research-article

Contextual modal type theory

Authors:

Aleksandar Nanevski,

Frank Pfenning,

Brigitte PientkaAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 9, Issue 3

Article No.: 23, Pages 1 - 49

https://doi.org/10.1145/1352582.1352591

Published: 12 June 2008 Publication History

Abstract

The intuitionistic modal logic of necessity is based on the judgmental notion of categorical truth. In this article we investigate the consequences of relativizing these concepts to explicitly specified contexts. We obtain contextual modal logic and its type-theoretic analogue. Contextual modal type theory provides an elegant, uniform foundation for understanding metavariables and explicit substitutions. We sketch some applications in functional programming and logical frameworks.

Supplementary Material

Nanevski Appendix (a23-nanevski-apndx.pdf)

Online appendix to Contextual modal type theory. The appendix supports the information on article 23.

Download
90.38 KB

References

[1]

Abadi, M., Cardelli, L., Curien, P.-L., and Lèvy, J.-J. 1990. Explicit substitutions. In Proceedings of the Symposium on Principles of Programming Languages (POPL'90, San Francisco, CA). 31--46.]]

Digital Library

[2]

Alechina, N., Mendler, M., de Paiva, V., and Ritter, E. 2001. Categorical and Kripke semantics for Constructive S4 modal logic. In International Workshop on Computer Science Logic, CSL'01, L. Fribourg, Ed. Lecture Notes in Computer Science, vol. 2142. Springer, Berlin, Germany, France, 292--307.]]

Digital Library

[3]

Attardi, G. and Simi, M. 1994. Proofs in context. In Principles of Knowledge Representation and Reasoning, J. Doyle, E. Sandewall, and P. Torasso, Eds. Morgan Kaufman, San Francisco, CA, 16--26.]]

[4]

Attardi, G. and Simi, M. 1995. A formalization of viewpoints. Fundamenta Informaticae 23, 3, 149--173.]]

Digital Library

[5]

Bjørner, N. and Muñoz, C. 2000. Absolute explicit unification. In International Conference on Rewriting Techniques and Applications, RTA'00. Lecture Notes in Computer Science, vol. 1833. Springer, Berlin, Germany, 31--46.]]

Digital Library

[6]

Bognar, M. and de Vrijer, R. 2001. A calculus of lambda calculus context. J. Automat. Reason. 27, 1, 29--59.]]

Digital Library

[7]

Buvač, S. 1996. Quantificational logic of context. In Proceedings of the Thirteenth National Conference on Artificial Intelligence. 600--606.]]

Digital Library

[8]

Buvač, S., Buvač, V., and Mason, I. 1995. Metamathematics of contexts. Fundamenta Informaticae 23, 3, 412--419.]]

Digital Library

[9]

Calcagno, C., Moggi, E., and Taha, W. 2004. ML-like inference for classifiers. In European Symposium on Programming, ESOP'04, D. Schmidt, Ed. Lecture Notes in Computer Science, 2986. Springer-Verlag, Berlin, Germany, 79--93.]]

[10]

Cartmell, J. 1986. Generalized algebraic theories and contextual categories. Ann. Pure Appl. Log. 32, 209--243.]]

[11]

Cervesato, I. and Pfenning, F. 2002. A linear logical framework. Inform. Computat. 179, 1 (Nov.), 19--75.]]

Digital Library

[12]

Dami, L. 1998. A lambda-calculus for dynamic binding. Theoret. Comput. Sci. 192, 2, 201--231.]]

Digital Library

[13]

Davies, R. 1996. A temporal logic approach to binding-time analysis. In Symposium on Logic in Computer Science (LICS'96), E. Clarke, Ed. IEEE Computer Society Press, Los Alamitos, CA, NJ, 184--195.]]

Digital Library

[14]

Davies, R. and Pfenning, F. 2001. A modal analysis of staged computation. J. ACM 48, 3 (May), 555--604.]]

Digital Library

[15]

de Groote, P. 2002. On the strong normalisation of intuitionistic natural deduction with permutation-conversions. Inform. Computat. 178, 2 (Nov.), 441--464.]]

