Method for solving quantifier linear equations based on the algebra of linear predicate operations

The subject involves structured approaches that extend the existing set of mathematical tools for processing complex relationships within databases and computational systems. This is particularly relevant for applications requiring efficient information retrieval, knowledge representation, and logical inference in automated decision-making environments. The task of this article is to develop a method for solving quantifier linear equations using the algebra of linear predicate operations, aimed at improving database query optimization and enhancing the capabilities of intelligent systems. The methods used in this research include algebraic techniques, logical operations, and matrix-based transformations to model and efficiently solve the predicate equations. By leveraging the algebra of finite predicates, the proposed approach enables a more systematic and scalable way to handle logical dependencies and optimize computational workflows. The method integrates linear logical operators, ensuring that complex queries and constraints in databases can be represented and processed through formal mathematical models. Additionally, it introduces a framework that enhances the structural representation of knowledge, facilitating intelligent data analysis. Because of the study, a formal method was developed to solve quantifier linear equations, enabling more effective query optimization, logical reasoning, and decision-support mechanisms within expert and automated information systems. The research demonstrates that algebraic approaches can significantly improve the efficiency of information retrieval processes, particularly in intelligent databases where relational constraints and dependencies play a crucial role. Benchmarks conducted on synthetic datasets validate the scalability of the method, showing that it maintains linear execution time growth even with increasing data complexity. Conclusion: the proposed method expands the mathematical foundation for solving logical equations in computational environments, providing a powerful tool for intelligent systems and database optimization. The ability to formalize and process complex logical relationships contributes to improved decision-making accuracy and automation efficiency.