# GeometricBooleanAlgebra

**Repository Path**: qingyuanqu/GeometricBooleanAlgebra

## Basic Information

- **Project Name**: GeometricBooleanAlgebra
- **Description**: 几何布尔代数：原点对称性的公理化恢复
Geometric Boolean Algebra: Axiomatic Restoration of Origin Symmetry
- **Primary Language**: Unknown
- **License**: Not specified
- **Default Branch**: main
- **Homepage**: None
- **GVP Project**: No

## Statistics

- **Stars**: 0
- **Forks**: 0
- **Created**: 2025-07-09
- **Last Updated**: 2025-07-10

## Categories & Tags

**Categories**: Uncategorized

**Tags**: None

## README

## Abstract  

This paper establishes *Geometric Boolean Algebra (GBA)*—a unified framework combining Boolean logic and geometric algebra. Key contributions:

### 1. Axiomatic Foundation  
- **Symmetric domain**: 𝔹ₛⁿ = {-1, 1}ⁿ  
- **Logic operations as arithmetic**:  
  - Negation: ¬x ≡ -x  
  - AND: x ∧ y ≡ min(x,y)  
  - OR: x ∨ y ≡ max(x,y)  

### 2. Theoretical Framework  
- **Isomorphism** to classical Boolean algebra (*Theorem 1*)  
- **Geometric embedding** Φ: 𝔹ₛⁿ → ℝⁿ preserves distances:  
  ‖Φ(𝐱) - Φ(𝐲)‖ = √[2n(1 - cosθ_𝐱𝐲)] (*Theorem 6*)  
- **All Boolean functions** become linearly separable in augmented spaces (*Theorem 7*)  

### 3. Computational Transformation  
- **Logic → Arithmetic conversion**:  
  Boolean → Geometric → Arithmetic  
- **Full adder implementation**:  
  - Sum: s = a ⊗ b ⊗ c_in  
  - Carry: c_out = sign(a + b + c_in)  
- **Hardware benefits**: Uniformity | Parallelism | Scalability  

*Breakthrough*: Solves XOR linear inseparability via geometric augmentation. Extensions to quantum systems and ML accelerators demonstrate cross-domain potential.


> Gitee: https://gitee.com/HeartOfDeepSeek/GeometricBooleanAlgebra
> Github: https://github.com/HeartOfDeepSeek/GeometricBooleanAlgebra