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Automated Theorem Generation in First-Order Logic by Construction from Triangular Standard Contradictions

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Briefings

A new framework for automated theorem generation in first-order logic uses triangular standard contradictions to construct minimally unsatisfiable clause sets. This approach generates theorems by keeping some clauses as premises and negating one, ensuring correctness by construction without separate proof procedures. The system extends an existing $\Delta_1$ generator into eight methods ($\Delta_2$ through $\Delta_8$), with four methods implemented and released. Combinatorial analysis shows the system can generate large numbers of distinct theorems systematically.

Key points

• The method constructs theorems by retaining selected first-order literals as premises and negating one clause, ensuring the remaining premises are satisfiable while the full set is unsatisfiable.
• Eight theorem-generation methods ($\Delta_2$ through $\Delta_8$) were developed, with four methods ($\Delta_1$ through $\Delta_4$) implemented and released.

Context

Automated theorem generation (ATG) aims to construct new theorems rather than just proving given conjectures. This research proposes a constructive framework based on triangular standard contradictions, extending a previously developed $\Delta_1$ generator into a unified family of eight methods.

Limitations

Evidence is limited to the arXiv preprint v1; no experimental results, performance metrics, or real-world applications are provided. The framework remains theoretical, and the specific number of theorems generated by the four implemented methods is not quantified.

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