research-article

Open access

Do-it-yourself type theory

Authors:

Roland Backhouse,

Erik SaamanAuthors Info & Claims

Formal Aspects of Computing, Volume 1, Issue 1

Pages 19 - 84

https://doi.org/10.1007/BF01887198

Published: 01 March 1989 Publication History

Abstract

This paper provides a tutorial introduction to a constructive theory of types based on, but incorporating some extensions to, that originally developed by Per Martin-Löf. The emphasis is on the relevance of the theory to the construction of computer programs and, in particular, on the formal relationship between program and data structure. Topics discussed include the principle of propositions as types, free types, congruence types, types with information loss and mutually recursive types. Several examples of program development within the theory are also discussed in detail.

References

References

[1]

Allen, S.: A Non-Type-Theoretic Semantics for Type-Theoretic Language. PhD thesis, Cornell University, September 1987.

[2]

Backhouse, R. C.: Notes on Martin-Löf's Theory of Types, parts 1 and 2. In:FACS ACTS, British Computer Society, 1986.

[3]

Backhouse, R. C.:On the Meaning and Construction of the Rules in Martin-Löf's Theory of Types. Computing Science Notes CS 8606, Department of Mathematics and Computing Science, University of Groningen, 1986.

[4]

Backhouse, R. C.:Program Construction and Verification. Prentice-Hall International, 1986.

[5]

Boyer, R. S. and Moore, J. S.:A Computational Logic. Academic Press, 1979.

[6]

Boyer, R. S. and Moore, J. S.:MJRTY — A Fast Majority Vote Algorithm. Technical Report ICSCA-CMP-32, Institute for Computing Science and Computer Application, University of Texas at Austin, 1982.

[7]

de Bruijn, N. G.: A Survey of the Project Automath. In:Essays in Combinatory Logic, Lambda Calculus, and Formalism, J. P. Seldin and J. R. Hindley, (eds.), pp. 589–606, Academic Press, 1980.

[8]

Chisholm P. Derivation of a Parsing Algorithm in Martin-Löf's Theory of Types Science of Computer Programming 1987 8 1-42

Digital Library

[9]

Chisholm P. Investigations into Martin-Löf Type Theory as a Programming Logic PhD thesis 1988 Edinburgh Department of Computer Science, Heriot-Watt University

[10]

Church A. Annals of Mathematical Studies The Calculi of Lambda-Conversion. Vol. 6 1951 Princeton Princeton University Press

[11]

Cleaveland, R. and Panangaden, P.:Type Theory and Concurrency. Technical Report TR 85–714, Department of Computer Science, Cornell University, December 1985.

[12]

Constable R. L. Constructive Mathematics as a Programming Logic 1: Some Principles of Theory Annals of Discrete Mathematics 1985 24 21-38

[13]

Constable R. L., Knoblock T. B., and Bates J. L. Writing Programs that Construct Proofs Journal of Automated Reasoning 1985 1 285-326

Digital Library

[14]

Constable, R. L. and Mendler, N. P.: Recursive Definitions in Type Theory. In:Proc. Logics of Programs Conference, LNCS 193, pp. 61–78, Springer-Verlag, 1985.

[15]

Constable, R. L. and Smith, S. F.: Partial Objects in Constructive Type Theory. In:Proc. IEEE Symp. on Logic in Computer Science, pp. 183–193, Computer Society Press of the IEEE, 1987.

[16]

Constable, R. L. et al.:Implementing Mathematics in the Nuprl Proof Development System. Prentice-Hall, 1986.

[17]

Coquand, T. and Huet, G: Constructions: a Higher Order Proof System for Mechanizing Mathematics. In:Proc. of EUROCAL 85, Linz, Austria, April 1985.

[18]

Curry, H. B. and Feys, R.:Combinatory Logic. Vol. 1. North-Holland, 1958.

[19]

Dahl, O.-J., Dijkstra, E. W. and Hoare, C. A. R.:Structured Programming. Academic Press, 1972.

