Abstract
To provide a formal framework for discussing specifications of algebraic abstract data types we introduce the notion of an algebraic institution. Our main results concern the problem of the existence of free constructions in algebraic institutions. We review a characterization of logical specification systems that guarantee the existence of initial models for any consistent set of axioms given by Mahr and Makowsky in [MM 83a, MM 83b]. Then the more general problem of the existence of free functors (left adjoints to forgetful functors) for any theory morphism is analysed. We give a construction of a free model of a theory over a model of a subtheory (with respect to an arbitrary theory morphism) which requires only the existence of initial models.
On leave from Institute of Computer Science, Polish Academy of Sciences, Warsaw.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
8 References
Goguen, J.A., Thatcher, J.W. and Wagner, E.G. An initial algebra approach to the specification, correctness, and implementation of abstract data types. Current Trends in Programming Methodology, Vol. 4: Data Structuring (R.T. Yeh, ed.). Prentice-Hall. pp. 80–149 (1978).
Barwise, K.J. Axioms for abstract model theory. Annals of Math. Logic 7, pp. 221–265.
Burstall, R.M. and Goguen, J.A. The semantics of Clear, a specification language. Proc. of Advanced Course on Abstract Software Specifications, Copenhagen. Springer LNCS 86, pp. 292–332.
Ehrig, H., Wagner, E.G. and Thatcher, J.W. Algebraic specifications with generating constraints. Proc. 10th ICALP, Barcelona. Springer LNCS 154, pp. 188–202.
Goguen, J.A. and Burstall, R.M. Introducing institutions. Proc. Logics of Programming Workshop. CMU.
Goguen, J.A., Meseguer, J. Completeness of many-sorted equational logic. SIGPLAN Notices 16(7), pp. 24–32, July 1981, extended version to appear in Houston Journal of Mathematics.
Goguen, J.A. and Meseguer, J. An initiality primer, to appear in Application of Algebra to Language Definition and Compliation (M. Nivat. J. Reynolds, editors). North Holland.
MacLane, S. Categories for the Working Mathematician. Springer.
Mahr, B. and Makowsky, J.A. Characterizing specification languages which admit initial semantics. Proc. 8th CAAP, L'Aquila, Italy. Springer LNCS 159, pp. 300–316.
Mahr, B. and Makowsky, J.A. An axiomatic approach to semantics of specification languages. Proc. 6th GI Conf. on Theoretical Computer Science, Dortmund. Springer LNCS 145.
Reichel, H. Initially restricting algebraic theories. in: Mathematical Foundations of Computer Science (Proc. 9th Symp. Rydzyna 1980. Poland, P. Dembinski, ed.), Lecture Notes in Computer Science 88, pp. 504–514, Springer-Verlag 1980.
Sannelia, D.T. and Tarlecki, A. Building specifications in an arbitrary institution, Proc. International Symposium on Semantics of Data Types. Sophia-Antipolis, June 1984, to appear.
Tarlecki, A. Free constructions in algebraic institutions. Report CSR-149-83. Dept. of Computer Science, Univ. of Edinburgh.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tarlecki, A. (1984). Free constructions in algebraic institutions. In: Chytil, M.P., Koubek, V. (eds) Mathematical Foundations of Computer Science 1984. MFCS 1984. Lecture Notes in Computer Science, vol 176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0030336
Download citation
DOI: https://doi.org/10.1007/BFb0030336
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13372-8
Online ISBN: 978-3-540-38929-3
eBook Packages: Springer Book Archive