research-article

First-class substitutions in contextual type theory

Authors:

Brigitte PientkaAuthors Info & Claims

LFMTP '13: Proceedings of the Eighth ACM SIGPLAN international workshop on Logical frameworks & meta-languages: theory & practice

Pages 15 - 24

https://doi.org/10.1145/2503887.2503889

Published: 23 September 2013 Publication History

Abstract

In this paper, we revisit the theory of first-class substitution in contextual type theory (CTT); in particular, we focus on the abstract notion of substitution variables. This forms the basis for extending Beluga, a dependently typed proof and programming language which already supports first-class contexts and contextual objects, with first-class substitutions. To illustrate the elegance and power of first-class substitution variables, we describe the implementation of a weak normalization proof for the simply-typed lambda-calculus in Beluga.

Supplementary Material

ZIP File (lfmtp01.zip)

contextual-substvars-spine.agda: Partial formalization of results. Includes simply-typed approximations of hereditary substitution, simultaneous substitution and expansion operations. Tested with Agda version 2.3.2

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References

[1]

Martin Abadi, Luca Cardelli, Pierre-Louis Curien, and Jean-Jacques Lèvy. Explicit substitutions. In 17th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 31--46. ACM Press, 1990.

Digital Library

[2]

Nick Benton, Chung-Kil Hur, AndrewJ. Kennedy, and Conor McBride. Strongly typed term representations in Coq. Journal of Automated Reasoning, 49 (2): 141--159, 2012.

Digital Library

[3]

Yves Bertot and Pierre Castéran. Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions. Springer, 2004.

Digital Library

[4]

Andrew Cave and Brigitte Pientka. Programming with binders and indexed data-types. In 39th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL'12), pages 413--424. ACM Press, 2012.

Digital Library

[5]

Iliano Cervesato and Frank Pfenning. A linear spine calculus. phJournal of Logic and Computation, 13 (5): 639--688, 2003.

[6]

Andrew Gacek. The Abella interactive theorem prover (system description). In 4th International Joint Conference on Automated Reasoning, volume 5195 of phLecture Notes in Artificial Intelligence, pages 154--161. Springer, August 2008.

Digital Library

[7]

David Gelernter, Suresh Jagannathan, and Thomas London. Environments as first class objects. In Proceedings of the 14th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL'87), pages 98--110, New York, NY, USA, 1987. ACM.

Digital Library

[8]

Robert Harper, Furio Honsell, and Gordon Plotkin. A framework for defining logics. Journal of the ACM, 40 (1): 143--184, January 1993.

Digital Library

[9]

Masatomo Hashimoto and Atsushi Ohori. A typed context calculus. Theoretical Computer Science, 266 (1-2): 249--272, 2001.

Digital Library

[10]

Chantal Keller and Thorsten Altenkirch. Hereditary substitutions for simple types, formalized. In Proceedings of the third ACM SIGPLAN workshop on Mathematically structured functional programming, MSFP '10, pages 3--10, New York, NY, USA, 2010.

Digital Library

[11]

Ian A. Mason. Computing with contexts. Higher-Order and Symbolic Computation, 12 (2): 171--201, 1999.

Digital Library

[12]

Aleksandar Nanevski, Frank Pfenning, and Brigitte Pientka. Contextual modal type theory. ACM Transactions on Computational Logic, 9 (3): 1--49, 2008.

Digital Library

[13]

Shin-Ya Nishizaki. A polymorphic environment calculus and its type-inference algorithm. Higher Order Symbol. Comput., 13 (3): 239--278, 2000.

Digital Library

[14]

Ulf Norell. Towards a practical programming language based on dependent type theory. PhD thesis, Department of Computer Science and Engineering, Chalmers University of Technology, September 2007. Technical Report 33D.

[15]

Frank Pfenning and Carsten Schürmann. System description: Twelf -- a meta-logical framework for deductive systems. In H. Ganzinger, editor, 16th International Conference on Automated Deduction (CADE-16), Lecture Notes in Artificial Intelligence (LNAI 1632), pages 202--206. Springer, 1999.

Digital Library

[16]

Brigitte Pientka. A type-theoretic foundation for programming with higher-order abstract syntax and first-class substitutions. In 35th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL'08), pages 371--382. ACM Press, 2008.

Digital Library

[17]

Brigitte Pientka and Joshua Dunfield. Beluga: a framework for programming and reasoning with deductive systems (System Description). In Jürgen Giesl and Reiner Haehnle, editors, 5th International Joint Conference on Automated Reasoning (IJCAR'10), Lecture Notes in Artificial Intelligence (LNAI 6173), pages 15--21. Springer-Verlag, 2010.

Digital Library

[18]

Adam B. Poswolsky and Carsten Schürmann. Practical programming with higher-order encodings and dependent types. In 17th European Symposium on Programming (ESOP '08), volume 4960, pages 93--107. Springer, 2008.

Digital Library

[19]

Christian Queinnec and David De Roure. Sharing code through first-class environments. In Robert Harper and Richard L. Wexelblat, editors, Proceedings of the 1st ACM SIGPLAN International Conference on Functional Programming (ICFP '96), pages 251--261. ACM, 1996.