[20]

Dijkstra, E. W.:A Discipline of Programming. Prentice-Hall, 1976.

[21]

Dijkstra, E. W. and Feijen, W. H.-J.:Een Methode van Programmeren. Academic Service, 1984. Now available asA Method of Programming, Addison-Wesley, 1988.

[22]

Dybjer, P.: Inductively Defined Sets in Martin-Löf's Set Theory. In:Workshop on General Logic, A. Avron, R. Harper, F. Honsell, I. Mason and G. Plotkin, (eds.), Report ECS-LFCS-88-52, Department of Computer Science, University of Edinburgh, February, 1987.

[23]

Dyckhoff, R.:Category Theory as an Extension of Martin-Löf Type Theory. Technical Report CS/86/3, Department of Computational Science, University of St. Andrews, 1985.

[24]

Gentzen, G.: Investigations into Logical Deduction. In:The Collected Papers of Gerhard Gentzen, M. E. Szabo, (ed.), pp. 68–213, North-Holland, 1969.

[25]

Glivenko V. Sur Quelques Points de la Logique de m. Brouwer Bulletins de la classe des sciences 1929 15 183-188

[26]

Gordon, M. J., Milner, R. and Wadsworth, C. P.:Edinburgh LCF. Springer-Verlag, 1979.

[27]

Gries, D.:The Science of Programming. Springer-Verlag, 1981.

[28]

Harper, R., Honsell, F. A. and Plotkin, G.: A Framework for Defining Logics. In:Proc. Second Annual Conf. on Logic in Computer Science, Cornell, December 1987.

[29]

Hehner, E. R. C.:The Logic of Programming. Prentice-Hall, 1984.

[30]

Harper, R., Milner, R. and Tofte, M.:The Definition of Standard ML: Version 2. Report No. ECS-LFC-88-62, Laboratory for Foundations of Computer Science, University of Edinburgh, 1988.

[31]

Hoare, C. A. R.: Notes on Data Structuring. In:Structured Programming, O.-J. Dahl, E. W. Dijkstra and C. A. R. Hoare, (eds.), Academic Press, 1972.

[32]

Howard, W. A.: The Formulas-as-Types Notion of Construction. In:To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism, J. P. Seldin and J. R. Hindley, (eds.), pp. 479–490, Academic Press, 1980.

[33]

Hughes, J.:Why Functional Programming Matters. Technical Report, Department of Computer Science, University of Göteborg/Chalmers, 1984.

[34]

Jackson, M. A.:Principles of Program Design. Academic Press, 1975.

[35]

Jensen, K. and Wirth, N.:PASCAL: User Manual and Report. Springer-Verlag, 1975.

[36]

Kleene S. C. Introduction to Metamathematics 1952 Amsterdam North-Holland

[37]

Knuth D. E., Morris J. H., and Pratt V. R. Fast Pattern Matching in Strings SIAM Journal of Computing 1977 6 325-350

[38]

Lambek, J. and Scott, P. J.:Studies in Advanced Mathematics Vol 7, Introduction to Higher Order Categorical Logic. Cambridge University Press, 1986.

[39]

Malcolm, G. R. and Chisholm, P.:Polymorphism and Information Loss in Martin-Löf's Type Theory. Report CS 8814, Department of Mathematics and Computing Science, University of Groningen, 1988.

[40]

Martin-Löf, P.: An Intuitionistic Theory of Types: Predicative Part. In:Logic Colloquium 1973, H. E. Rose and J. C. Shepherdson (eds.), pp. 73–118, North-Holland, 1975.

[41]

Martin-Löf, P.: Constructive Mathematics and Computer Programming. In:Logic, Methodology and Philosophy of Science, IV, L. J. Cohen, J. Los, H. Pfeiffer and K.-P. Podewski (eds.) pp. 153–175, North-Holland, 1982.

[42]

Martin-Löf, P.: Constructive Mathematics and Computer Programming. In:Mathematical Logic and Computer Programming, C. A. R. Hoare and J. C. Shepherdson (eds.), pp. 167–184, Prentice-Hall, 1984.