[16]

de Paiva, V. 2003. Natural deduction and context as (constructive) modality. In Modelling and Using Context, P. Blackburn, C. Ghidini, R. M. Turner, and F. Giunchiglia, Eds. Lecture Notes in Artificial Inteligence, vol. 2680. Springer, Berlin, Germany, 116--129.]]

Digital Library

[17]

Dowek, G., Hardin, T., and Kirchner, C. 1995. Higher-order unification via explicit substitutions. In Symposium on Logic in Computer Science (LICS'95). D. Kozen, Ed. IEEE Computer Society Press, Los Alamitos, CA, 366--374.]]

Digital Library

[18]

Dowek, G., Hardin, T., Kirchner, C., and Pfenning, F. 1996. Unification via explicit substitutions: The case of higher-order patterns. In Proceedings of the Joint International Conference and Symposium on Logic Programming, M. Maher, Ed. MIT Press, Cambridge, MA, 259--273.]]

Digital Library

[19]

Geuvers, H. and Jojgov, G. I. 2002. Open proofs and open terms: a basis for interactive logic. In International Workshop on Computer Science Logic, CSL'02, J. C. Bradfield, Ed. Lecture Notes in Computer Science, vol. 2471. Springer, Berlin, Germany, 537--552.]]

Digital Library

[20]

Giunchiglia, F. 1993. Contextual reasoning. Epistemologica 16, 345--364.]]

[21]

Giunchiglia, F. and Serafini, L. 1994. Multilanguage hierarchical logics, or: How to do without modal logics. Artific. Intell. 65, 1, 29--70.]]

Digital Library

[22]

Harper, R., Honsell, F., and Plotkin, G. 1993. A framework for defining logics. JACM. 40, 1 (Jan.), 143--184.]]

Digital Library

[23]

Hashimoto, M. and Ohori, A. 2001. A typed context calculus. Theoret. Comput. Sci. 266, 1--2, 249--272.]]

Digital Library

[24]

Hodas, J. and Miller, D. 1994. Logic programming in a fragment of intuitionistic linear logic. Inform. Computat. 110, 2, 327--365.]]

Digital Library

[25]

Jojgov, G. I. 2003. Holes with binding power. In Types for Proofs and Programs, Second International Workshop, TYPES 2002, Berg en Dal, The Netherlands, April 24-28, 2002, Selected Papers, H. Geuvers and F. Wiedijk, Eds. Lecture Notes in Computer Science, vol. 2646. Springer, Berlin, Germany, 162--181.]]

Digital Library

[26]

Kripke, S. 1959. A completeness theorem in modal logic. J. Symbol. Log. 24, 1--14.]]

[27]

Lee, S.-D. and Friedman, D. P. 1996. Enriching the lambda calculus with contexts: Toward a theory of incremental program construction. In Proceedings of the International Conference on Functional Programming (ICFP'96). 239--250.]]

Digital Library

[28]

Liang, C. and Nadathur, G. 2002. Tradeoffs in the intensional representation of lambda terms. In International Conference on Rewriting Techniques and Applications, RTA'02, S. Tison, Ed. Lecture Notes in Computer Science, vol. 2378. Springer-Verlag, Berlin, Germany, 192--206.]]

Digital Library

[29]

Magnusson, L. 1995. The implementation of ALF—a proof editor based on Martin-Löf's monomorphic type theory with explicit substitutions. Ph.D. dissertation. Chalmers University of Technology and Göteborg University, Göteborg, Sweden.]]

[30]

Martin-Löf, P. 1996. On the meanings of the logical constants and the justifications of the logical laws. Nordic J. Philosoph. Log. 1, 1, 11--60.]]

[31]

Mason, I. A. 1999. Computing with contexts. Higher-Order Symbol. Computat. 12, 2, 171--201.]]

Digital Library

[32]

McCarthy, J. 1987. Generality in artificial intelligence. Commun. ACM 30, 12, 1030--1035.]]

Digital Library

[33]

McCarthy, J. 1993. Notes on formalizing contexts. In Proceedings of the International Joint Conference on Artificial Intelligence (Chambery, France). 555--560.]]

