Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Linear \( \lambda \mu \) is \( \textsc {CP} \) (more or less)

  • Chapter
  • First Online:
A List of Successes That Can Change the World

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9600))

  • 1441 Accesses

Abstract

In this paper we compare Wadler’s \( \textsc {CP} \) calculus for classical linear processes to a linear version of Parigot’s \( \lambda \mu \) calculus for classical logic. We conclude that linear \( \lambda \mu \) is “more or less” \( \textsc {CP} \), in that it equationally corresponds to a polarized version of \( \textsc {CP} \). The comparison is made by extending a technique from Melliès and Tabareau’s tensor logic that correlates negation with polarization. The polarized \( \textsc {CP} \), which is written \( \textsc {CP}^{\pm } \) and pronounced “\( \textsc {CP} \) more or less,” is an interesting bridge in the landscape of Curry-Howard interpretations of logic.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Pfenning and Griffith (2015) have also studied polarization in their work on intuitionistic session types, where it distinguishes synchronous and asynchronous communication.

  2. 2.

    To give the definitions in this section precisely, it would be necessary to use named holes and substitutions to fill in the holes. For the purposes of this paper we leave these operations informal.

  3. 3.

    The operational semantics we present here is quite a bit different from Selinger (2001) and Wadler (2005). Their \( \mu -\beta \) rules are encompassed by our notion of \( \mu \)-capturing. Their \(\xi \) rule corresponds to the procedure of \( \mu \)-capturing inside a term described here.

  4. 4.

    Drawing the connection to category theory, every object \(A \in \mathcal {C}\) has a dual object \({A}^{\text{ op }} \in {\mathcal {C}}^{\text{ op }}\). Thus A and its dual live in distinct categories.

References

  • Beffara, E.: A concurrent model for linear logic. Electron. Notes Theor. Comput. Sci. 155, 147–168 (2006)

    Article  MATH  Google Scholar 

  • Bellin, G., Scott, P.: On the \(\pi \)-calculus and linear logic. Theor. Comput. Sci. 135(1), 11–65 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Bernardy, J.P., Rosn, D., Smallbone, N.: Compiling linear logic using continuations (2014)

    Google Scholar 

  • Caires, L., Pfenning, F.: Session types as intuitionistic linear propositions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 222–236. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  • Curien, P.L., Herbelin, H.: The duality of computation. ACM SIGPLAN Not. 35(9), 233–243 (2000)

    Article  MATH  Google Scholar 

  • Felleisen, M.: The theory and practice of first-class prompts. In: Proceedings of the 15th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 1988, pp. 180–190. ACM, New York (1988)

    Google Scholar 

  • Gay, S.J., Vasconcelos, V.T.: Linear type theory for asynchronous session types. J. Funct. Program. 20, 19–50 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Girard, J.Y.: Linear logic. Theor. Comput. Sci. 50(1), 1–101 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Girard, J.Y.: A new constructive logic: classic logic. Math. Struct. Comput. Sci. 1, 255–296 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Griffin, T.G.: A formulae-as-type notion of control. In: Proceedings of the 17th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 1990, pp. 47–58. ACM, New York (1990)

    Google Scholar 

  • de Groote, P.: On the relation between the \(\lambda \mu \)-calculus and the syntactic theory of sequential control. In: Pfenning, F. (ed.) Logic Programming and Automated Reasoning. Lecture Notes in Computer Science, vol. 822, pp. 31–43. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  • Honda, K.: Types for dyadic interaction. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 509–523. Springer, Heidelberg (1993)

    Google Scholar 

  • Honda, K., Vasconcelos, V.T., Kubo, M.: Language primitives and type discipline for structured communication-based programming. In: Hankin, C. (ed.) ESOP 1998. LNCS, vol. 1381, pp. 122–138. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  • Kobayashi, N., Pierce, B.C., Turner, D.N.: Linearity and the \(\pi \)-calculus. ACM Trans. Program. Lang. Syst. 21(5), 914–947 (1999)

    Article  Google Scholar 

  • Lafont, Y., Reus, B., Streicher, T.: Continuation semantics or expressing implication by negation. Technical report 9321, Ludwig-Maximilians-Universitt (1993)

    Google Scholar 

  • Laurent, O.: Polarized proof-nets and \(\lambda \mu \)-calculus. Theor. Comput. Sci. 290(1), 161–188 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Laurent, O., Regnier, L.: About translations of classical logic into polarized linear logic. In: Logic in Computer Science, pp. 11–20. IEEE (2003)

