## Abstract

In its modern form the algebra of relations has been under investigation by mathematicians since Tarski’s seminal paper of (1941). The main line of development has been the study of a class of algebras called *relation algebras* (Chin and Tarski 1951, Jónsson 1982), in parallel with developments such as Boolean algebras with operators (Jónsson and Tarski 1951/1952) and cylindric algebras (Henkin, Monk and Tarski 1985). Since the early seventies the algebra of relations has increasingly become of interest to computer scientists. Just as the notion of a partial function provides a natural model for deterministic programs, so the more general notion of a (binary) relation provides a natural model for nondeterministic programs. This idea has been exploited by various authors. For example, it is evident in Floyd-Hoare logic for program verification, it has been extended to specification in Hoare and He, Jifeng (1987), it figures in logics of programs such as dynamic logic (Parikh 1981, Harel 1984), and it was used in the early seventies to model recursive procedures (de Bakker and de Roever 1973, Hitchcock and Park 1972). Recently the algebra of relations has been extensively used in a graph-theoretic approach to programs by Schmidt and Ströhlein (1991). In modal logic, relation algebra features strongly in the Dutch-Hungarian cooperation on van Benthem’s (1991) new *arrow logic* (see *Logic at Work, Proceedings of the Applied Logic Conference* (1992)). Venema (1992) is another interdisciplinary study of relation algebra and multi-modal logic. The proof theory of relations is also of interest to computer scientists, and several relational inference systems are available (Wadge 1975, Hennessy 1980, Maddux 1983, Orlowska 1991).

## Keywords

Modal Logic Boolean Algebra Relation Algebra Dynamic Logic Early Seventy## Preview

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## References

- Brink, C. (1981), Boolean modules,
*Journal of Algebra***71**(2), 291–313.CrossRefzbMATHMathSciNetGoogle Scholar - Brink, C. and Schmidt, R. A. (1992), Subsumption computed algebraically,
*Computers and Mathematics with Applications***23**(2-9), 329–342.CrossRefzbMATHGoogle Scholar - Chin, L. H. and Tarski, A. (1951), Distributive and modular laws in the arithmetic of relation algebras,
*Univ. Calif. Publ. Math.***1**(9), 341–384.MathSciNetGoogle Scholar - de Bakker, J. W. and de Roever, W. P. (1973), A calculus for recursive program schemes,
*in*M. Nivat (ed.),*Symposium on Automata, Formal Languages and Programming*,North Holland, Amsterdam.Google Scholar - Harel, D. (1984), Dynamic logic,
*in*D. Gabbay and F. Guenther (eds),*Handbook of Philosophical*Logic,Vol. II, Reidel Publ. Co., Dordrecht, Holland, pp. 497–604.Google Scholar - Henkin, L., Monk, J. D. and Tarski, A. (1985),
*Cylindric Algebras: Part II*,Vol. 115 of*Studies in Logic and the Foundations of Mathematics*,North-Holland, Amsterdam.Google Scholar - Hennessy, M. C. B. (1980), A proof-system for the first-order relational calculus,
*Journal of Computer and System Sciences***20**, 96–110.CrossRefzbMATHMathSciNetGoogle Scholar - Hitchcock, P. and Park, D. (1972), Induction rules and termination proofs,
*in*M. Nivat (ed.),*Automata, Languages and Programming*,North-Holland, Amsterdam.Google Scholar - Hoare, C. A. R. and He, Jifeng (1987), The weakest prespecification,
*Information Processing Letters***24**, 127–132.CrossRefzbMATHMathSciNetGoogle Scholar - Hoare, C. A. R., He, Jifeng and Sanders, J. W. (1987), Prespecification in data refinement,
*Information Processing Letters***25**, 71–76.CrossRefzbMATHMathSciNetGoogle Scholar - Jónsson, B. (1982), Varieties of relation algebras,
*Algebra Universalis***15**(3), 273–298.CrossRefzbMATHMathSciNetGoogle Scholar - Jónsson, B. and Tarski, A. (1951/1952), Boolean algebras with operators, Part I/II,
*American Journal of Mathematics***73/74**, 891–939/127-162.CrossRefGoogle Scholar - Kozen, D. (1980), A representation theorem for models of *-free PDL,
*in*J. de Bakker and J. van Leeuwen (eds),*Automata, Languages and Programming*,Vol. 85 of*Lecture Notes in Computer Science*,Springer-Verlag, Berlin, pp. 351–362.Google Scholar *Logic at Work, Proceedings of the Applied Logic Conference*(1992), University of Amsterdam. Preprint. To appear.Google Scholar- Maddux, R. D. (1983), A sequent calculus for relation algebras,
*Annals of Pure and Applied Logic***25**, 73–101.CrossRefzbMATHMathSciNetGoogle Scholar - Maddux, R. D. (1990). Personal communication with C. Brink.Google Scholar
- Orlowska, E. (1991), Relational interpretation of modal logic,
*in*H. Andréka, J. D. Monk and I. Németi (eds),*Algebraic Logic*,Vol. 54 of*Colloquia Mathematica Societatis János Bolyai*,North-Holland, Amsterdam, pp. 443–471.Google Scholar - Parikh, D. (1981), Propositional dynamic logic of programs: A survey,
*in*E. Engeler (ed.),*Logic of Programs*,Vol. 125 of*Lecture Notes in Computer Science*,Springer-Verlag, Berlin, pp. 102–144.Google Scholar - Schmidt, G. and Ströhlein, T. (1991),
*Relations and Graphs*,Springer-Verlag, Berlin. Schmidt, R. A. (1993), Terminological representation, natural language & relation algebra,*in*H. J. Ohlbach (ed.),*Proceedings of the sixteenth German AI Conference (GWAI-92)*,Vol. 671 of*Lecture Notes in Artificial Intelligence*,Springer-Verlag, Berlin, pp. 357–371.Google Scholar - Schmidt-Schauß, M. and Smolka, G. (1991), Attributive concept description with complements,
*Artificial Intelligence***48**, 1–26.CrossRefzbMATHMathSciNetGoogle Scholar - Suppes, P. (1976), Elimination of quantifiers in the semantics of natural language by use of extended relation algebras,
*Rev. Int. de Philosophie***30**(3-4), 243–259.MathSciNetGoogle Scholar - Tarski, A. (1941), On the calculus of relations,
*Journal of Symbolic Logic***6**(3), 73–89.CrossRefzbMATHMathSciNetGoogle Scholar - van Benthem, J. (1991), Logic and the flow of information,
*Technical Report, ILLC Prepub-lication Series for Logic, Semantics and Philosophy of Language LP-92-11*,Institute for Logic, Language and Computation, University of Amsterdam. To appear.Google Scholar - Venema, Y. (1992),
*Many-Dimensional Modal Logic*,PhD thesis, University of Amsterdam.Google Scholar - Wadge, W. W. (1975), A complete natural deduction system for the relational calculus,
*Theory of Computation Report 5*,University of Warwick.Google Scholar - Woods, W. A. and Schmölze, J. G. (1992), The KL-ONE family,
*Computers and Mathematics with Applications***23**(2-5), 133–177.CrossRefzbMATHGoogle Scholar