research-article

Maximal sharing in the Lambda calculus with letrec

Authors:

Clemens Grabmayer,

Jan RochelAuthors Info & Claims

ACM SIGPLAN Notices, Volume 49, Issue 9

Pages 67 - 80

https://doi.org/10.1145/2692915.2628148

Published: 19 August 2014 Publication History

Abstract

Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the lambda calculus with letrec. We introduce a notion of 'maximal compactness' for λ_letrec-terms among all terms with the same infinite unfolding. Instead of defined purely syntactically, this notion is based on a graph semantics. λ_letrec-terms are interpreted as first-order term graphs so that unfolding equivalence between terms is preserved and reflected through bisimilarity of the term graph interpretations. Compactness of the term graphs can then be compared via functional bisimulation.

We describe practical and efficient methods for the following two problems: transforming a λ_letrec-term into a maximally compact form; and deciding whether two λ_letrec-terms are unfolding-equivalent. The transformation of a λ_letrec-terms L into maximally compact form L₀ proceeds in three steps: (i) translate L into its term graph G = [[L]] ; (ii) compute the maximally shared form of G as its bisimulation collapse G₀ ; (iii) read back a λ_letrec-term L₀ from the term graph G₀ with the property [[L₀]] = G₀. Then L₀ represents a maximally shared term graph, and it has the same unfolding as L.

The procedure for deciding whether two given λ_letrec-terms L₁ and L₂ are unfolding-equivalent computes their term graph interpretations [[L₁]] and [[L₂]], and checks whether these are bisimilar.

For illustration, we also provide a readily usable implementation.

Supplementary Material

ZIP File (icfp38.zip)

"Maximal Sharing in the Lambda Calculus with letrec" for ICFP 2014 by Clemens Grabmayer and Jan Rochel.

Download
688.33 KB

References

[1]

A. Asperti and S. Guerrini. The Optimal Implementation of Functional Programming Languages. Cambridge University Press, 1998.

Digital Library

[2]

T. Balabonski. A unified approach to fully lazy sharing. In Proceedings of POPL '12, pages 469--480, New York, NY, USA, 2012. ACM.

Digital Library

[3]

R. S. Bird and R. Patterson. de Bruijn notation as a nested datatype. Journal of Functional Programming, 9(1):77--91, 1999.

Digital Library

[4]

S. Blom. Term Graph Rewriting - Syntax and Semantics. PhD thesis, Vrije Universiteit Amsterdam, 2001.

[5]

M. v. d. Brand and P. Klint. ATERMs for manipulation and exchange of structured data: It's all about sharing. Information and Software Technology, 49(1):55--64, 2007.

Digital Library

[6]

O. Chitil. Common Subexpressions Are Uncommon in Lazy Functional Languages. In Selected Papers from the 9th International Workshop IFL (IFL '97), pages 53--71, London, UK, UK, 1998. Springer-Verlag.

Digital Library

[7]

O. Danvy and U. P. Schultz. Lambda-lifting in quadratic time. Journal of Functional and Logic Programming, 2004, 2004.

[8]

N. G. de Bruijn. Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, with Applic. to the Church-Rosser Theorem. Indagationes Mathematicae, 34:381--392, 1972.

[9]

A. L. de Medeiros Santos. Compilation by Transformation in Non-Strict Functional Languages. PhD thesis, University of Glasgow, 1995.

[10]

B. Goldberg and P. Hudak. Detecting Sharing of Partial Applications in Functional Programs. Technical Report YALEU/DCS/RR-526, Department of Computer Science, Yale University, March 1987.

[11]

C. Grabmayer and J. Rochel. Expressibility in the Lambda Calculus with Letrec. Technical report, arXiv, August 2012. arXiv:1208.2383.

[12]

C. Grabmayer and J. Rochel. Term Graph Representations for Cyclic Lambda Terms. In Proceedings of TERMGRAPH 2013, number 110 in EPTCS, 2013. For an extended report see: arXiv:1308.1034.

[13]

C. Grabmayer and J. Rochel. Expressibility in the Lambda Calculus withμ. In Proceedings of RTA 2013, 2013. Report: arXiv:1304.6284.

