes in Computer Science, vol. 789, Springer, 555–574.Google Scholar
Gentzen, G. (1969). The collected papers of Gerhard Gentzen. In: Szabo, M. E. (ed.) Studies in Logic and the Foundations of Mathematics, North-Holland Pub. Co.Google Scholar
Ghilezan, S., Ivetic, J., Lescanne, P. and Likavec, S. (2011). Intersection types for the resource control lambda calculi. In: Antonio Cerone, P. P. (ed.) Proceedings of the 8th International Colloquium on Theoretical Aspects of Computing (ICTAC), Lecture Notes in Computer Science, vol. 6916, Springer-Verlag, 116–134.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50 1–102.Google Scholar
Girard, J.-Y. (1996). Proof-nets: The parallel syntax for proof-theory. In: Aglianò, P. and Ursini, A. (eds.) Logic and Algebra, Lecture Notes in Pure and Applied Mathematics, vol. 180, CRC Press, 97–124.Google Scholar
Herbelin, H. (1995). A lambda-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L. and Tiuryn, J. (eds.) The 8th International Workshop on Computer Science Logic (CSL), Lecture Notes in Computer Science, vol. 933, Springer-Verlag, 61–75.Google Scholar
Kesner, D. and Ventura, D. (2014a). Quantitative types for intuitionistic calculi. Technical Report hal-00980868, Paris Cité Sorbonne.Google Scholar
Kesner, D. and Ventura, D. (2014b). Quantitative types for the linear substitution calculus. In: IFIP Theoretical Computer Science (TCS), LNCS, Springer-Verlag, 296–310.Google Scholar
Kesner, D. and Ventura, D. (2015). A resource aware computational interpretation for herbelin's syntax. In: Leucker, M., Rueda, C. and Valencia, F. D. (eds.) 12th International Colloquium Theoretical Aspects of Computing – ICTAC 2015, Lecture Notes in Computer Science, vol. 9399, Springer-Verlag, 388–403.Google Scholar
Kfoury, A. (1996). A linearization of the lambda-calculus and consequences. Technical report, Boston Universsity.Google Scholar
Kfoury, A. and Wells, J. B. (2004). Principality and type inference for intersection types using expansion variables. Theoretical Computer Science 311 (1–3) 1–70.Google Scholar
Kikuchi, K. (2014). Uniform proofs of normalisation and approximation for intersection types. In Proceedings of the 7th Workshop on Intersection Types and Related Systems (ITRS), Vienna, Austria.Google Scholar
Krivine, J.-L. (1993). Lambda-Calculus, Types and Models, Ellis Horwood.Google Scholar
Lengrand, S., Lescanne, P., Dougherty, D., Dezani-Ciancaglini, M. and van Bakel, S. (2004). Intersection types for explicit substitutions. Information and Computation 189 (1) 17–42.Google Scholar
Miller, D., Nadathur, G., Pfenning, F. and Scedrov, A. (1991). Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51 (1–2) 125–157.Google Scholar
Milner, R. (2007). Local bigraphs and confluence: Two conjectures: (extended abstract). Electronic Notes in Theoretical Computer Science 175 (3) 65–73.Google Scholar
Neergaard, P. M. and Mairson, H. G. (2004). Types, potency, and idempotency: why nonlinearity and amnesia make a type system work. In: Okasaki, C. and Fisher, K. (eds.) Proceedings of the 9th ACM SIGPLAN International Conference on Functional Programming (ICFP), ACM, 138–149.Google Scholar
Ong, L. and Ramsay, S. J. (2011). Verifying higher-order functional programs with pattern matching algebraic data types. In: Ball, T. and Sagiv, M. (eds.) Proceedings of the 38th Annual ACM Symposium on Principles of Programming Languages (POPL), ACM, 587–598.Google Scholar
Pagani, M. and Ronchi Della Rocca, S. (2011). Solvability in resource lambda-calculus. In: Ong, L. (ed.) Foundations of Software Science and Computation Structures (FOSSACS), Lecture Notes in Computer Science, vol. 6014, Springer-Verlag, 358–373.Google Scholar
Pottinger, G. (1977). Normalization as a homomorphic image of cut-elimination. Annals of Mathematical Logic 12 323–357.Google Scholar
Pottinger, G. (1980). A type assignment for the strongly normalizable λ-terms. In: Seldin, J. P. and Hindley, J. R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, 561–578.Google Scholar
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Phd thesis, Stockholm University.Google Scholar
Regnier, L. (1994). Une équivalence sur les lambda-termes. Theoretical Computer Science 2 (126) 281–292.Google Scholar
Urzyczyn, P. (1999). The emptiness problem for intersection types. Journal of Symbolic Logic 64 (3) 1195–1215.Google Scholar
Valentini, S. (2001). An elementary proof of strong normalization for intersection types. Archive of Mathematical Logic 40 (7) 475–488.Google Scholar
van Bakel, S. (1992). Complete restrictions of the intersection type discipline. Theoretical Computer Science 102 (1) 135–163.Google Scholar
Zucker, J. (1974). The correspondence between cut-elimination and normalization I. Annals of Mathematical Logic 7 (1) 1–112.Google Scholar