[43]

Martin-Löf, P.:Intuitionistic Type Theory. Bibliopolis, 1984. Notes by Giovanni Sambin of a series of lectures given in Padova.

[44]

Meertens, L.: Algorithmics — Towards Programming as a Mathematical Activity. In:Proc. CWI Symp. on Mathematics and Computer Science, pp. 289–334, North-Holland, 1986.

[45]

Mendler, N. P.: Inductive Definitions in Type Theory. PhD thesis, Cornell University, September 1987.

[46]

Milner R. A Theory of Type Polymorphism in Programming Journal of Computer System Sciences 1977 17 348-375

[47]

Misra J. and Gries D. Finding Repeated Elements Science of Computer Programming 1982 2 143-152

[48]

Mitchell, J. and Plotkin, G.: Abstract Types have Existential Types. In:Proc. 12th ACM Symp. on Principles of Programming Languages, pp. 37–51, 1985.

[49]

Nordström, B.: Multilevel Functions in Type Theory. In:Programs as Data Objects, N. Jones (ed.), Springer-Verlag, LNCS 217, 1985.

[50]

Nordström, B.:Terminating General Recursion. Technical Report, Programming Methodology Group, University of Göteborg/Chalmers, September 1987.

[51]

Nordström, B. and Petersson, K.: Types and Specifications. In:IFIP'83, R. E. Mason (ed.), pp. 915–920, Elsevier Science Publishers, 1983.

[52]

Nordström, B. and Petersson, K.:The Semantics of Module Specifications in Martin-Löf's Type Theory. Technical Report 36, Programming Methodology Group, University of Göteborg/Chalmers, October 1985.

[53]

Nordström, B., Petersson, K. and Smith, J.:An Introduction to Martin-Löf's Theory of Types. Technical Report, Programming Methodology Group, University of Göteborg/Chalmers, 1986.

[54]

Paulson L. C. Constructing Recursion Operators in Intuitionistic Type Theory Journal of Symbolic Computation 1986 2 325-355

Digital Library

[55]

Petersson, K. and Synek, D.,A Set Constructor for Trees in Intuitionistic Type Theory. Technical Report, Department of Computer Science, University of Göteborg/Chalmers, August 1987.

[56]

Prawitz, D.: Proofs and the Meaning and Completeness of the Logical Constants. In:Essays on Mathematical and Philosophical Logic, J. Hintikka, I. Niiniluoto and E. Saarinen (eds.), pp. 25–40, Reidel, 1979.

[57]

Reynolds, J. C.:The Craft of Programming. Prentice-Hall, 1981.

[58]

Saaman, E. and Malcolm, G. R.:Well-founded Recursion in Type Theory. Computing Science Notes CS 8701, Department of Mathematics and Computer Science, University of Groningen, 1987.

[59]

Schmidt, D.:Denotational Semantics: A Methodology for Language Development. Allyn and Bacon, 1986.

[60]

Schröder-Heister, P.: A Natural Extension of Natural Deduction.The Journal of Symbolic Logic,49, 1984.

[61]

Smith, J.: On a Nonconstructive Type Theory and Program Derivation. In:Mathematical Logic and its Applications, D. G. Skordev (ed.), pp. 331–340, Plenum Publishing Corporation, 1987.

[62]

Stoy, J.:Denotational Semantics. The MIT Press, 1977.

Cited By

Dybjer P(2022)Inductive familiesFormal Aspects of Computing10.1007/BF012113086:4(440-465)Online publication date: 2-Jan-2022
https://dl.acm.org/doi/10.1007/BF01211308
Pitts A(2020)Typal Heterogeneous Equality TypesACM Transactions on Computational Logic10.1145/337944721:3(1-10)Online publication date: 19-Apr-2020
https://dl.acm.org/doi/10.1145/3379447
Hatcliff JLeavens GLeino KMüller PParkinson M(2012)Behavioral interface specification languagesACM Computing Surveys10.1145/2187671.218767844:3(1-58)Online publication date: 14-Jun-2012
https://dl.acm.org/doi/10.1145/2187671.2187678
Show More Cited By

Index Terms

Do-it-yourself type theory

Index terms have been assigned to the content through auto-classification.

