research-article

Type inference in intuitionistic linear logic

Authors:

Patrick Baillot,

Martin HofmannAuthors Info & Claims

PPDP '10: Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming

Pages 219 - 230

https://doi.org/10.1145/1836089.1836118

Published: 26 July 2010 Publication History

Abstract

We study the type checking and type inference problems for intuitionistic linear logic: given a System F typed λ-term, (i) for an alleged linear logic type, determine whether there exists a corresponding typing derivation in linear logic (type checking) ii) provide a concise description of all possible corresponding linear logic typings (type inference).

We solve these problems using a novel algorithmic type system for linear logic whose typing rules carry arithmetic side conditions describing essentially the nesting depth of (proof-net) boxes. By understanding these side conditions as unknowns we then reduce type inference to solving a system of arithmetic constraints. We show that these constraint systems fall into a tractable class hence leading to an efficient (polynomial-time) solution.

There are two important restrictions: first, our source language is typed System F rather than untyped lambda calculus; this is necessary because type inference for System F is known to be undecidable. Second, we assume that sharing is made explicit in the input, thus we do not try to automatically infer opportunities for sharing identical subterms. Relieving the latter restriction is left as a challenge for future work.

References

[1]

V. Atassi, P. Baillot, and K. Terui. Verification of Ptime reducibility for system F terms: Type inference in dual light affine logic. Logical Methods in Computer Science, 3(4), 2007.

[2]

P. Baillot and K. Terui. A Feasible Algorithm for Typing in Elementary Affine Logic. In Proceedings of TLCA'05, volume 3461 of LNCS, pages 55--70. Springer, 2005.

Digital Library

[3]

E. Barendsen and S. Smetsers. Uniqueness type inference. In M. V. Hermenegildo and S. D. Swierstra, editors, PLILP, volume 982 of Lecture Notes in Computer Science, pages 189--206. Springer, 1995. ISBN 3-540-60359-X.

Digital Library

[4]

P. Coppola and S. Martini. Typing lambda terms in elementary logic with linear constraints. In Proceedings of TLCA'01, volume 2044 of LNCS, pages 76--90, 2001.

Digital Library

[5]

P. Coppola and S. Martini. Optimizing optimal reduction. a type inference algorithm for elementary affine logic. ACM Transactions on Computational Logic, 7(2):219--260, 2006.

Digital Library

[6]

P. Coppola and S. Ronchi Della Rocca. Principal typing for lambda calculus in elementary affine logic. Fundam. Inform., 65(1-2):87--112, 2005.

Digital Library

[7]

V. Danos, J.-B. Joinet, and H. Schellinx. On the linear decoration of intuitionistic derivations. Archive for Mathematical Logic, 33(6), 1994.

[8]

B. Dutertre and L. de Moura. The YICES SMT Solver. Available at: yices.csl.sri.com/tool-paper.pdf.

[9]

R. Ennals, R. Sharp, and A. Mycroft. Linear types for packet processing. In D. A. Schmidt, editor, ESOP, volume 2986 of Lecture Notes in Computer Science, pages 204--218. Springer, 2004. ISBN 3-540-21313-9.

[10]

M. Fluet, G. Morrisett, and A. J. Ahmed. Linear regions are all you need. In P. Sestoft, editor, ESOP, volume 3924 of Lecture Notes in Computer Science, pages 7--21. Springer, 2006. ISBN 3-540-33095-X.

Digital Library

[11]

J.-Y. Girard. Linear logic. Theor. Comput. Sci., 50:1--102, 1987.

Digital Library

[12]

S. Guerrini, S. Martini, and A. Masini. An analysis of (linear) exponentials based on extended sequents. Logic Journal of the IGPL, 6(5): 735--753, 1998.

[13]

M. Hofmann. A type system for bounded space and functional in-place update. Nordic Journal of Computing, 7(4):258--289, 2000.

Digital Library

[14]

P. Jonsson and C. Bäckström. A linear-programming approach to temporal reasoning. In Proceedings of the 13th (US) National Conference on Artificial Intelligence (AAAI-96), pages 1235--1240, Portland, OR, USA, 1996.

Digital Library

[15]

I. Mackie. Lilac: A functional programming language based on linear logic. J. Funct. Program., 4(4):395--433, 1994.

[16]

S. Martini and A. Masini. On the fine structure of the exponential rule. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic (Proc. of the Workshop on Linear Logic, Cornell University, June 1993), number 222. Cambridge University Press, 1995. URL citeseer.ist.psu.edu/martini93fine.html.

Digital Library

[17]

F. Pfenning and H.-C. Wong. On a modal lambda calculus for S4. Electr. Notes Theor. Comput. Sci., 1, 1995.

[18]

L. Roversi. A compiler from Curry-typed λ-terms to linear-λ-terms. In Theoretical Computer Science: Proceedings of the Fourth Italian Conference, pages 330--344, L'Aquila (Italy), October 1992. World Scientific.

