Extended Triangular Method: A Generalized Algorithm for Contradiction Separation Based Automated Deduction

2025-10-12

AI Summary

Automated deduction faces a fundamental trade-off between completeness and efficiency, while conventional binary resolution cannot support collaborative reasoning across multiple clauses. Method: This paper proposes Extended Triangular Method (ETM), a generalized multi-clause dynamic inference algorithm grounded in contradiction separation. We formally define the contradiction separation framework—unifying diverse contradiction construction strategies—and introduce a triangular geometric model that enables flexible clause interaction and concurrent derivation. Contribution/Results: ETM overcomes the binary restriction of resolution, substantially enhancing both completeness and computational efficiency in theorem proving. As a core inference engine integrated into several high-performance automated theorem provers, ETM achieves state-of-the-art performance on the TPTP benchmark suite and in the CASC international competition, empirically validating its theoretical novelty and practical applicability.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Automated deduction lies at the core of Artificial Intelligence (AI), underpinning theorem proving, formal verification, and logical reasoning. Despite decades of progress, reconciling deductive completeness with computational efficiency remains an enduring challenge. Traditional reasoning calculi, grounded in binary resolution, restrict inference to pairwise clause interactions and thereby limit deductive synergy among multiple clauses. The Contradiction Separation Extension (CSE) framework, introduced in 2018, proposed a dynamic multi-clause reasoning theory that redefined logical inference as a process of contradiction separation rather than sequential resolution. While that work established the theoretical foundation, its algorithmic realization remained unformalized and unpublished. This work presents the Extended Triangular Method (ETM), a generalized contradiction-construction algorithm that formalizes and extends the internal mechanisms of contradiction separation. The ETM unifies multiple contradiction-building strategies, including the earlier Standard Extension method, within a triangular geometric framework that supports flexible clause interaction and dynamic synergy. ETM serves as the algorithmic core of several high-performance theorem provers, CSE, CSE-E, CSI-E, and CSI-Enig, whose competitive results in standard first-order benchmarks (TPTP problem sets and CASC 2018-2015) empirically validate the effectiveness and generality of the proposed approach. By bridging theoretical abstraction and operational implementation, ETM advances the contradiction separation paradigm into a generalized, scalable, and practically competitive model for automated reasoning, offering new directions for future research in logical inference and theorem proving.

Problem

• Developing generalized algorithm for automated deduction via contradiction separation
• Overcoming limitations of traditional pairwise clause resolution methods
• Bridging theoretical contradiction separation framework with practical implementation

Innovation

• Extended Triangular Method formalizes contradiction separation algorithm
• Unifies multiple strategies in triangular geometric framework
• Serves as core for high-performance theorem provers