Recommendations

Material Dialogues for First-Order Logic in Constructive Type Theory
Logic, Language, Information, and Computation
Abstract
Material dialogues are turn taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity ...
Towards Formalizing Categorical Models of Type Theory in Type Theory

This note is about work in progress on the topic of ''internal type theory'' where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category ...
On Building Constructive Formal Theories of Computation Noting the Roles of Turing, Church, and Brouwer
LICS '12: Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science

In this article I will examine a few key concepts and design decisions that account for the high value of implemented constructive type theories in computer science. I'll stress the historical fact that these theories, and the proof assistants that ...

Comments

Information & Contributors

Information

Published In

cover image Formal Aspects of Computing

Formal Aspects of Computing Volume 1, Issue 1

Mar 1989

410 pages

ISSN:0934-5043

EISSN:1433-299X

Issue’s Table of Contents

© BCS 1989.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 March 1989

Accepted: 15 November 1988

Received: 15 October 1988

Published in FAC Volume 1, Issue 1

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

12
Total Citations
View Citations
74
Total Downloads

Downloads (Last 12 months)25
Downloads (Last 6 weeks)5

Reflects downloads up to 04 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Dybjer P(2022)Inductive familiesFormal Aspects of Computing10.1007/BF012113086:4(440-465)Online publication date: 2-Jan-2022
https://dl.acm.org/doi/10.1007/BF01211308
Pitts A(2020)Typal Heterogeneous Equality TypesACM Transactions on Computational Logic10.1145/337944721:3(1-10)Online publication date: 19-Apr-2020
https://dl.acm.org/doi/10.1145/3379447
Hatcliff JLeavens GLeino KMüller PParkinson M(2012)Behavioral interface specification languagesACM Computing Surveys10.1145/2187671.218767844:3(1-58)Online publication date: 14-Jun-2012
https://dl.acm.org/doi/10.1145/2187671.2187678
Forsberg FSetzer A(2010)Inductive-inductive definitionsProceedings of the 24th international conference/19th annual conference on Computer science logic10.5555/1887459.1887496(454-468)Online publication date: 23-Aug-2010
https://dl.acm.org/doi/10.5555/1887459.1887496
Valentini S(2003)A cartesian closed category in Martin-Löf's intuitionistic type theoryTheoretical Computer Science10.1016/S0304-3975(01)00309-7290:1(189-219)Online publication date: 1-Jan-2003
https://dl.acm.org/doi/10.1016/S0304-3975%2801%2900309-7
Nogin A(2002)Quotient TypesProceedings of the 15th International Conference on Theorem Proving in Higher Order Logics10.5555/646529.695217(263-280)Online publication date: 20-Aug-2002
https://dl.acm.org/doi/10.5555/646529.695217
Fu Y(1996)RECURSIVE MODELS OF GENERAL INDUCTIVE TYPESFundamenta Informaticae10.5555/2379532.237953426:2(115-131)Online publication date: 1-Apr-1996
https://dl.acm.org/doi/10.5555/2379532.2379534
Larsen PHansen B(1996)Semantics of under-determined expressionsFormal Aspects of Computing10.1007/BF012110508:1(47-66)Online publication date: 1-Jan-1996
https://dl.acm.org/doi/10.1007/BF01211050
Meertens L(1992)ParamorphismsFormal Aspects of Computing10.1007/BF012113914:5(413-424)Online publication date: 1-Sep-1992
https://dl.acm.org/doi/10.1007/BF01211391
Hedberg M(1991)Normalising the associative law: An experiment with Martin-Löf's type theoryFormal Aspects of Computing10.1007/BF012456323:3(218-252)Online publication date: 1-Jul-1991
https://dl.acm.org/doi/10.1007/BF01245632
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Get Access

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

Media

Figures

Other

Tables

View Issue’s Table of Contents