Free constructions in algebraic institutions
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.
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- [ADJ 76]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).Google Scholar
- [Bar 74]Barwise, K.J. Axioms for abstract model theory. Annals of Math. Logic 7, pp. 221–265.Google Scholar
- [BG 80]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.Google Scholar
- [EWT 83]Ehrig, H., Wagner, E.G. and Thatcher, J.W. Algebraic specifications with generating constraints. Proc. 10th ICALP, Barcelona. Springer LNCS 154, pp. 188–202.Google Scholar
- [GB 83]Goguen, J.A. and Burstall, R.M. Introducing institutions. Proc. Logics of Programming Workshop. CMU.Google Scholar
- [GM 81]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.Google Scholar
- [GM 83]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.Google Scholar
- [MacL 71]MacLane, S. Categories for the Working Mathematician. Springer.Google Scholar
- [MM 83a]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.Google Scholar
- [MM 83b]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.Google Scholar
- [Rei 80]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.Google Scholar
- [ST 83]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.Google Scholar
- [Tar 83]Tarlecki, A. Free constructions in algebraic institutions. Report CSR-149-83. Dept. of Computer Science, Univ. of Edinburgh.Google Scholar