Digital Library

[20]

Didier Rémy. Type inference for records in a natural extension of ML. Research Report 1431, Institut National de Recherche en Informatique et Automatisme, Rocquencourt, BP 105, 78 153 Le Chesnay Cedex, France, May 1991.

[21]

Masahiko Sato, Takafumi Sakurai, and Rod Burstall. Explicit environments. phFundamenta Informaticae, 45 (1-2): 79--115, 2001.

Digital Library

[22]

Masahiko Sato, Takafumi Sakurai, and Yukiyoshi Kameyama. A simply typed context calculus with first-class environments. Journal of Functional and Logic Programming, 2002 (4): 1--41, March 2002.

[23]

Carsten Schürmann and Jeffrey Sarnat. Structural logical relations. In 23rd Annual Symposium on Logic in Computer Science (LICS), Pittsburgh, PA, USA, pages 69--80. IEEE Computer Society, 2008.

Digital Library

Cited By

Gaulin APientka B(2023)Contextual Refinement TypesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.396.2396(4-19)Online publication date: 17-Nov-2023
https://doi.org/10.4204/EPTCS.396.2
Murase YNishiwaki YIgarashi A(2023)Contextual Modal Type Theory with Polymorphic ContextsProgramming Languages and Systems10.1007/978-3-031-30044-8_11(281-308)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30044-8_11
Zyuzin NNanevski A(2021)Contextual modal types for algebraic effects and handlersProceedings of the ACM on Programming Languages10.1145/34735805:ICFP(1-29)Online publication date: 19-Aug-2021
https://dl.acm.org/doi/10.1145/3473580
Show More Cited By

Index Terms

First-class substitutions in contextual type theory
1. Software and its engineering
  1. Software notations and tools
    1. Formal language definitions
2. Theory of computation
  1. Formal languages and automata theory

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Published In

cover image ACM Conferences

LFMTP '13: Proceedings of the Eighth ACM SIGPLAN international workshop on Logical frameworks & meta-languages: theory & practice

September 2013

74 pages

ISBN:9781450323826

DOI:10.1145/2503887

Program Chairs:
Alberto Momigliano
Università di Milano, Italy
,
Brigitte Pientka
McGill University, Canada
,
Randy Pollack
Harvard University, USA

Copyright © 2013 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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SIGPLAN: ACM Special Interest Group on Programming Languages

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 September 2013

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ICFP'13

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SIGPLAN

ICFP'13: ACM SIGPLAN International Conference on Functional Programming

September 23, 2013

Massachusetts, Boston, USA

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LFMTP '13 Paper Acceptance Rate 5 of 7 submissions, 71%;

Overall Acceptance Rate 11 of 20 submissions, 55%

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Cited By

Gaulin APientka B(2023)Contextual Refinement TypesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.396.2396(4-19)Online publication date: 17-Nov-2023
https://doi.org/10.4204/EPTCS.396.2
Murase YNishiwaki YIgarashi A(2023)Contextual Modal Type Theory with Polymorphic ContextsProgramming Languages and Systems10.1007/978-3-031-30044-8_11(281-308)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30044-8_11
Zyuzin NNanevski A(2021)Contextual modal types for algebraic effects and handlersProceedings of the ACM on Programming Languages10.1145/34735805:ICFP(1-29)Online publication date: 19-Aug-2021
https://dl.acm.org/doi/10.1145/3473580
Errington JJang JPientka B(2021)Harpoon: Mechanizing Metatheory InteractivelyAutomated Deduction – CADE 2810.1007/978-3-030-79876-5_38(636-648)Online publication date: 5-Jul-2021
https://doi.org/10.1007/978-3-030-79876-5_38
Pientka BThibodeau DAbel AFerreira FZucchini RBouyer P(2019)A type theory for defining logics and proofsProceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3470152.3470181(1-13)Online publication date: 24-Jun-2019
https://dl.acm.org/doi/10.5555/3470152.3470181
Pientka BThibodeau DAbel AFerreira FZucchini R(2019)A Type Theory for Defining Logics and Proofs2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2019.8785683(1-13)Online publication date: Jun-2019
https://doi.org/10.1109/LICS.2019.8785683
ABEL AALLAIS GHAMEER APIENTKA BMOMIGLIANO ASCHÄFER SSTARK K(2019)POPLMark reloaded: Mechanizing proofs by logical relationsJournal of Functional Programming10.1017/S095679681900017029Online publication date: 10-Dec-2019
https://doi.org/10.1017/S0956796819000170
CAVE APIENTKA B(2018)Mechanizing proofs with logical relations – Kripke-styleMathematical Structures in Computer Science10.1017/S096012951800015428:09(1606-1638)Online publication date: 2-Aug-2018
https://doi.org/10.1017/S0960129518000154
Georges AMurawska AOtis SPientka B(2017)LINCX: A Linear Logical Framework with First-Class ContextsProgramming Languages and Systems10.1007/978-3-662-54434-1_20(530-555)Online publication date: 19-Mar-2017
https://doi.org/10.1007/978-3-662-54434-1_20
Cave APientka B(2015)A Case Study on Logical Relations using Contextual TypesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.185.3185(33-45)Online publication date: 27-Jul-2015
https://doi.org/10.4204/EPTCS.185.3
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