[Paper Review] Craig Interpolation and Access Interpolation with Clausal First-Order Tableaux

Christoph Wernhard|arXiv (Cornell University)|Feb 14, 2018

Advanced Database Systems and Queries2 references3 citations

TL;DR

This paper presents a foundation for computing Craig-Lyndon interpolants and access interpolants using clausal first-order tableaux, enabling efficient integration with modern theorem provers. It introduces a calculus-independent lifting method based on Herbrand’s theorem and shows how structural constraints for access interpolation can be enforced via preprocessing or tableau construction, supporting both top-down and bottom-up theorem proving approaches with practical implementation potential.

ABSTRACT

We develop foundations for computing Craig interpolants and similar intermediates of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient machine-oriented systems based on calculi of two families: goal-oriented like model elimination and the connection method, and bottom-up like the hyper tableau calculus. The presented method for Craig-Lyndon interpolation involves a lifting step where terms are replaced by quantified variables, similar as known for resolution-based interpolation, but applied to a differently characterized ground formula and proven correct more abstractly on the basis of Herbrand's theorem, independently of a particular calculus. Access interpolation is a recent form of interpolation for database query reformulation that applies to first-order formulas with relativized quantifiers and constrains the quantification patterns of predicate occurrences. It has been previously investigated in the framework of Smullyan's non-clausal tableaux. Here, in essence, we simulate these with the more machine-oriented clausal tableaux through structural constraints that can be ensured either directly by bottom-up tableau construction methods or, for closed clausal tableaux constructed with arbitrary calculi, by postprocessing with restructuring transformations.

Motivation & Objective

• To establish a theoretical and practical foundation for computing Craig-Lyndon interpolants using clausal first-order tableaux, applicable to a wide range of modern theorem provers.
• To extend this framework to access interpolation, a form of interpolation relevant to database query reformulation with relativized quantifiers.
• To enable the use of efficient, machine-oriented theorem proving systems—especially those based on clausal tableaux—for interpolant computation without modifying their core algorithms.
• To show that interpolant lifting can be performed abstractly via Herbrand’s theorem, independent of a specific calculus, thereby unifying the treatment of interpolants across different proof systems.

Proposed method

• The paper introduces a lifting transformation that replaces terms in a ground formula with quantified variables, converting a ground interpolant into a first-order interpolant, justified via Herbrand’s theorem.
• It demonstrates that this lifting method is sound and complete for Craig-Lyndon interpolation, regardless of the underlying calculus, provided the tableau is closed and unsatisfiable.
• For access interpolation, the method enforces structural constraints on the tableau to preserve quantifier scope patterns, ensuring that only symbols from the target language appear in the interpolant.
• It distinguishes between 'local' inferences (preprocessing on individual formulas) and 'non-local' inferences (joint processing in the tableau), enabling modular interpolation computation.
• For bottom-up tableaux (e.g., hyper tableaux), structural constraints can be enforced during construction; for other calculi, a postprocessing restructuring step transforms arbitrary closed tableaux into valid ones.
• The approach is general and applies to both goal-oriented (e.g., model elimination, connection method) and data-oriented (e.g., hyper tableau) theorem proving methods.

Experimental results

Research questions

• RQ1 Can Craig-Lyndon interpolants be computed efficiently using clausal tableaux, independent of the underlying calculus, via a Herbrand-based lifting method?
• RQ2 How can access interpolation—requiring constrained quantifier scopes—be supported within the clausal tableau framework?
• RQ3 What is the role of 'local' versus 'non-local' inferences in tableau-based interpolation, and how does this differ from resolution-based approaches?
• RQ4 Can structural constraints for access interpolation be enforced during tableau construction or only via costly postprocessing?
• RQ5 Is there a fundamental advantage of clausal tableaux over resolution in interpolant computation due to their preservation of clause structure?

Key findings

• The proposed lifting method for Craig-Lyndon interpolants is sound and complete, and can be applied uniformly to any clausal tableau prover without modifying its internal logic.
• The method enables the use of any first-order theorem prover that constructs closed clausal tableaux for interpolant computation, leveraging existing optimizations and data structures.
• For access interpolation, structural constraints can be enforced natively in bottom-up methods like the hyper tableau calculus, avoiding costly restructuring.
• When using calculi that do not naturally enforce structural constraints, a postprocessing transformation can restructure the tableau to meet requirements, though this may involve subtableaux duplication.
• The framework supports both top-down (goal-oriented) and bottom-up (data-oriented) theorem proving methods, broadening the applicability of interpolation techniques.
• A first implementation of Craig-Lyndon interpolation using a top-down prover (PIE/CM) is already available, demonstrating the practical feasibility of the approach.