research-article

A hierarchy of mendler style recursion combinators: taming inductive datatypes with negative occurrences

Authors:

Tim SheardAuthors Info & Claims

ICFP '11: Proceedings of the 16th ACM SIGPLAN international conference on Functional programming

Pages 234 - 246

https://doi.org/10.1145/2034773.2034807

Published: 19 September 2011 Publication History

Abstract

The Mendler style catamorphism (which corresponds to weak induction) always terminates even for negative inductive datatypes. The Mendler style histomorphism (which corresponds to strong induction) is known to terminate for positive inductive datatypes. To our knowledge, the literature is silent on its termination properties for negative datatypes. In this paper, we prove that histomorphisms do not always termintate by showing a counter-example. We also enrich the Mendler collection of recursion combinators by defining a new form of Mendler style catamorphism (msfcata), which terminates for all inductive datatypes, that is more expressive than the original. We organize the collection of combinators by placing them into a hierarchy of ever increasing generality, and describing the termination properties of each point on the hierarchy. We also provide many examples (including a case study on a negative inductive datatype), which illustrate both the expressive power and beauty of the Mendler style. One lesson we learn from this work is that weak induction applies to negative inductive datatypes but strong induction is problematic. We provide a proof of weak induction by exhibiting an embedding of our new combinator into F_ω. We pose the open question: Is there a safe way to apply strong induction to negative inductive datatypes?

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Cited By

Abreu PDelaware BHubers AJenkins CMorris JStump A(2023)A Type-Based Approach to Divide-and-Conquer Recursion in CoqProceedings of the ACM on Programming Languages10.1145/35711967:POPL(61-90)Online publication date: 11-Jan-2023
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Stump AJenkins CSpahn SMcDonald C(2020)Strong functional pearl: Harper’s regular-expression matcher in CedilleProceedings of the ACM on Programming Languages10.1145/34090044:ICFP(1-25)Online publication date: 3-Aug-2020
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Devesas Campos MFiore M(2020)Classical logic with Mendler inductionJournal of Logic and Computation10.1093/logcom/exaa004Online publication date: 3-Mar-2020
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A hierarchy of mendler style recursion combinators: taming inductive datatypes with negative occurrences

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A hierarchy of mendler style recursion combinators: taming inductive datatypes with negative occurrences
ICFP '11

The Mendler style catamorphism (which corresponds to weak induction) always terminates even for negative inductive datatypes. The Mendler style histomorphism (which corresponds to strong induction) is known to terminate for positive inductive datatypes. ...
Boxes go bananas: encoding higher-order abstract syntax with parametric polymorphism

Higher-order abstract syntax is a simple technique for implementing languages with functional programming. Object variables and binders are implemented by variables and binders in the host language. By using this technique, one can avoid implementing ...
Boxes go bananas: encoding higher-order abstract syntax with parametric polymorphism
ICFP '03: Proceedings of the eighth ACM SIGPLAN international conference on Functional programming

Higher-order abstract syntax is a simple technique for implementing languages with functional programming. Object variables and binders are implemented by variables and binders in the host language. By using this technique, one can avoid implementing ...

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cover image ACM Conferences

ICFP '11: Proceedings of the 16th ACM SIGPLAN international conference on Functional programming

September 2011

470 pages

ISBN:9781450308656

DOI:10.1145/2034773

General Chairs:
Manuel Chakravarty
University of New South Wales, Australia
,
Zhenjiang Hu
National Institute of Informatics, Japan
,
Program Chair:
Olivier Danvy
Aarhus University, Denmark

ACM SIGPLAN Notices Volume 46, Issue 9
ICFP '11
September 2011
456 pages
ISSN:0362-1340
EISSN:1558-1160
DOI:10.1145/2034574
Issue’s Table of Contents

Copyright © 2011 ACM.

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Published: 19 September 2011

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ICFP '11: ACM SIGPLAN International Conference on Functional Programming

September 19 - 21, 2011

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ICFP '11 Paper Acceptance Rate 33 of 92 submissions, 36%;

Overall Acceptance Rate 333 of 1,064 submissions, 31%

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Cited By

Abreu PDelaware BHubers AJenkins CMorris JStump A(2023)A Type-Based Approach to Divide-and-Conquer Recursion in CoqProceedings of the ACM on Programming Languages10.1145/35711967:POPL(61-90)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3571196
Stump AJenkins CSpahn SMcDonald C(2020)Strong functional pearl: Harper’s regular-expression matcher in CedilleProceedings of the ACM on Programming Languages10.1145/34090044:ICFP(1-25)Online publication date: 3-Aug-2020
https://dl.acm.org/doi/10.1145/3409004
Devesas Campos MFiore M(2020)Classical logic with Mendler inductionJournal of Logic and Computation10.1093/logcom/exaa004Online publication date: 3-Mar-2020
https://doi.org/10.1093/logcom/exaa004
Firsov DStump A(2018)Generic derivation of induction for impredicative encodings in CedilleProceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 201810.1145/3176245.3167087(215-227)Online publication date: 2018
https://doi.org/10.1145/3176245.3167087
Firsov DStump AAndronick JFelty A(2018)Generic derivation of induction for impredicative encodings in CedilleProceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3167087(215-227)Online publication date: 8-Jan-2018
https://dl.acm.org/doi/10.1145/3167087
Torrini P(2016)Modular Dependent Induction in Coq, Mendler-StyleInteractive Theorem Proving10.1007/978-3-319-43144-4_25(409-424)Online publication date: 7-Aug-2016
https://doi.org/10.1007/978-3-319-43144-4_25
Ahn KVezzosi A(2016)Executable Relational Specifications of Polymorphic Type Systems Using PrologFunctional and Logic Programming10.1007/978-3-319-29604-3_8(109-125)Online publication date: 21-Feb-2016
https://doi.org/10.1007/978-3-319-29604-3_8
Devesas Campos MFiore M(2015)Classical Logic with Mendler InductionLogical Foundations of Computer Science10.1007/978-3-319-27683-0_4(43-59)Online publication date: 10-Dec-2015
https://doi.org/10.1007/978-3-319-27683-0_4
Casinghino CSjöberg VWeirich S(2014)Combining proofs and programs in a dependently typed languageACM SIGPLAN Notices10.1145/2578855.253588349:1(33-45)Online publication date: 8-Jan-2014
https://dl.acm.org/doi/10.1145/2578855.2535883
Casinghino CSjöberg VWeirich SJagannathan SSewell P(2014)Combining proofs and programs in a dependently typed languageProceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages10.1145/2535838.2535883(33-45)Online publication date: 11-Jan-2014
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