[34]

Miller, D. 1991. A logic programming language with lambda-abstraction, function variables, and simple unification. J. Log. Computat. 1, 4, 497--536.]]

[35]

Montague, R. 1963. Syntactical treatment of modalities, with corollaries on reflexion principles and finite axiomatizability. Acta Philosophica Fennica 16, 153--167.]]

[36]

Moody, J. 2003. Modal logic as a basis for distributed computation. Tech. rep. CMU-CS-03-194. Carnegie Mellon University, Pittsburgh, PA.]]

[37]

Muñoz, C. 2001. Dependent types and explicit substitutions. Math. Struct. Comput. Sci. 11, 1 (Feb.), 91--129.]]

Digital Library

[38]

Murphy VII, T., Crary, K., Harper, R., and Pfenning, F. 2004. A symmetric modal lambda calculus for distributed computing. In Symposium on Logic in Computer Science, H. Ganzinger, Ed. (LICS 04, Turku, Finland). 286--295.]]

Digital Library

[39]

Nanevski, A. 2003. From dynamic binding to state via modal possibility. In Proceedings of the International Conference on Principles and Practice of Declarative Programming (PPDP'03. Uppsala, Sweden). 207--218.]]

Digital Library

[40]

Nanevski, A. 2004. Functional programming with names and necessity. Ph.D. dissertation. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA.]]

Digital Library

[41]

Nanevski, A. and Pfenning, F. 2005. Staged computation with names and necessity. J. Funct. Programm. 15, 6 (Nov.), 837--891.]]

Digital Library

[42]

Nanevski, A., Pientka, B., and Pfenning, F. 2003. A modal foundation for meta variables. In Proceedings of MERλIN 2003 (Uppsala, Sweden).]]

Digital Library

[43]

Pfenning, F. 2000. Structural cut elimination I. Intuitionistic and classical logic. Inform. Computat. 157, 1/2 (Mar.), 84--141.]]

Digital Library

[44]

Pfenning, F. 2001. Intensionality, extensionality, and proof irrelevance in modal type theory. In Symposium on Logic in Computer Science (LICS'01), J. Halpern, Ed. IEEE Computer Society Press, Los Alamitos, CA, 221--230.]]

Digital Library

[45]

Pfenning, F. and Davies, R. 2001. A judgmental reconstruction of modal logic. Math. Struct. Comput. Sci. 11, 4, 511--540.]]

Digital Library

[46]

Pfenning, F. and Schürmann, C. 1999. System description: Twelf—a meta-logical framework for deductive systems. In International Conference on Automated Deduction, CADE'99, H. Ganzinger, Ed. Lecture Notes in Artificial Inteligence, vol. 1632. Springer-Verlag, Berlin, Germany, 202--206.]]

Digital Library

[47]

Pientka, B. 2003. Tabled higher-order logic programming. Ph.D. dissertation. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA.]]

Digital Library

[48]

Pientka, B. and Pfennning, F. 2003. Optimizing higher-order pattern unification. In International Conference on Automated Deduction, CADE'03, F. Baader, Ed. Lecture Notes in Computer Science, vol. 2741. Springer, Berlin, Germany, 473--487.]]

[49]

Sato, M., Sakurai, T., and Burstall, R. 2001. Explicit environments. Fundamenta Informaticae 45, 1-2, 79--115.]]

Digital Library

[50]

Sato, M., Sakurai, T., and Kameyama, Y. 2002. A simply typed context calculus with first-class environments. J. Funct. Log. Programm. 2002, 4 (Mar.).]]

Digital Library

[51]

Sato, M., Sakurai, T., Kameyama, Y., and Igarashi, A. 2003. Calculi of meta-variables. In International Workshop on Computer Science Logic, CSL'03, M. Baaz and J. A. Makowsky, Eds. Lecture Notes in Computer Science, vol. 2803. Springer, Berlin, Germany, 484--497.]]

[52]

Schürmann, C., Poswolsky, A., and Sarnat, J. 2005. The ∇-calculus: Functional programming with higher-order encodings. In International Conference on Typed Lambda Calculus and Applications, TLCA'05, P. Urzyczyn, Ed. Lecture Notes in Computer Science, vol. 3461. Springer, Berlin, Germany, 339--353.]]

