Abstract
There are several notions of type with different semantics in computer science. The approach in this paper considers an extension of algebraic type specifications with respect to functional sorts and tries to give a suitable semantics for them.
The basic constituent of the theory is an extended notion of signature, which now consists of sorts, constructors and axioms. For sorts and constructors the semantics is defined by coherent Grothendieck topoi. Then it can be shown that initial topoi always exist.
Since each topos defines a canonical theory of a higher-order (intuitionistic) logic, the axioms in the signature define a theory. It is known that models of such theories are uniquely defined by logical functors, which define the models of type specifications in general.
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References
S. Abiteboul, M. Vardi, V. Vianu: Computing with Infinitary Logic, in J. Biskup, R. Hull (Eds.): Proc. 4th Int. Conf. on Database Theory, Springer LNCS 646, 1992, 113–123
J. R. Abrial: A Formal Approach to Large Software Construction, in J. L. A. Van de Snepscheut (Ed.), Mathematics of Program Construction, Springer LNCS, vol. 375, 1989, 1–20
R. M. Amadio, L. Cardelli: Subtyping Recursive Types, ACM TOPLAS, vol. 15 (4), 1993, 575–631
M. P. Atkinson, P. Buneman: Types and Persistence in Database Programming Languages, ACM Comp. Surveys, vol. 19 (2), 1987, 105–190
M. Barr, C. Wells: Category Theory for Computing Science, Prentice-Hall 1990
M. Barr, C. Wells: Toposes, Triples and Theories, Springer Grundlehren der mathematischen Wissenschaften 278, 1985
D. Bjørner, C. B. Jones: Formal Specification and Software Development, Prentice Hall, 1982
A. Boileau, A. Joyal: La Logique des Topos, Journal of Symbolic Logic, vol. 46 (1), 1981, 6–16
M. Broy: Equational Specification of Partial Higher Order Algebras, in M. Broy (Ed.): Logic of Programming and Calculi of Discrete Design, Springer, NATO ASI Series F, vol. 36, 1986, 185–242
K. B. Bruce, A. R. Meyer: The Semantics of Second Order Polymorphic Lambda Calculus, in G. Kahn, D. B. MacQueen, G. Plotkin (Eds.): Semantics of Data Types, Springer LNCS 173, 1984, pp. 131–144
P. Cousot: Methods and Logics for Proving Programs, in J. van Leeuwen (Ed.): The Handbook of Theoretical Computer Science, vol B: “Formal Models and Semantics”, Elsevier, 1990, 841–993
E. W. Dijkstra, C. S. Scholten: Predicate Calculus and Program Semantics, Springer Texts and Monographs in Computer Science, 1989
H.-D. Ehrich, U. Lipeck: Algebraic Domain Equations, Theoretical Computer Science, vol. 27, 1983, 167–196
H. Ehrig, B. Mahr: Fundamentals of Algebraic Specification 1, Springer EATCS Monographs, vol. 6, 1985
M. P. Fourman: The Logic of Topoi, in J. Barwise (Ed.); Handbook of Mathematical Logic, North-Holland Studies in Logic, vol. 90, 1977, 1053–1090
J. A. Goguen: Types as Theories, Oxford University, 1990
J. A. Goguen, R. M. Burstall: Institutions: Abstract Model Theory for Specification and Programming, J.ACM, vol. 39 (1), 1992, 95–146
R. Goldblatt: Topoi—The Categorial Analysis of Logic, North-Holland, Studies in Logic, vol. 98, 1984
P. Johnstone: Topos Theory, Academic Press, 1977
A. Kock, G. Reyes: Doctrines in Categorial Logic, in J. Barwise (Ed.): Handbook of Mathematical Logic, North-Holland Studies in Logic, vol. 90, 1977, 283–313
S. Mac Lane: Categories for the Working Mathematician, Springer GTM, vol. 5, 1972
S. Mac Lane, I. Moerdijk: Sheaves in Geometry and Logic — A first Introduction to Topos Theory, Springer Universitext, 1992
M. Makkai, R. Paré: Accessible Categories: The Foundations of Categorial Model Theory, Contemporary Mathematics, vol. 104, AMS, Providence (Rhode Island), 1989
J. C. Mitchell: Type Systems for Programming Languages, in J. van Leeuwen (Ed.): The Handbook of Theoretical Computer Science, vol B: “Formal Models and Semantics”, Elsevier, 1990, 365–58
G. Nelson: A Generalization of Dijkstra's Calculus, ACM TOPLAS, vol. 11 (4 1989, 517–561
J. Palomäki: Towards a Foundation of Concept Theory, in H. Kangassalo, H. Jaakkola: Information Modelling and Knowledge Bases V, IOS Press, Amsterdam, 1994
J. C. Reynolds: Polymorphism is not Set-Theoretic, in G. Kahn, D. B. MacQueen, G. Plotkin (Eds.): Semantics of Data Types, Springer LNCS 173, 1984, 145–156
K.-D. Schewe, B. Thalheim: Fundamental Concepts of Object Oriented Databases, Acta Cybernetica, vol. 11 (4), 1993, 49–85
K.-D. Schewe, J. W. Schmidt, I. Wetzel: Identification, Genericity and Consistency in Object-Oriented Databases, in J. Biskup, R. Hull (Eds.): Proc. 4th Int. Conf. on Database Theory, Springer LNCS 646, 1992, 341–356
K.-D. Schewe: Specification of Data-Intensive Application Systems, Habilitationsschrift, TU Cottbus, 1994
J. M. Spivey: Understanding Z, A Specification Language and its Formal Semantics, Cambridge University Press, 1988
C. Tuijn, M. Gyssens: Views and Decompositions of Databases from a Categorial Perspective, in J. Biskup, R. Hull (Eds.): Proc. 4th Int. Conf. on Database Theory, Springer LNCS 646, 1992, 99–112
W. Wechler: Universal Algebra for Computer Scientists, Springer EATCS Monographs, vol. 25, 1992
M. Wirsing: Algebraic Specification, in in J. van Leeuwen (Ed.): The Handbook of Theoretical Computer Science, vol B: “Formal Models and Semantics”, Elsevier, 1990, pp. 675–788
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Schewe, KD. (1995). Functional sorts in data type specifications. In: Reichel, H. (eds) Fundamentals of Computation Theory. FCT 1995. Lecture Notes in Computer Science, vol 965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60249-6_74
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DOI: https://doi.org/10.1007/3-540-60249-6_74
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