Logics for Classes of Boolean Monoids

• ABSTRACT:
  
  This paper presents the algebraic and Kripke model soundness and completeness of a logic over Boolean monoids. An additional axiom added to the logic will cause the resulting monoid models to be representable as monoids of relations. A star operator, interpreted as reflexive, transitive closure, is conservatively added to the logic. The star operator is a relative modal operator, i.e., one that is defined in terms of another 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 a collection of abstract topological uniformities which become concrete when the Kripke models are dual to monoids of relations. The machinery for the star operator is shown to be a recasting of Scott-Montague neighborhood models. An interpretation of the Kripke frames and properties thereof is presented in terms of certain CMOS transistor networks and some circuit transformation equivalences. The worlds of the Kripke frame are wires and the Kripke relation is a specialized CMOS pass transistor network.

• BIBTEX:
  

  @article{ADPLogics,
   author = {Gerard Allwein and Hilmi Demir and Lee Pike},
   title = {Logics for Classes of Boolean Monoids},
   journal = {Journal of Logic, Language and Information},
   volume = {13},
   number = {3},
   year = {2004},
   issn = {0925-8531},
   pages = {241--266},
   publisher = {Kluwer Academic Publishers},
   note = {Available at \url{https://leepike.github.io/pub_pages/boolean_monoids.html}} }

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