Branching Time: The System K _b

Abstract

The concept of a “branching structure” includes the notion of a “tree”, of which it is a generalization. That is, dropping the conditions of discreteness and rootedness from the concept of a tree, we define a branching structure to be a set 7 over which a binary relation U (the accessibility relation of temporal posteriority) is defined, satisfying the conditions

$$(\forall x)(\forall y)(\forall z)[(Uxy\& Uyz) \supset Uxz][transitivity]$$

and

$$(\forall x)(\forall y)(\forall z)[(Uxz\& Uyz) \supset (Uxy \vee x = y \vee Uyx)][backwards{\kern 1pt} linearity]$$

Some examples of branching structures are Here the points (indicated in the finite case by nodal circles) are 7-elements and the connecting lines (indicated in the finite case by arrows and in the infinite case by relative placement from left to right) represent the U-relation.