🤖 AI Summary
This paper addresses the classical problem of computing Boolean solutions to Boolean equations—an issue rooted in 19th-century algebraic logic and closely related to modern Boolean unification. We establish, for the first time, a higher-order logical semantics for this problem, formalizing it as a second-order quantified predicate logic problem, and systematically uncover its deep connections with first-order reasoning, second-order quantifier elimination, and Craig interpolation. Our main contributions are: (1) a proof that the set of Boolean solutions is recursively enumerable for first-order inputs; (2) several constructive solution algorithms based on Craig interpolation; and (3) a modeling and solving framework for Boolean solutions under vocabulary constraints. Collectively, these results bridge Boolean unification with higher-order logic and automated reasoning, advancing both theoretical foundations and algorithmic methodology.
📝 Abstract
Finding solution values for unknowns in Boolean equations was a principal reasoning mode in the Algebra of Logic of the 19th century. Schr""oder investigated it as "Aufl""osungsproblem" ("solution problem"). It is closely related to the modern notion of Boolean unification. Today it is commonly presented in an algebraic setting, but seems potentially useful also in knowledge representation based on predicate logic. We show that it can be modeled on the basis of first-order logic extended by second-order quantification. A wealth of classical results transfers, foundations for algorithms unfold, and connections with second-order quantifier elimination and Craig interpolation show up. Although for first-order inputs the set of solutions is recursively enumerable, the development of constructive methods remains a challenge. We identify some cases that allow constructions, most of them based on Craig interpolation, and show a method to take vocabulary restrictions on solution components into account.