Formal Aspects of Computing

, Volume 6, Issue 3, pp 339–358 | Cite as

Peirce algebras

  • Chris Brink
  • Katarina Britz
  • Renate A. Schmidt


We present a two-sorted algebra, called aPeirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a set-forming operator on relations (the Peirce product of Boolean modules) and a relation-forming operator on sets (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the so-calledterminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.


Boolean modules Relation algebras Terminological logics Weakest prespecifications 


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  1. [BlJ72]
    Blyth, T. S. and Janowitz, M. F..:Residuation Theory. Pergamon Press, Oxford, England, 1972.Google Scholar
  2. [Bli87]
    Blikle, A.:MetaSoft Primer, Lecture Notes in Computer Science 288. Springer-Verlag, 1987.Google Scholar
  3. [Böt92a]
    Böttner, M.: State transition semantics.Theoretical Linguistics, 18(2/3):239–286, 1992.Google Scholar
  4. [Böt92b]
    Böttner, M.: Variable-free semantics for anaphora.Journal of Philosophical Logic, 21:375–390, 1992.Google Scholar
  5. [Bri81]
    Brink, C.: Boolean modules.Journal of Algebra, 71(2):291–313, 1981.Google Scholar
  6. [Bri88]
    Brink, C.: On the application of relations.South African Journal of Philosophy, 7(2):105–112, 1988.Google Scholar
  7. [Bri88]
    Britz, K.: Relations and programs. Master's thesis, Department of Computer Science, University of Stellenbosch, Stellenbosch, South Africa, 1988.Google Scholar
  8. [BrS85]
    Brachman, R. J. and Schmolze, J. G.: An overview of theKl-ONE knowledge representation system.Cognitive Science, 9(2):171–216, 1985.Google Scholar
  9. [BrS92]
    Brink, C. and Schmidt, R. A.: Subsumption computed algebraically.Computers and Mathematics with Applications, 23:329–342, 1992.Google Scholar
  10. [ChT51]
    Chin, L. H. and Tarski, A.: Distributive and modular laws in the arithmetic of relation algebras.University of California Publications in Mathematics, 1(9):341–384, 1951.Google Scholar
  11. [dBd73]
    de Bakker, J. W. and de Roever, W. P.: A calculus for recursive program schemes. In M. Nivat, editor,Symposium on Automata, Formal Languages and Programming. North Holland, Amsterdam, 1973.Google Scholar
  12. [DLN91]
    Donini, F. M. Lenzerini, M. Nardi, D. and Nutt. W.: The complexity of concept languages. InProceedings of the Second International Conference on Principles of Knowledge Representation and Reasoning, pages 151–162. Morgan Kaufmann, San Mateo, California, 1991.Google Scholar
  13. [Har84]
    Harel, D.: Dynamic logic. In D. Gabbay and F. Guenthner, editors,Handbook of Philosophical Logic, volume II, pages 497–604. Reidel, Dordrecht, Holland, 1984.Google Scholar
  14. [Hen80]
    Hennessey, M. C. B.: A proof system for the first-order relational calculus.Journal of Computer and System Sciences, 20:96–110, 1980.Google Scholar
  15. [HoJ87]
    Hoare, C. A. R. and Jifeng, He.: The weakest prespecification.Information Processing Letters, 24:127–132, 1987.Google Scholar
  16. [HJS87]
    Hoare, C. A. R. Jifeng, He and Sanders, J. W.: Prespecification in data refinement.Information Processing Letters, 25:71–76, 1987.Google Scholar
  17. [HMT85]
    Henkin, L. Monk, J. D. and Tarski, A.:Cylindric Algebras I, volume 64 ofStudies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1985.Google Scholar
  18. [HNS90]
    Hollunder, B. Nutt, W. and Schmidt-Schauß. M.: Subsumption algorithms for concept description languages. InProceedings of the 9th European Conference on Artificial Intelligence, pages 348–353, 1990.Google Scholar
  19. [HiP72]
    Hitchcock, P. and Park, D.: Induction rules and termination proofs. In M. Nivat, editor,Automata, Languages and Programming. North-Holland, Amsterdam, 1972.Google Scholar
  20. [Jón82]
    Jónsson, B.: Varieties of relation algebras.Algebra Universalis, 15:273–298, 1982.Google Scholar
  21. [JóT51]
    Jónsson, B. and Tarski, A.: Boolean algebras with operators, Part I.American Journal of Mathematics, 73:891–939, 1951.Google Scholar
  22. [JóT52]
    Jónsson, B. and Tarski, A.: Boolean algebras with operators, Part II.American Journal of Mathematics, 74:127–162, 1952.Google Scholar
  23. [Koz80]
    Kozen, D.: A representation theorem for models of *-free PDL. In J. de Bakker and J. van Leeuwen, editors,Automata, Languages and Programming, volume 85 ofLecture Notes in Computer Science, pages 351–36. Springer-Verlag, 1980.Google Scholar
  24. [Koz81]
    Kozen, D.: On the duality of dynamic algebras and Kripke models. In E. Engeler, editor,Logic of Programs, volume 125 ofLecture Notes in Computer Science, pages 1–11. Springer-Verlag, 1981.Google Scholar
  25. [Mad83]
    Maddux, R. D.: A sequent calculus for relation algebras.Annals of Pure and Applied Logic, 25:73–101, 1983.Google Scholar
  26. [Mad90]
    Maddux, R. D. 1990.: Personal communication with C. Brink.Google Scholar
  27. [Orl91a]
    Orlowska, E.: Relational interpretation of modal logic. In H. Andréka, J. D. Monk, and I. Németi, editors,Algebraic Logic, volume 54 ofColloquia Mathematica Societatis János Bolyai, pages 443–471. North-Holland, Amsterdam, 1991.Google Scholar
