[Paper Review] The Boolean Solution Problem from the Perspective of Predicate Logic -- Extended Version

Christoph Wernhard|arXiv (Cornell University)|Jun 26, 2017

Logic, Reasoning, and Knowledge3 references3 citations

TL;DR

This paper reinterprets the classical Boolean solution problem—originally studied in 19th-century algebra of logic—within first-order logic extended by second-order quantification. It establishes a formal framework linking solution construction to second-order quantifier elimination and Craig interpolation, identifies conditions under which solutions can be constructively generated (especially via interpolation), and shows how vocabulary restrictions can be modeled through two-stage solution problems.

ABSTRACT

Finding solution values for unknowns in Boolean equations was a principal reasoning mode in the Algebra of Logic of the 19th century. Schröder investigated it as Auflösungsproblem (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 become apparent. 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.

Motivation & Objective

• To formalize the Boolean solution problem in first-order logic with second-order quantification, enabling a systematic treatment beyond propositional logic.
• To identify constructive methods for solving Boolean equations in first-order settings, particularly through connections with Craig interpolation and definability.
• To model vocabulary restrictions on solution components by reducing them to two related solution problems.
• To establish a foundation for integrating solution solving with automated reasoning techniques such as second-order quantifier elimination and theorem proving.
• To explore the limits of constructability and identify special cases where solutions can be systematically generated, especially in relational monadic fragments.

Proposed method

• Models the solution problem using second-order formulas, where unknowns are replaced by existentially quantified predicate variables.
• Applies the method of successive eliminations to reduce n-ary solution problems to unary ones, abstracting a classical technique from Boole and Schröder.
• Uses Skolemization to transform formulas with second-order prefixes into quantifier-free forms, enabling analysis via Herbrand-style expansions.
• Leverages Craig interpolation to construct solutions in cases where a formula is definable, particularly when the interpolant serves as a witness for the solution.
• Introduces a two-stage approach to model vocabulary restrictions: first solve for a general solution, then refine it to satisfy constraints on the vocabulary of unknowns.
• Employs definitional normal forms and relaxed substitutability conditions to enable solution construction in relational monadic fragments.

Experimental results

Research questions

• RQ1 Under what conditions can solutions to first-order Boolean equations be constructively generated using standard proof-theoretic techniques?
• RQ2 How can second-order quantifier elimination and Craig interpolation be systematically applied to solve Boolean equations in first-order logic?
• RQ3 In which fragments of first-order logic (e.g., relational monadic) can solutions be effectively constructed, and what relaxations of substitutability allow this?
• RQ4 How can vocabulary restrictions on solution components be formally modeled and enforced within the solution framework?
• RQ5 Can the solution problem be reduced to entailment checking via Skolemization and Herbrand expansion, and under what conditions does this yield effective algorithms?

Key findings

• The set of solutions to a first-order Boolean solution problem is recursively enumerable, but not necessarily decidable, reflecting the undecidability of first-order validity.
• Solutions can be constructed in special cases—particularly when the solution formula corresponds to a Craig interpolant or a definientia of a formula.
• The method of successive eliminations can be abstracted into a general algorithmic framework that reduces n-ary to unary solution problems.
• A solution $ H $ that is free of certain function symbols and satisfies $ H \models F^h $ (where $ F^h $ is a quantifier-free Skolemized form) is also a solution to the original formula $ F $, under the $ \mathsf{CLEAN}(F) $ condition.
• The concept of Schröder’s reproductive interpolant, while foundational in Boolean algebra, does not generalize directly to first-order logic due to the lack of full distributivity and closure under substitution.
• Vocabulary restrictions on solution components can be modeled by formulating two consecutive solution problems: one for the general solution and one for the restricted variant, enabling modular construction.