Abstract
The aim of this paper is to present a new semantics of higher-order order-sorted types for functional programming, data type specification and program transformation. Our type discipline unifies higher-order functions, overloading and subtype polymorphism in a very simple way. The new approach can be considered as an extension of order-sorted algebra with higher-order functions. We show the existence of initial algebras and give a sound and complete equational deduction system.
The research has been partially supported by the Commission of the European Communities under the ESPRIT Programme in the PROSPECTRA Project, ref #390, and the ESPRIT Basic Research Action COMPASS, ref #3264.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Breazu-Tannen, V. [88]: "Combing algebra and higher-order types." In: Proc. LICS, Edinburgh. (1988)
Cardelli,L. and Wegner,P. [85]: "On understanding types, data abstraction and polymorphism." In: ACM Computing Surveys, 17. (1985)
Cardelli,L. [84]: "A Semantics of Multiple Inheritance." In: Semantics of Data Types, eds. G.Kahn, D.B.MacQueen, G.Plotkin, LNCS 173, Springer-Verlag. (1984) 51–67. Also in: Information and Computation 76. (1988) 130–164.
Ehrig,H., Mahr,B. [85]: "Fundamentals of Algebraic Specification: Equations and Initial Semantics" Springer-Verlag 1985
Gogolla,M. [84]: "Partially Ordered Sorts in Algebraic Specifications." Proc. 9th CAAP, Cambridge University Press, 139–153. (1984)
Goguen,J.A. [78]: "Order-Sorted Algebra. Semantics and Theory of Computation." Report No. 14, UCLA computer Science Dept. 1978.
Goguen, J.A. and Meseguer, J. [82]: "Completeness of Many-sorted Equational Logic." In: SIGPLAN Notices Vol.17, No.1 (1982) 9–17
Goguen,J.A. and Meseguer,J. [89]: "Order-Sorted Algebra I: Equational Deduction for Multiple Inheritance, Polymorphism, and Partial Operations." Tech. Report SRI (1989).
Goguen,J.A., Thatcher,J.W. and Wagner,E.G. [78]: "An Initial Algebra Approach to the Specification, Correctness, and Implementation of Abstract Data Types." In R.T.Yeh, Current Trends in Programming Methodology, Vol IV, Data Structuring; Prentice Hall (1978), 80–149.
Hanus, M. [89]: "Horn Clause Programs with Polymorphic Types: semantics and Resolution." In: Proc. TAPSOFT'89 Barcelona, Apain. LNCS 352 (1989) 225–240.
Hindley, J., and Seldin, J. [86]: "Introduction to Combinators and Lambda Calculus, Cambridge University Press. 1986
Kreowski,H.-J. and Qian, Zh. [88]: "Relation-Sorted Algebraic Specifications with Built-in Coercers:Basic Notions and Results." PROSPECTRA-Report M.1.1.S3-SN-44, 1988.
Krieg-Brückner, B. [89]: "Algebraic Specification and Functionals for Transformational Program and Meta Program Development." In: Proc. TAPSOFT, 1989, LNCS 352, 36–59.
MacQueen, B.D. and Sannella, D.T. [82]: "Completeness of proof Systems for Equational Specifications" IEEE Transactions on Software Engineering, Vol. SE-11, No.5 (May, 1985) 454–461.
Maibaum, T.S.E. and Lucena, C.L. [82]: "Higher-order data types" Int. Journal of Computer Science 9. (1980) 31–53.
Meyer, A.R. [82]: "What is a model of the lambda calculus?" Information and Control 52. (1982) 87–122
Möller,B. [87]: "Higher-Order Algebraic Specifications." Habilitationsschrift, TU München (1987). See also "Algebraic Specifications with Higher-Order Operators" In L.Meetens Proc. IFIP TC2 Working Conf. on Program Specification and Transformation, Bad Tälz. (1986) North-Holland.
Poigne, A. [86]: "On specifications, theories, and models with higher types." Information and Control 68. (1986) 1–46.
Poigne, A. [90]: "Parameterization for Order-Sorted Algebraic Specification." In: Journal of Computer and System Sciences Vol. 40, No. 2 (1990) 229–268.
Qian, Zh. [89]: "Relation-Sorted Algebraic Specifications with Built-in Coercers: Parameterization and Parameter Passing." In: Proc. Categorical Methods in Computer Science with Aspects from Topology, LNCS 393.
Qian, Zh. [90a]: "Extensions of Algebraic Specifications: Suptype Polymorphism, Higher Order Functions and Parameterisation." Dissertation, Univ. of Bremen. (To appear 1990)
Reynolds, J. [80]: "Using category theory to design implicit conversions and generic operations." In: Semantics-Directed Compiler Generation, LNCS 94. (1980) 211–258
Scott, D. [76]: "Data types as lattices." In: SIAM Journal of Computing, Vol 5, No 3. (1976) 523–587
Smolka,G., Nutt,W., Goguen,J.A. and Meseguer,J. [87]: "Order-Sorted Equational Computation." In H.Ait-Kaci and M.Nivat, Resolution of Equations in Algebraic Structures, Academic Press.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Qian, Z. (1990). Higher-order order-sorted algebras. In: Kirchner, H., Wechler, W. (eds) Algebraic and Logic Programming. ALP 1990. Lecture Notes in Computer Science, vol 463. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53162-9_32
Download citation
DOI: https://doi.org/10.1007/3-540-53162-9_32
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53162-3
Online ISBN: 978-3-540-46738-0
eBook Packages: Springer Book Archive