Journal of Logic, Language and Information

, Volume 13, Issue 3, pp 241–266 | Cite as

Logics for Classes of Boolean Monoids

  • Gerard Allwein
  • Hilmi Demir
  • Lee Pike


This paper presents the algebraic and Kripke modelsoundness and completeness ofa logic over Boolean monoids. An additional axiom added to thelogic will cause the resulting monoid models to be representable as monoidsof relations. A star operator, interpreted as reflexive, transitiveclosure, is conservatively added to the logic. The star operator isa relative modal operator, i.e., one that is defined in terms ofanother modal operator. A further example, relative possibility,of this type of operator is given. A separate axiom,antilogism, added to the logic causes the Kripke models to support acollection of abstract topological uniformities which become concretewhen the Kripke models are dual to monoids of relations. The machineryfor the star operator is shownto be a recasting of Scott-Montague neighborhood models. An interpretationof the Kripke frames and properties thereof is presented in terms ofcertain CMOS transister networks and some circuit transformation equivalences.The worlds of the Kripke frame are wires and the Kripke relation is a specializedCMOS pass transistor network.

algebras of relations Boolean monoids CMOS circuits correspondence theory Kripke frames relative modalities 


  1. Birkhoff, G., 1967, Lattice Theory, Providence, RI: American Mathematical Society.Google Scholar
  2. Blackburn, P., de Rijke, M., and Venema, Y., 2001, Modal Logic, Cambridge Tracts in Theoretical Computer Science, Vol. 53, Cambridge: Cambridge University Press.Google Scholar
  3. Chellas, B.F., 1980, Modal Logic: An Introduction, Cambridge: Cambridge University Press.Google Scholar
  4. Dunn, J.M., 1990, "Gaggle theory: An abstraction of Galois connections and residuation with ap-plications to negation and various logical operations," in Logics in AI, Proceedings European Workshop JELIA, Lecture Notes in Computer Science, Vol. 478, Berlin: Springer-Verlag.Google Scholar
  5. Dunn, J.M., 2001, Algebraic Methods in Philosophical Logic, Oxford Logic Guides, Vol. 41, Oxford: Oxford University Press.Google Scholar
  6. Fletcher, P. and Lindgren, W.F., 1982, Quasi-Uniform Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 77, New York: Marcel Dekker.Google Scholar
  7. Graetzer, G., 1979, Universal Algebra, Berlin: Springer-Verlag.Google Scholar
  8. James, I.M., 1987, Topological and Uniform Spaces, Berlin: Springer-Verlag.Google Scholar
  9. Jónsson, B. and Tarski, A., 1951–1952, "Boolean algebras with operators," American Journal of Mathematics 73–74, 891–939, 127–162.Google Scholar
  10. Kozen, D., 1990, "A completeness theorem for Kleene algebras and the algebra of regular events," Technical Report 90-1123, Cornell University.Google Scholar
  11. Ng, K.C., 1984, "Relation algebras with transitive closure," Ph.D. Thesis, University of California, Berkeley.Google Scholar
  12. Ng, K.C. and Tarski, A., 1977, "Relation algebras with transitive closure," Notices of the American Mathematics Society 24, A29–A30.Google Scholar
  13. Pratt, V., 1990a, "Action logic and pure induction," pp. 97–120 in Proceedings, Logic in AI: European Workshop JELIA 1990, Lecture Notes in Computer Science, Vol. 478, Berlin: Springer-Verlag.Google Scholar
  14. Pratt, V., 1990b, "Dynamic algebras as a well behaved fragment of relation algebras," in Proceedings, Algebra and Computer Science, Lecture Notes in Computer Science, Berlin: Springer-Verlag.Google Scholar
  15. Routley, R. and Meyer, R.K., 1973, "The semantics of entailment," pp. 194–243 in Truth, Syntax, and Modality, H. Leblanc, ed., Amsterdam: North-Holland.Google Scholar
  16. Sahlqvist, H., 1975, "Completeness and correspondence in the first and second order semantics for modal logic," pp. 110–143 in Proceedings of the Third Scandanavian Logic Symposium, Uppsala, 1973, Amsterdam: North-Holland.Google Scholar
  17. Stone, M.H., 1936, "The theory of representation for Boolean algebras," Transactions of the American Mathematical Society 40, 37–111.Google Scholar
  18. van Benthem, J.F.A.K., 1984, Modal Correspondence Theory, Vol. II, pp. 167–247, Dordrecht: D. Reidel.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Gerard Allwein
    • 1
  • Hilmi Demir
    • 1
  • Lee Pike
    • 1
  1. 1.Naval Research LaboratoryWashingtonU.S.A.

Personalised recommendations