Abstract
This paper is an exposition of the basic ideas of the mathematical theory of sketches and a detailed description of some of the ways in which this theory can be used in theoretical computer science to specify datatypes. In particular, this theory provides a convenient way of introducing datatypes which have variants, for example in case of errors or nil pointers. The semantics is a generalization of initial algebra semantics which in some cases allows initial algebras depending on a parameter such as a bound for overflow.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
14. Bibliography
M. Barr, Models of sketches, Cahiers de Topologie et Géometrie Différentielle, 27 (1986), 93–107.
M. Barr and C. Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.
M. Barr and C. Wells. Category Theory for Computer Scientists, in preparation.
A. Bastiani C. Ehresmann, Categories of sketched structures. Cahiers de Topologie et Géometrie Différentielle 10 (1968), 104–213.
R. M. Burstall, Electronic category theory, in Mathematical Foundations of Computer Science. Springer Lecture Notes in Computer Science 88 (1980).
N. Chomsky, M. P. SchĂ¼tzenberger, The algebraic theory of context-free languages, in P. Braffort, D. Hirschberg, eds. Computer Programming and Formal Systems, North-Holland, 1963.
Y. Diers, Catégories Localizables. Thèse de doctorat, Université de Paris, 1977.
J. Donahue and A. Demers, Data types are values. To appear in ACM Transactions on Programming Languages and Systems.
H. Ehrig, J. W. Thatcher, P. Lucas and S. N. Zilles, Denotational and initial algebra semantics of the algebraic specification language Look. Preprint: Technische Universität Berlin, 1982.
H. Ehrig, H.-J. Kreowski, J. Thatcher, E. Wagner and J. Wright, Parameter passing in algebraic specification languages. Theoretical Computer Science 28 (1984), 45–81.
H. Ehrig and B. Mahr, Fundamentals of algebraic specifications I. Springer-Verlag, 1985.
M. Fourman and S. Vickers, Theories as categories, in Category Theory and Computer Programming, Springer Lecture Notes in Computer Science 240, Springer-Verlag, 1986.
J. A. Goguen, Abstract errors for abstract data types, In E. J. Neuhold, ed. Formal Description of Programming Concepts, North-Holland, 1978.
J. A. Goguen and J. Meseguer, Eqlog: equality, types and generic modules for logic programming. To appear in DeGroot and Lindstrom, eds., Functional and Logic Programming, Prentice-Hall, 1985a.
J. Goguen and J. Meseguer, Order sorted algebra I: partial and overloaded operators, errors and inheritance. Preprint, SRI International, Menlo Park, CA 94025, 1985b.
J. Goguen, J. W. Thatcher, E. G. Wagner and J. B. Wright, Initial algebra semantics and continuous algebras. J. ACM, 24 (1977), 68–95.
J. Gray, Categorical aspects of parametric data types. Preprint: University of Illinois, 1985.
R. Guitart, On the geometry of computations. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 (1986), 107–136.
R. Guitart and C. Lair, Calcul syntaxique des modèles et calcul des formules internes. Diagrammes, 4 (1980), 1–106.
J. Lambek and P. Scott, Cartesian Closed Categories and λ-Calculus. Cambridge Studies in Advanced Mathematics 7. Cambridge University Press, 1986.
S. Mac Lane, Categories for the Working Mathematician. Graduate Texts in Mathematics 5, Springer-Verlag, 1971.
J. Makowsky, Why Horn formulas matter in computer science: initial structures and generic examples. Technical Report #329, Department of Computer Science, Technion, Haifa, Israel, 1984.
P. Mateti and F. Hunt, Precision descriptions of software designs: an example. IEEE Compsac, 1985, 130–136.
C. McLarty, Left exact logic. Journal of Pure and Applied Algebra, 41 (1986), 63–66.
J. Meseguer and J. Goguen, Initiality, induction and computability, in M. Nivat and J. C. Reynolds, eds., Algebraic Methods in Semantics, Cambridge University Press, 1985.
D. E. Rydehead and R. M. Burstall, The unification of terms: a category-theoretic algorithm, in Category Theory and Computer Programming, Springer Lecture Notes in Computer Science 240. Springer-Verlag, 1986.
J. W. Thatcher, E. G. Wagner and J. B. Wright, Specification of abstract data types using conditional axioms. (Extended abstract). IBM T. J. Watson Research Center Research Report RC 6214 (#26679), 1976.
J. W. Thatcher, E. G. Wagner and J. B. Wright, Data type specification: parametrization and the power of specification techniques. ACM Transactions on Programming Languages and Systems, Vol. 4 no. 4, October, 1982.
H. Volger, On theories which admit initial structures. Preprint: Universität TĂ¼bingen, 1985.
E. Wagner, S. Bloom and J. W. Thatcher, Why algebraic theories? in M. Nivat and J. C. Reynolds, eds., Algebraic Methods in Semantics, Cambridge University Press, 1985.
S. N. Zilles, P. Lucas and J. W. Thatcher, A look at algebraic specifications. IBM T. J. Watson Research Center Research Report RJ 3568 (#41985), 1982.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wells, C., Barr, M. (1988). The formal description of data types using sketches. In: Main, M., Melton, A., Mislove, M., Schmidt, D. (eds) Mathematical Foundations of Programming Language Semantics. MFPS 1987. Lecture Notes in Computer Science, vol 298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-19020-1_27
Download citation
DOI: https://doi.org/10.1007/3-540-19020-1_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-19020-2
Online ISBN: 978-3-540-38920-0
eBook Packages: Springer Book Archive