Contradictions

2025-09-07

AI Summary

Classical binary resolution suffers from limited expressiveness and difficulty handling multi-clause conflicts in automated theorem proving. To address this, this paper proposes a novel framework for automated deduction based on contradiction separation. The core innovation lies in systematically constructing two classes of canonical contradiction structures: maximal triangular canonical contradictions and triangular canonical contradictions, along with exact combinatorial formulas for counting their embedded sub-contradictions. By integrating formal deductive system modeling, structural analysis of clause sets, and combinatorial enumeration techniques, we develop the first algorithm for deciding clause-set satisfiability based on maximal canonical contradictions. This approach overcomes the binary restriction inherent in resolution, significantly enhancing dynamic multi-clause reasoning capability. It establishes a new theoretical foundation and a practically implementable methodology for highly expressive automated reasoning systems.

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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Trustworthy AI requires reasoning systems that are not only powerful but also transparent and reliable. Automated Theorem Proving (ATP) is central to formal reasoning, yet classical binary resolution remains limited, as each step involves only two clauses and eliminates at most two literals. To overcome this bottleneck, the concept of standard contradiction and the theory of contradiction-separation-based deduction were introduced in 2018. This paper advances that framework by focusing on the systematic construction of standard contradictions. Specially, this study investigates construction methods for two principal forms of standard contradiction: the maximum triangular standard contradiction and the triangular-type standard contradiction. Building on these structures, we propose a procedure for determining the satisfiability and unsatisfiability of clause sets via maximum standard contradiction. Furthermore, we derive formulas for computing the number of standard sub-contradictions embedded within both the maximum triangular standard contradiction and the triangular-type standard contradiction. The results presented herein furnish the methodological basis for advancing contradiction-separation-based dynamic multi-clause automated deduction, thereby extending the expressive and deductive capabilities of automated reasoning systems beyond the classical binary paradigm.

Problem

• Overcoming limitations of classical binary resolution in automated theorem proving
• Systematic construction of standard contradictions for advanced deduction
• Determining satisfiability of clause sets via maximum standard contradiction

Innovation

• Systematic construction of standard contradictions
• Procedure for clause set satisfiability via maximum standard contradiction
• Formulas for computing embedded standard sub-contradictions