Digital Library

[53]

Serafini, L. and Bouquet, P. 2004. Comparing formal theories of context in AI. Artific. Intell. 155, 1--2, 41--67.]]

Digital Library

[54]

Serafini, L. and Giunchiglia, F. 2002. ML system: A proof theory for contexts. J. Log. Lang. Inform. 11, 4, 471--518.]]

Digital Library

[55]

Simpson, A. K. 1994. The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh.]]

[56]

Strecker, M. 1999. Construction and deduction in type theories. Ph.D. dissertation. Universität Ulm, Ulm, Germany.]]

[57]

Taha, W. and Nielsen, M. F. 2003. Environment classifiers. In Proceedings of the Symposium on Principles of Programming Languages (POPL'03, New Orleans, LA). G. Morrisett, Ed. ACM Press, New York, NY, 26--37.]]

Digital Library

[58]

Taha, W. and Sheard, T. 1997. Multi-stage programming with explicit annotations. In Proceedings of the Workshop on Partial Evaluation and Semantics-Based Program Manipulation (PEPM'97. Amsterdam, The Netherland). 203--217.]]

Digital Library

[59]

Thomason, R. H. 1999. Type theoretic foundations for context, part 1: Contexts as complex type-theoretic objects. In Proceedings of the Second International and Interdisciplinary Conference on Modeling and Using Contexts (CONTEXT'99, Trento, Italy), P. Bouquet, L. Serafini, P. Brézillon, M. Benerecetti, and F. Castellani, Eds. 352--374.]]

Digital Library

[60]

Thomason, R. H. 2003. Dynamic contextual intensional logic: Logical foundations and an application. In Modeling and Using Context, P. Blackburn, C. Ghidini, R. M. Turner, and F. Giunchiglia, Eds. Lecture Notes in Artificial Inteligence, vol. 2680. Springer, Berlin, Germany, 328--341.]]

Digital Library

[61]

Watkins, K., Cervesato, I., Pfenning, F., and Walker, D. 2002. A concurrent logical framework I: Judgments and properties. Tech. rep. CMU-CS-02-101, Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA. Revised May 2003.]]

[62]

Weyhrauch, R. W. 1979. Prolegomena to a theory of mechanized formal reasoning. Artific. Intell. 13, 1--2, 133--170.]]

[63]

Wickline, P., Lee, P., and Pfenning, F. 1998a. Run-time code generation and Modal-ML. In Proceedings of the Conference on Programming Language Design and Implementation (PLDI'98, Montreal, P. Q., Canada), K. D. Cooper, Ed. ACM Press, 224--235.]]

Digital Library

[64]

Wickline, P., Lee, P., Pfenning, F., and Davies, R. 1998b. Modal types as staging specifications for run-time code generation. ACM Comput. Surv. 30, 3es.]]

Digital Library

Cited By

Hu JPientka B(2025)A Dependent Type Theory for Meta-programming with Intensional AnalysisProceedings of the ACM on Programming Languages10.1145/37048519:POPL(416-445)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704851
Pientka BAllais GLiu Y(2025)A Type-Theoretic Framework for Certified Meta-programming (Invited Talk Extended Abstract)Proceedings of the 2025 ACM SIGPLAN International Workshop on Partial Evaluation and Program Manipulation10.1145/3704253.3706136(10-11)Online publication date: 10-Jan-2025
https://dl.acm.org/doi/10.1145/3704253.3706136
Zackon DSano CMomigliano APientka BStark KTimany ABlazy STabareau N(2025)Split Decisions: Explicit Contexts for Substructural LanguagesProceedings of the 14th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3703595.3705888(257-271)Online publication date: 10-Jan-2025
https://dl.acm.org/doi/10.1145/3703595.3705888
Show More Cited By

Index Terms

Contextual modal type theory
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features
        Frameworks
2. Theory of computation
  1. Logic
    1. Modal and temporal logics