[14]

C. Grabmayer and J. Rochel. Confluent Let-Floating. In Proceedings of IWC 2013 (2nd International Workshop on Confluence), 2013.

[15]

C. Grabmayer and J. Rochel. Maximal Sharing in the Lambda Calculus with letrec. Technical report, 2014. arXiv:1401.1460.

[16]

C. Grabmayer and V. van Oostrom. Nested Term Graphs. Technical report, arXiv, May 2014. arXiv:1405.6380.

[17]

D. Hendriks and V. van Oostrom. λ. In F. Baader, editor, Proceedings CADE-19, volume 2741 of LNAI, pages 136--150. Springer, 2003.

[18]

J. Hopcroft. An nlog n Algorithm for Minimizing States in a Finite Automata. Technical report, Stanford University, CA, USA, 1971.

Digital Library

[19]

J. Hopcroft and R. Karp. A Linear Algorithm for Testing Equivalence of Finite Automata. Technical report, Cornell University, 1971.

[20]

R. Hughes. Supercombinators: A new implementation method for applicative languages. In LFP '82: Proceedings of the 1982 ACM symposium on LISP and functional programming, pages 1--10, 1982.

Digital Library

[21]

T. Johnsson. Lambda lifting: Transforming programs to recursive equations. In FPCA, pages 190--203, 1985.

Digital Library

[22]

J. Ketema and J. G. Simonsen. Infinitary Combinatory Reduction Systems. Information and Computation, 209(6):893--926, 2011.

Digital Library

[23]

R. Milner. Communicating and mobile systems: the π-calculus. Cambridge University Press, 1999.

Digital Library

[24]

M. T. Morazán and U. P. Schultz. Optimal lambda lifting in quadratic time. In Workshop IAFL 2007, number 5083 in LNCS. Springer, 2008.

Digital Library

[25]

D. A. Norton. Algorithms for Testing Equivalence of Finite Automata. Master's thesis, Dept. of Computer Science, Rochester Institute of Technology, 2009. https://ritdml.rit.edu/handle/1850/8712.

[26]

V. v. Oostrom, K.-J. van de Looij, and M. Zwitserlood. Lambdascope. Extended Abstract, Workshop ALPS, Kyoto, April 10th 2004, 2004.

[27]

J. Palsberg and M. Schwartzbach. Binding-time analysis: abstract interpretation versus type inference. In Int. Conf. on Computer Languages, 1994, pages 289--298, 1994.

[28]

S. L. Peyton Jones. The Implementation of Functional Programming Languages. Prentice-Hall, Inc., 1987.

Digital Library

[29]

D. Plump. Evaluation of Functional Expressions by Hypergraph Rewriting. PhD thesis, Universität Bremen, 1993.

[30]

Terese. Term Rewriting Systems, volume 55 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 2003.

[31]

C. P. Wadsworth. Semantics and Pragmatics of the Lambda-Calculus. PhD thesis, University of Oxford, 1971.

Cited By

Grabmayer C(2021)Structure-Constrained Process Graphs for the Process Semantics of Regular ExpressionsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.334.3334(29-45)Online publication date: 8-Feb-2021
https://doi.org/10.4204/EPTCS.334.3
Grabmayer CFokkink WHermanns HZhang LKobayashi NMiller D(2020)A Complete Proof System for 1-Free Regular Expressions Modulo BisimilarityProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394744(465-478)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394744
Accattoli BCoen C(2015)On the Relative Usefulness of FireballsProceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2015.23(141-155)Online publication date: 6-Jul-2015
https://dl.acm.org/doi/10.1109/LICS.2015.23
Show More Cited By

Index Terms

Maximal sharing in the Lambda calculus with letrec
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features
        Recursion
2. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Functional constructs

Recommendations

Maximal sharing in the Lambda calculus with letrec
ICFP '14: Proceedings of the 19th ACM SIGPLAN international conference on Functional programming

Increasing sharing in programs is desirable to compactify the code, and to avoid duplication of reduction work at run-time, thereby speeding up execution. We show how a maximal degree of sharing can be obtained for programs expressed as terms in the ...
The Intensional Lambda Calculus
LFCS '07: Proceedings of the international symposium on Logical Foundations of Computer Science

We introduce a natural deduction formulation for the Logic of Proofs, a refinement of modal logic S4 in which the assertion □ A is replaced by [[ s ]] A whose intended reading is " s is a proof of A ". A term calculus for this formulation yields a typed ...
The Lambda Lambda-Bar calculus: a dual calculus for unconstrained strategies
POPL '13: Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages

We present a calculus which combines a simple, CCS-like representation of finite behaviors, with two dual binders λ and λ¯. Infinite behaviors are obtained through a syntactical fixed-point operator, which is used to give a translation of λ-terms. The ...

