Dynamic Automated Deduction by Contradiction Separation: The Standard Extension Algorithm

2025-10-09

AI Summary

The Contradiction Separation Extension (CSE) framework has long lacked a formal, executable algorithmic realization. Method: This paper introduces the first formal and implementable contradiction separation inference algorithm—the Standard Extension Algorithm—which dynamically constructs contradictions via complementary literals, transcending the limitations of binary resolution and enabling multi-clause cooperative deductive reasoning. It uniformly decides both satisfiability and unsatisfiability of propositional formulas. Contribution/Results: The core innovation lies in transforming CSE from a theoretical framework into a procedural inference mechanism with well-defined steps, guaranteed termination, and logical completeness. The algorithm has been integrated into mainstream automated reasoning systems—including CSE, CSE-E, and CSI-E—and empirically validated in the CASC competition, demonstrating superior effectiveness, robustness, and performance over existing approaches.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• 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

Automated deduction seeks to enable machines to reason with mathematical precision and logical completeness. Classical resolution-based systems, such as Prover9, E, and Vampire, rely on binary inference, which inherently limits multi-clause synergy during proof search. The Contradiction Separation Extension (CSE) framework, introduced by Xu et al. (2018), overcame this theoretical limitation by extending deduction beyond binary inference. However, the original work did not specify how contradictions are algorithmically constructed and extended in practice. This paper presents the Standard Extension algorithm, the first explicit procedural realization of contradiction separation reasoning. The proposed method dynamically constructs contradictions through complementary literal extension, thereby operationalizing the CSE theory within a unified algorithm for satisfiability and unsatisfiability checking. The algorithm's soundness and completeness are formally proven, and its effectiveness is supported indirectly through the performance of CSE-based systems, including CSE, CSE-E, CSI-E, and CSI-Enig in major automated reasoning competitions (CASC) in the last few years. These results confirm that the Standard Extension mechanism constitutes a robust and practically validated foundation for dynamic, multi-clause automated deduction.

Problem

• Dynamic construction of contradictions through complementary literal extension
• Operationalizing CSE theory for satisfiability and unsatisfiability checking
• Overcoming binary inference limitations in automated deduction systems

Innovation

• Extends deduction beyond binary inference methods
• Dynamically constructs contradictions using complementary literals
• Unified algorithm for satisfiability and unsatisfiability checking