Branching Time Axiomatized With the Use of Change Operators

Abstract

Abstract We present a temporal logic of branching time with four primitive operators: $\exists {\mathcal {C}}$ – it may change whether; $\forall {\mathcal {C}} $ – it must change whether; $\exists \Box $ – it may be endlessly unchangeable that; and $\forall \Box $ – it must be endlessly unchangeable that. Semantically, operator $\forall {\mathcal {C}}$ expresses a change in the logical value of the given formula in every state that may be an immediate successor of the one considered, while $\exists {\mathcal {C}}$ expresses a change in the logical value of the given formula in some state that is a possible immediate successor. $\forall {\mathcal {C}} $ and $\exists {\mathcal {C}}$ are not normal operators, they are not mutually definable and have unusual properties: $\forall {\mathcal {C}} A\leftrightarrow \forall {\mathcal {C}} \neg A$ and $\exists {\mathcal {C}} A \leftrightarrow \exists {\mathcal {C}} \neg A$ are axioms. Operators $\exists \Box $ and $\forall \Box $ express endless unchangeability in some and all paths, respectively. Our axiomatization contains two infinitary rules that allow one to obtain $\exists \Box A$ from infinitely many combinations of formulas with $A, \neg , \forall {\mathcal {C}}, \land $, and to get $\forall \Box A$ from infinitely many combinations of formulas with $A, \neg , \exists {\mathcal {C}}, \land $. We show that our formalism is complete by presenting a bridge between our logic and logic $\textsf {UB}$, which is a fragment of $\textsf {CTL}$ in which until operators does not occur.