Modal Theory of Arrows: Arrows Logics 1

摘要

The notion of Arrow Structure (AS) is introduced as an algebraic version of thenotion of directed multigraph. By means of a special kind of a representation theorem for arrow structures, it is shown that the whole information of an AS is contained in the set of its arrows equipped with four binary relations describing the four possibilites for two arrows to have a common point. This makes possible the use of arrow structures as a semantic base for a special polymodal logic called BAL (Basic Arrow Logic). BAL and various kinds of its extensions are used for expressing in a modal setting different properties of arrow structures. Several kinds of completeness theorems for BAL and some other arrow logics are proved, including completeness with respect to classes of finite models. Some open problems and possibilities for further development of the arrow approach are formulated.