# Theories of Point Relations and Relational Model Checking

## Abstract

In this chapter we consider logics providing a means of relational reasoning in the theories which refer to individual objects of their domains. There are two relational formalisms for coping with the objects. A logic\({\mathsf{RL}}_{\mathit{a}x}(\mathbb{C})\)presented in Sect. 3.2is a purely relational formalism, where objects are introduced through point relations which, in turn, are presented axiomatically with a well known set of axioms. The axioms say that a binary relation is a point relation whenever it is a non-empty right ideal relation with one-element domain. We recall that a binary relation*R*on a set*U*is right ideal whenever*R*; 1 =*R*, where 1 =*U*×*U*. In other words such an*R*is of the form*X*×*U*, for some*X*⊆*U*. We may think of right ideal relations as representing sets, they are sometimes referred to as vectors (see[SS93]). If the domain of a right ideal relation is a singleton set, the relation may be seen as a representation of an individual object. A logic\({\mathsf{RL}}_{\mathit{df }}(\mathbb{C})\)presented in Sect. 3.3includes object constants in its language interpreted as singletons. Moreover, associated with each object constant*c*is a binary relation*C*, such that its meaning in every model is defined as a right ideal relation with the domain consisting of the single element being the meaning of*c*. The logic\({\mathsf{RL}}_{\mathit{a}x}(\mathbb{C})\)will be applied in Sect. 16.5 to the relational representation of some temporal logics. The logic\({\mathsf{RL}}_{\mathit{df }}(\mathbb{C})\)will be applied in Chap. 15 to the relational representation of the logic for order of magnitude reasoning. In Sects. 3.4and 3.5we present the methods of model checking and verification of satisfaction of a formula by some given objects in a finite model, respectively. The methods are based on the development of a relational logic which enables us to replace the problems of model checking and verification of satisfaction by the problems of verification of validity of some formulas of this logic. The logic is obtained fromRL(1, 1*′*)-logic by an appropriate choice of object constants and relational constants in its language and by some specific postulates concerning its models. Then, a dual tableau for the logic is obtained from theRL(1, 1*′*)-dual tableau by adapting it to this language and by adding the rules which reflect these specific semantic postulates.