Algebraic Manipulation of Modalities

Unlike knowledge representation in modal logic, human tends to introduce new modalities on ad hoc basis in communication. Similar challenges arise from applications of distributed computing and hybrid systems that involve several time domains. This paper presents a parameterized approach for algebraic manipulation of modalities. Each pair of modalities of possibility and necessity is determined by an accessibility relation based on Kripke semantics (or a restriction that the accessibility relation satisfies). Traditional low-level pointwise arguments for correspondences between modal axioms and Kripke frame conditions can then be lifted to a higher level of reasoning with algebraic laws. A number of well-known frame conditions are studied, including some complicated frame conditions (such as weak connectedness), which can be treated neatly. Each frame condition is shown to be equivalent to a semantic condition in a shape similar to the corresponding axiom schema. The technique is first presented semantically. Once the algebraic correspondence between an axiom schema and a frame condition is established, it can be shown that any theorem of the modal system must be valid in any Kripke model satisfying the frame condition. For completeness of the method, we prove that any well-formed formula whose semantics satisfies the semantic condition that is equivalent to the frame condition must be a theorem of the modal system. The technique is applied to solve the puzzle of unfaithful wives in knowledge representation.