[19]

H. Schellinx. The Noble Art of Linear Decorating. ILLC Dissertation Series 1994-1, Institute for Language, Logic and Computation, University of Amsterdam, 1994.

[20]

P.Wadler. Is there a use for linear logic? In Proceedings of Symposium on Partial Evaluation and Semantics-Based Program Manipulation, PEPM'91, volume 26(9) of SIGPLAN Notices, pages 255--273, 1991.

Digital Library

[21]

D. Walker. Substructural Type Systems. In B. C. Pierce, editor, Advanced Topics in Types and Programming Languages, chapter 1. MIT Press, 2005.

Cited By

Matsuda K(2020)Modular Inference of Linear Types for Multiplicity-Annotated ArrowsProgramming Languages and Systems10.1007/978-3-030-44914-8_17(456-483)Online publication date: 18-Apr-2020
https://doi.org/10.1007/978-3-030-44914-8_17
Dunchev CCoen CTassi EAlves SDowek GLicata D(2016)Implementing HOL in an Higher Order Logic Programming LanguageProceedings of the Eleventh Workshop on Logical Frameworks and Meta-Languages: Theory and Practice10.1145/2966268.2966272(1-10)Online publication date: 23-Jun-2016
https://dl.acm.org/doi/10.1145/2966268.2966272
Bugliesi MCalzavara SEigner FMaffei M(2015)Affine Refinement Types for Secure Distributed ProgrammingACM Transactions on Programming Languages and Systems10.1145/274301837:4(1-66)Online publication date: 13-Aug-2015
https://dl.acm.org/doi/10.1145/2743018
Show More Cited By

Index Terms

Type inference in intuitionistic linear logic
1. Mathematics of computing
  1. Continuous mathematics
    1. Calculus
      1. Lambda calculus
  2. Mathematical analysis
    1. Calculus
      1. Lambda calculus
2. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Type structures

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Information

Published In

cover image ACM Other conferences

PPDP '10: Proceedings of the 12th international ACM SIGPLAN symposium on Principles and practice of declarative programming

July 2010

266 pages

ISBN:9781450301329

DOI:10.1145/1836089

Conference Chairs:
Temur Kutsia
RISC, Johannes Kepler University Linz, Austria
,
Wolfgang Schreiner
RISC, Johannes Kepler University Linz, Austria
,
Program Chair:
Maribel Fernández
King's College London, UK

Copyright © 2010 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Johannes Kepler University, Linz, Austria
SIGPLAN: ACM Special Interest Group on Programming Languages

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 26 July 2010

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PPDP '10

PPDP '10: Principles and Practice of Declarative Programming

July 26 - 28, 2010

Hagenberg, Austria

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PPDP '10 Paper Acceptance Rate 21 of 57 submissions, 37%;

Overall Acceptance Rate 230 of 486 submissions, 47%

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Cited By

Matsuda K(2020)Modular Inference of Linear Types for Multiplicity-Annotated ArrowsProgramming Languages and Systems10.1007/978-3-030-44914-8_17(456-483)Online publication date: 18-Apr-2020
https://doi.org/10.1007/978-3-030-44914-8_17
Dunchev CCoen CTassi EAlves SDowek GLicata D(2016)Implementing HOL in an Higher Order Logic Programming LanguageProceedings of the Eleventh Workshop on Logical Frameworks and Meta-Languages: Theory and Practice10.1145/2966268.2966272(1-10)Online publication date: 23-Jun-2016
https://dl.acm.org/doi/10.1145/2966268.2966272
Bugliesi MCalzavara SEigner FMaffei M(2015)Affine Refinement Types for Secure Distributed ProgrammingACM Transactions on Programming Languages and Systems10.1145/274301837:4(1-66)Online publication date: 13-Aug-2015
https://dl.acm.org/doi/10.1145/2743018
D'Antoni LGaboardi MGallego Arias EHaeberlen APierce BLazarus RKfoury ABeal J(2013)Sensitivity analysis using type-based constraintsProceedings of the 1st annual workshop on Functional programming concepts in domain-specific languages10.1145/2505351.2505353(43-50)Online publication date: 22-Sep-2013
https://dl.acm.org/doi/10.1145/2505351.2505353
Eigner FMaffei M(2013)Differential Privacy by Typing in Security ProtocolsProceedings of the 2013 IEEE 26th Computer Security Foundations Symposium10.1109/CSF.2013.25(272-286)Online publication date: 26-Jun-2013
https://dl.acm.org/doi/10.1109/CSF.2013.25
Bugliesi MCalzavara SEigner FMaffei M(2013)Logical foundations of secure resource management in protocol implementationsProceedings of the Second international conference on Principles of Security and Trust10.1007/978-3-642-36830-1_6(105-125)Online publication date: 16-Mar-2013
https://dl.acm.org/doi/10.1007/978-3-642-36830-1_6

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