  28. [Orl91b]
    Orlowska, E.: Relational proof systems for some AI logics. In Ph. Jorrand and J. Kelemen, editors,Fundamentals of Artificial Intelligence Research, volume 535 ofLecture Notes in Artificial Intelligence, pages 33–47. Springer-Verlag, Berlin, 1991.Google Scholar
  29. [Par81]
    Parikh, D.: Propositional dynamic logic of programs: A survey. In E. Engeler, editor,Logic of Programs, volume 125 ofLecture Notes in Computer Science, pages 102–144. Springer-Verlag, 1981.Google Scholar
  30. [Pra90]
    Pratt, V. R.: Dynamic algebras as a well-behaved fragment of relation algebras. In C. H. Bergman, R. D. Maddux, and D. L. Pigozzi, editors,Algebraic Logic and Universal Algebra in Computer Science, volume 425 ofLecture Notes in Computer Science, pages 77–110. Springer-Verlag, 1990.Google Scholar
  31. [Pre90]
    Pretorius, J. P. G.: The algebra and topology of Boolean modules. Master's thesis, Department of Mathematics, University of Cape Town, Cape Town, South Africa, 1990.Google Scholar
  32. [PaS87]
    Patel-Schneider, P. F.:Decidable, Logic-Based Knowledge Representation. PhD thesis, University of Toronto, 1987.Google Scholar
  33. [PaS89]
    Patel-Schneider, P. F.: A four-valued semantics for terminological logics.Artificial Intelligence, 38:319–351, 1989.Google Scholar
  34. [San81]
    Sanderson, J. G.:A Relational Theory of Computing, volume 82 ofLecture Notes in Computer Science. Springer-Verlag, 1981.Google Scholar
  35. [Sch91]
    Schmidt, R. A.: Algebraic terminological representation. Master's thesis, Department of Mathematics, University of Cape Town, Cape Town, South Africa, 1991. Available as Thesis-Reprint TR 011. Also as Technical Report MPI-I-91-216, Max-Planck-Institut für Informatik, Saarbrücken, Germany.Google Scholar
  36. [Sch93]
    Schmidt, R. A.: Terminological representation, natural language & relation algebra. In H. J. Ohlbach, editor,Proceedings of the sixteenth German AI Conference (GWAI-92), volume 671 ofLecture Notes in Artificial Intelligence, pages 357–371. Springer-Verlag, Berlin, 1993.Google Scholar
  37. [ScM91]
    Schmolze, J. G. and Mark. W. S.: The NIKL experience.Computational Intelligence, 6:48–69, 1991.Google Scholar
  38. [ScS93]
    Schmidt, G. and Ströhlein, T.:Relations and Graphs. Springer-Verlag, Berlin-Heidelberg, 1993.Google Scholar
  39. [SSS91]
    Schmidt-Schauß, M. and Smolka, G.: Attributive concept description with complement.Artificial Intelligence, 48:1–26, 1991.Google Scholar
  40. [Sup76]
    Suppes, P.: Elimination of quantifiers in the semantics of natural language by use of extended relation algebra.Revue Internationale de Philosophie, 30:243–259, 1976.Google Scholar
  41. [Sup79]
    Suppes, P.: Variable-free semantics for negations with prosodic variation. In E. Saarinen, R. Hilpinen, I. Niiniluoto, and M. P. Hintikka, editors,Essays in Honour of Jaakko Hintikka, pages 49–59. Reidel, Dordrecht, Holland, 1979.Google Scholar
  42. [Sup81]
    Suppes, P.: Direct inference in English.Teaching Philosophy, 4:405–418, 1981.Google Scholar
  43. [Tar41]
    Tarski, A.: On the calculus of relations.Journal of Symbolic Logic, 6:73–89, 1941.Google Scholar
  44. [Tar55]
    Tarski, A.: Contributions to the theory of models, Part III.Indagationes Mathematicae, 17:56–64, 1955.Google Scholar
  45. [Uni92]
    University of Amsterdam.Logic at Work, Proceedings of the Applied Logic Conference, Amsterdam, The Netherlands, December 1992. Preprint. To appear.Google Scholar
  46. [vBe91]
    van Benthem, J.: Logic and the flow of information. ILLC Prepublication Series for Logic, Semantics and Philosophy of Language LP-92-11, Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands, 1991. To appear in D. Prawitz, B. Skyrms, and D. Westerstahl, editors,Proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science, Uppsala 1991. North-Holland, Amsterdam.Google Scholar
  47. [vBe92]
    van Benthem, J.: A note on dynamic arrow logic. ILLC Prepublication Series for Logic, Semantics and Philosophy of Language LP-92-11, Institute for Language, Logic and Computation, University of Amsterdam, Amsterdam, The Netherlands, September 1992. To appear in J. van Eyck, and A. Visser, editors,Logic and Information Flow. Studies in Logic, Language and Information, Kluwer, Dordrecht.Google Scholar
  48. [Ven92]
    Venema, Y.:Many-Dimensional Modal Logic. PhD thesis, University of Amsterdam, Amsterdam, The Netherlands, 1992.Google Scholar
  49. [Wad75]
    Wadge, W. W.: A complete natural deduction system for the relational calculus. Theory of Computation Report, University of Warwick, 1975.Google Scholar
  50. [WoS92]
    Woods, W. A. and Schmolze, J. G.: TheKl-one family.Computers and Mathematics with Applications, 23:133–177, 1992.Google Scholar

Copyright information

© BCS 1994

Authors and Affiliations

  • Chris Brink
    • 1
  • Katarina Britz
    • 1
  • Renate A. Schmidt
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

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