    Google Scholar 

  • Lindley, S., Morris, J.G.: Sessions as propositions. Electron. Proc. Theor. Comput. Sci. 155, 9–16 (2014)

    Article  Google Scholar 

  • Lindley, S., Morris, J.G.: A semantics for propositions as sessions. In: Vitek, J. (ed.) ESOP 2015. LNCS, vol. 9032, pp. 560–584. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  • Mazurak, K., Zdancewic, S.: Lolliproc: To concurrency from classical linear logic via Curry-Howard and control. SIGPLAN Not. 45(9), 39–50 (2010)

    Article  MATH  Google Scholar 

  • Melliès, P.A., Tabareau, N.: Resource modalities in tensor logic. Ann. Pure Appl. Logic 161(5), 632–653 (2010). The Third Workshop on Games for Logic and Programming Languages (GaLoP), 5–6 April 2008

    Article  MathSciNet  MATH  Google Scholar 

  • Milner, R.: Functions as processes. Math. Struct. Comput. Sci. 2, 119–141 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Munch-Maccagnoni, G.: Focalisation and classical realisability. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 409–423. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  • Parigot, M.: \(\lambda \mu \)-calculus: An algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) Logic Programming and Automated Reasoning. Lecture Notes in Computer Science, vol. 624, pp. 190–201. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  • Paykin, J., Zdancewic, S., Krishnaswami, N.R.: Linear temporal type theory for event-based reactive programming (2015)

    Google Scholar 

  • Pérez, J.A., Caires, L., Pfenning, F., Toninho, B.: Linear logical relations and observational equivalences for session-based concurrency. Inf. Comput. 239, 254–302 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Pfenning, F., Griffith, D.: Polarized substructural session types. In: Pitts, A. (ed.) FOSSACS 2015. LNCS, vol. 9034, pp. 3–22. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  • Sabry, A., Felleisen, M.: Reasoning about programs in continuation-passing style. LISP Symbolic Comput. 6(3–4), 289–360 (1993)

    Article  Google Scholar 

  • Selinger, P.: Control categories and duality: On the categorical semantics of the lambda-mu calculus. Math. Struct. Comput. Sci. 11, 207–260 (2001). http://journals.cambridge.org/article_S096012950000311X

    Article  MathSciNet  MATH  Google Scholar 

  • Spiwack, A.: A dissection of L (2014)

    Google Scholar 

  • Wadler, P.: Call-by-value is dual to call-by-name. In: Proceedings of the Eighth ACM SIGPLAN International Conference on Functional Programming, ICFP 2003, pp. 189–201. ACM, New York (2003)

    Google Scholar 

  • Wadler, P.: Call-by-value is dual to call-by-name reloaded. In: Giesl, J. (ed.) Term Rewriting and Applications. LNCS, vol. 3467, pp. 185–203. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  • Wadler, P.: Propositions as sessions. SIGPLAN Not. 47(9), 273–286 (2012)

    Article  MATH  Google Scholar 

  • Wadler, P.: Propositions as sessions. J. Funct. Program. 24, 384–418 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Zeilberger, N.: On the unity of duality. Ann. Pure Appl. Logic 153(13), 66–96 (2008). Special Issue: Classical Logic and Computation (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Thanks to Neel Krishnaswami, Sam Lindley, and the paper’s reviewers for their valuable insights. We would also like to thank Phil Wadler for inspiring a fascination with the Curry-Howard correspondence. This material is based in part upon work supported by the NSF Graduate Research Fellowship under Grant No. DGE-1321851 and by NSF Grant No. CCF-1421193.

Author information

Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, PA, 19104, USA

    Jennifer Paykin & Steve Zdancewic

Authors
  1. Jennifer Paykin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steve Zdancewic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jennifer Paykin .

Editor information

Editors and Affiliations

  1. Informatics Forum, University of Edinburgh Informatics Forum, Edinburgh, United Kingdom

    Sam Lindley

  2. University of Strathclyde , Glasgow, United Kingdom

    Conor McBride

  3. University of Glasgow , Glasgow, United Kingdom

    Phil Trinder

  4. Informatics Forum, University of Edinburgh Informatics Forum, Edinburgh, United Kingdom

    Don Sannella

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Paykin, J., Zdancewic, S. (2016). Linear \( \lambda \mu \) is \( \textsc {CP} \) (more or less). In: Lindley, S., McBride, C., Trinder, P., Sannella, D. (eds) A List of Successes That Can Change the World. Lecture Notes in Computer Science(), vol 9600. Springer, Cham. https://doi.org/10.1007/978-3-319-30936-1_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30936-1_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30935-4

  • Online ISBN: 978-3-319-30936-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation