[Paper Review] SCL(FOL) Revisited

Martin Bromberger, Simon Schwarz+1 more|arXiv (Cornell University)|Feb 12, 2023

Logic, programming, and type systems4 citations

TL;DR

This paper revisits the SCL(FOL) calculus for first-order logic without equality by introducing a stronger regularity definition that ensures non-redundant clause learning without exhaustive propagation, and adds trail and model bounding to guarantee termination. The result is a refutationally complete, non-redundant, and terminating decision procedure for logics with the finite model property, with rigorous soundness and completeness proofs.

ABSTRACT

This paper presents an up-to-date and refined version of the SCL calculus for first-order logic without equality. The refinement mainly consists of the following two parts: First, we incorporate a stronger notion of regularity into SCL(FOL). Our regularity definition is adapted from the SCL(T) calculus. This adapted definition guarantees non-redundant clause learning during a run of SCL. However, in contrast to the original presentation, it does not require exhaustive propagation. Second, we introduce trail and model bounding to achieve termination guarantees. In previous versions, no termination guarantees about SCL were achieved. Last, we give rigorous proofs for soundness, completeness and clause learning guarantees of SCL(FOL) and put SCL(FOL) into context of existing first-order calculi.

Motivation & Objective

• To address the lack of termination guarantees in prior SCL(FOL) versions by introducing trail and model bounding mechanisms.
• To refine the regularity condition in SCL(FOL) to ensure non-redundant clause learning without requiring exhaustive propagation.
• To provide a formal, rigorous proof of soundness, completeness, and clause learning guarantees for SCL(FOL).
• To position SCL(FOL) within the broader landscape of first-order calculi and clarify its theoretical foundations.
• To establish SCL(FOL) as a decision procedure for logics with the finite model property, including the Bernays-Schoenfinkel class.

Proposed method

• Introduces a refined regularity definition adapted from SCL(T), ensuring non-redundant clause learning without exhaustive propagation.
• Implements trail bounding by restricting the set of literals considered based on a well-founded ordering, limiting the search space.
• Introduces a grow rule to incrementally increase the bound β, enabling fair exploration of larger literal sets.
• Uses a ground trail to maintain a partial candidate model, with propagation and decision rules applied to literals in a trail-ordered manner.
• Applies conflict-driven clause learning via resolution and factoring on conflicting ground instances, producing non-redundant, non-ground learned clauses.
• Employs a fixed subset ordering ≺⊆ to compare clauses and ensure well-foundedness of the clause learning process.

Experimental results

Research questions

• RQ1 Can SCL(FOL) be extended to guarantee non-redundant clause learning without requiring exhaustive propagation?
• RQ2 How can termination be ensured in SCL(FOL) without sacrificing completeness or non-redundancy?
• RQ3 What formal guarantees can be provided for soundness, completeness, and clause learning in SCL(FOL)?
• RQ4 Can SCL(FOL) be made a decision procedure for the Bernays-Schoenfinkel class and other logics with the finite model property?
• RQ5 How does the new regularity definition compare to previous notions in terms of efficiency and correctness?

Key findings

• The refined regularity condition ensures that all learned clauses are non-redundant with respect to the input and previously learned clauses, even without exhaustive propagation.
• Trail and model bounding via β-limiting guarantees termination for any logic with the finite model property, including the Bernays-Schoenfinkel class.
• The calculus is refutationally complete: if an input clause set is unsatisfiable, the empty clause is eventually derived.
• The SCL(FOL) calculus is sound and complete, with rigorous formal proofs now provided for these properties.
• The grow rule enables fair exploration of larger literal sets, allowing refutation to be reached even when initial bounds are insufficient.
• The number of non-redundant ground clauses is finite under the bounded setting, due to the well-foundedness of the subset ordering ≺⊆.