Comments

Information & Contributors

Information

Published In

cover image ACM SIGPLAN Notices

ACM SIGPLAN Notices Volume 49, Issue 9

ICFP '14

September 2014

361 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/2692915

Editors:
Mark W. Bailey
Hamilton College, Clinton, NY
,
Rajeev Balasubramonian
University of Utah
,
Al Davis
University of Utah
,
Sarita Adve
University of Illinois at Urbana-Champ

Issue’s Table of Contents

ICFP '14: Proceedings of the 19th ACM SIGPLAN international conference on Functional programming
August 2014
390 pages
ISBN:9781450328739
DOI:10.1145/2628136
General Chair:
Johan Jeuring
Universiteit Utrecht, The Netherlands
,
Program Chair:
Manuel M.T. Chakravarty
University of New South Wales, Australia

Copyright © 2014 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 the author(s) 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].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 19 August 2014

Published in SIGPLAN Volume 49, Issue 9

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Nederlandse Organisatie voor Wetenschappelijk Onderzoek

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

9
Total Citations
View Citations
285
Total Downloads

Downloads (Last 12 months)9
Downloads (Last 6 weeks)1

Reflects downloads up to 13 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Grabmayer C(2021)Structure-Constrained Process Graphs for the Process Semantics of Regular ExpressionsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.334.3334(29-45)Online publication date: 8-Feb-2021
https://doi.org/10.4204/EPTCS.334.3
Grabmayer CFokkink WHermanns HZhang LKobayashi NMiller D(2020)A Complete Proof System for 1-Free Regular Expressions Modulo BisimilarityProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394744(465-478)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394744
Accattoli BCoen C(2015)On the Relative Usefulness of FireballsProceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2015.23(141-155)Online publication date: 6-Jul-2015
https://dl.acm.org/doi/10.1109/LICS.2015.23
Grabmayer C(2024)From Compactifying Lambda-Letrec Terms to Recognizing Regular-Expression ProcessesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.408.2408(21-41)Online publication date: 1-Oct-2024
https://doi.org/10.4204/EPTCS.408.2
Blaauwbroek LOlšák MGeuvers H(2024)Hashing Modulo Context-Sensitive 𝛼-EquivalenceProceedings of the ACM on Programming Languages10.1145/36564598:PLDI(2027-2050)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656459
Grabmayer C(2019)Modeling Terms by Graphs with Structure Constraints (Two Illustrations)Electronic Proceedings in Theoretical Computer Science10.4204/EPTCS.288.1288(1-13)Online publication date: 6-Feb-2019
https://doi.org/10.4204/EPTCS.288.1
Condoluci AAccattoli BCoen CKomendantskaya E(2019)Sharing Equality is LinearProceedings of the 21st International Symposium on Principles and Practice of Declarative Programming10.1145/3354166.3354174(1-14)Online publication date: 7-Oct-2019
https://dl.acm.org/doi/10.1145/3354166.3354174
Grabmayer Cvan Oostrom V(2015)Nested Term Graphs (Work In Progress)Electronic Proceedings in Theoretical Computer Science10.4204/EPTCS.183.4183(48-65)Online publication date: 26-May-2015
https://doi.org/10.4204/EPTCS.183.4
Accattoli BCoen C(2015)On the Relative Usefulness of FireballsProceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2015.23(141-155)Online publication date: 6-Jul-2015
https://dl.acm.org/doi/10.1109/LICS.2015.23
Grabmayer CRochel J(2014)Maximal sharing in the Lambda calculus with letrecACM SIGPLAN Notices10.1145/2692915.262814849:9(67-80)Online publication date: 19-Aug-2014
https://doi.org/10.1145/2692915.2628148

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents