Quantales and Temporal Logics

Abstract:
We provide an algebraic semantics for the temporal logic CTL^* and simplify it for its sublogics CTL and LTL. We abstractly represent state and path formulas over transition systems in Boolean left quantales. These are complete lattices with an operation of multiplication that is completely disjunctive in its left argument and isotone in its right argument. On these quantales, the semantics of CTL^* formulas can be encoded via finite and infinite iteration operators, the CTL and LTL operators can be related to domain operators. This yields interesting new connections between representations as known from the modal μ-calculus and Kleene/ω-algebraic ones.