Inverse Function Method for Boolean Satisfiability
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Solution Overview
Problem
Current algorithms for Boolean Satisfiability (SAT) cannot operate in polynomial time, leading to inefficiencies in solving SAT problems.
Innovation Solution
The implementation of inverse function, knowledge learning, and cognitive logic reasoning technologies using hierarchical mirrored memories and 3-D relation methods to convert SAT formulas into complement conjunction form, eliminate unnecessary operations, and determine satisfiability in polynomial (kn to kn3) time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional algorithms are used to solve Boolean Satisfiability problems, then the problems can be solved with existing methods, but the solving time cannot operate in polynomial time leading to inefficiency
Solution Approach 1:
The patent applies inversion by converting the traditional SAT problem formulation into its complement form. Instead of directly satisfying the original formula, the method transforms it into a complement conjunction form where the satisfiability of the original formula corresponds to the unsatisfiability of the complement. This inversion allows the use of rejection rules to eliminate unsatisfiable assignments systematically, achieving polynomial-time solving by working backwards from the complement rather than forwards from the original formulation.
Solution Approach 2:
The patent segments the SAT solving process into distinct modular components: (1) conversion of the SAT formula into complement conjunction form, (2) storage in hierarchical mirrored memories organized by element-class relations, (3) application of rejection rules to eliminate unsatisfiable assignments, and (4) extraction of satisfiable assignments from remaining data. This segmentation allows each component to be optimized independently and contributes to the overall polynomial-time complexity.
2Device complexity
If the traditional Boolean OR operation is used in SAT solving, then the standard logical operations can be applied, but unnecessary operations increase the complexity and time required
Solution Approach 1:
The patent extracts and eliminates the Boolean OR operation from the SAT solving process by transforming the disjunctive form into conjunctive form. The conversion process removes all OR operations, replacing them with AND operations in the complement formulation. This extraction of the problematic OR operation reduces operational complexity and allows the solving process to proceed more efficiently using only conjunction operations combined with rejection rules.
3Ease of manufacture
If trial and error methods are used to determine satisfiability, then simple algorithms can be implemented, but unnecessary trial and error increases the time complexity beyond polynomial
Solution Approach 1:
The patent applies preliminary action by pre-converting the SAT formula into complement conjunction form and pre-organizing the data in hierarchical mirrored memories before the actual solving process begins. This preliminary transformation and organization eliminates the need for trial and error during execution, as the rejection rules can systematically and deterministically eliminate unsatisfiable assignments based on the pre-processed structure, achieving polynomial-time complexity.
Solution Approach 2:
The patent introduces hierarchical mirrored memories as an intermediary data structure between the input SAT formula and the solving process. These memories store element-class relations in a structured format that enables efficient application of rejection rules. The intermediary structure transforms the raw formula into an organized representation that eliminates the need for trial and error, allowing systematic deduction of satisfiability in polynomial time.
Data Source
AI summary
A computer system uses an inverse function method to solve Boolean Satisfiability problems. The system benefits from the system disclosed in the US patent “Knowledge Acquisition and Retrieval Apparatus and Method” (U.S. Pat. No. 6,611,841). The system applies a learning function to access iterative set relations among variables, literals, words and clauses as knowledge; and applies deduction and reduction functions to retrieve relations as reasoning. The system uses knowledge learning (KL) and knowledge reasoning algorithms (KRA). The system abandons the “OR” operation of Boolean logic and processes only set relations on data. The system leverages the reversibility of deduction and reduction to determine whether 3-SAT formulas are satisfiable.