Evaluation Trees for Proposition Algebra - The Case for Free and Repetition-Proof Valuation Congruence

Proposition algebra is based on Hoare’s conditional connective, which is a ternary connective comparable to if-then-else and used in the setting of propositional logic. Conditional statements are provided with a simple semantics that is based on evaluation trees and that characterizes so-called free valuation congruence: two conditional statements are free valuation congruent if, and only if, they have equal evaluation trees. Free valuation congruence is axiomatized by the four basic equational axioms of proposition algebra that define the conditional connective. Valuation congruences that identify more conditional statements than free valuation congruence are repetition-proof, contractive, memorizing, and static valuation congruence. Each of these valuation congruences is characterized using a transformation on evaluation trees: two conditional statements are C-valuation congruent if, and only if, their C-transformed evaluation trees are equal. These transformations are simple and natural, and only for static valuation congruence a slightly more complex transformation is used. Also, each of these valuation congruences is axiomatized in proposition algebra. A spin-off of our approach is “basic form semantics for proposition algebra”: for each valuation congruence C considered, two conditional statements are C-valuation congruent if, and only if, they have equal C-basic forms, where C-basic forms are obtained by a syntactic transformation of conditional statements, which is a form of normalization.