Recommendations

Reasoning about Object-based Calculi in (Co)Inductive Type Theory and the Theory of Contexts
Abstract
We illustrate a methodology for formalizing and reasoning about Abadi and Cardelli’s object-based calculi, in (co)inductive type theory, such as the Calculus of (Co)Inductive Constructions, by taking advantage of natural deduction semantics and ...
Correspondence Theory for Modal Fairtlough–Mendler Semantics of Intuitionistic Modal Logic
Abstract
We study the correspondence theory of intuitionistic modal logic in modal Fairtlough–Mendler semantics (modal FM semantics) (Fairtlough and Mendler in Inf Comput 137(1):1–33, 1997), which is the intuitionistic modal version of possibility ...
MOIN: A Nested Sequent Theorem Prover for Intuitionistic Modal Logics (System Description)
Automated Reasoning
Abstract
We present a simple Prolog prover for intuitionistic modal logics based on nested sequent proof systems. We have implemented single-conclusion systems (Gentzen-style) and multi-conclusion systems (Maehara-style) for all logics in the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 9, Issue 3

June 2008

289 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/1352582

Issue’s Table of Contents

Copyright © 2008 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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 12 June 2008

Accepted: 01 February 2007

Revised: 01 February 2007

Received: 01 September 2005

Published in TOCL Volume 9, Issue 3

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

National Science Foundation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

215
Total Citations
View Citations
1,022
Total Downloads

Downloads (Last 12 months)74
Downloads (Last 6 weeks)7

Reflects downloads up to 24 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Hu JPientka B(2025)A Dependent Type Theory for Meta-programming with Intensional AnalysisProceedings of the ACM on Programming Languages10.1145/37048519:POPL(416-445)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704851
Pientka BAllais GLiu Y(2025)A Type-Theoretic Framework for Certified Meta-programming (Invited Talk Extended Abstract)Proceedings of the 2025 ACM SIGPLAN International Workshop on Partial Evaluation and Program Manipulation10.1145/3704253.3706136(10-11)Online publication date: 10-Jan-2025
https://dl.acm.org/doi/10.1145/3704253.3706136
Zackon DSano CMomigliano APientka BStark KTimany ABlazy STabareau N(2025)Split Decisions: Explicit Contexts for Substructural LanguagesProceedings of the 14th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3703595.3705888(257-271)Online publication date: 10-Jan-2025
https://dl.acm.org/doi/10.1145/3703595.3705888
Chen APorter TOmar C(2025)Polymorphism with Typed HolesTrends in Functional Programming10.1007/978-3-031-74558-4_7(134-159)Online publication date: 10-Jan-2025
https://doi.org/10.1007/978-3-031-74558-4_7
Ângelo PIgarashi AT. Vasconcelos V(2024)Linear Contextual Metaprogramming and Session TypesElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.401.1401(1-10)Online publication date: 6-Apr-2024
https://doi.org/10.4204/EPTCS.401.1
Hu JPientka B(2024)A Layered Approach to Intensional Analysis in Type TheoryACM Transactions on Programming Languages and Systems10.1145/370720346:4(1-43)Online publication date: 5-Dec-2024
https://dl.acm.org/doi/10.1145/3707203
Carnier DPottier FKeuchel S(2024)Type Inference LogicsProceedings of the ACM on Programming Languages10.1145/36897868:OOPSLA2(2125-2155)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689786
Blinn ALi XKim JOmar C(2024)Statically Contextualizing Large Language Models with Typed HolesProceedings of the ACM on Programming Languages10.1145/36897288:OOPSLA2(468-498)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689728
Palmer ZFilardo NWu K(2024)Intensional FunctionsProceedings of the ACM on Programming Languages10.1145/36897148:OOPSLA2(87-112)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689714
Gao CParreaux LChiba SThüm T(2024)Seamless Scope-Safe Metaprogramming through Polymorphic Subtype Inference (Short Paper)Proceedings of the 23rd ACM SIGPLAN International Conference on Generative Programming: Concepts and Experiences10.1145/3689484.3690733(121-127)Online publication date: 21-Oct-2024
https://dl.acm.org/doi/10.1145/3689484.3690733
Show More Cited By

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 Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents