Logics for Classes of Boolean Monoids

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 transister 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. frames, relative modalities 1. Boolean Monoids (BM) Boolean monoids consist of a Boolean lattice and a monoid operation that is ordered by the lattice. The unit of the monoid is not necessarily the top to the lattice. Boolean monoids are simpler structures than propositional dynamic algebras but slightly more complicated than action algebras (Pratt, 1990a, 1990b). It turns out in both cases, however, Boolean monoids have simpler dual Kripke frames. Propositional dynamic logic has two sorts, a relational sort and a propositional sort. The relational sort is to represent bits of program at a fairly high level of generality, while the propositional sort is to express pre and post conditions “between” bits of program. A relation then is thought to be an input-output relation of information, namely the information which makes the pre and post conditions true. In practice, usually only the post conditions are used. The expressions are of the form: [a]p, where [a] is a relation or modality and p is a Boolean proposition or formula in classical propositional logic, depending upon whether the dynamic algebras or