BPT Dimensional Analysis Operators

Database-Driven Operator System

Our operator system uses database configuration for dynamic behavior control, eliminating hardcoded limitations and providing flexible operator management.

Operator Types
EQUATION, MATHEMATICAL, LOGICAL, FUNCTIONAL, COMPARISON
Precedence Levels
1 (lowest) → 4 (highest)
Processing Modes
Sequential, Binary Tree, Duplication Format
Associativity
LEFT, RIGHT, NONE

Operator Precedence Hierarchy

1
Equation & Comparison
→, ⇔, ≈, =, >, <, ≥, ≤, ≠, ⟷
2
Addition & Subtraction
+, -, ±, ∓
3
Multiplication & Division
×, ÷, /, ·
4
Functions & Grouping
( ), [ ], functions, powers

Operator Categories

Equation Operators (Type: EQUATION)
Symbol Name Usage Example Output Format
→ Implies Logical implication TestZinf → Test↑ [ℨ] → [↑] = [ℨ] → [↑]
⇔ If and only if Bidirectional implication Test𝔸 ⇔ Test♆ [𝔸] ⇔ [↑·𝔸] = [𝔸] ⇔ [↑·𝔸]
≈ Approximately equal Approximate equality TestZinf ≈ Test⋄ [ℨ] ≈ [ℨ·↑·2ᵇ] = [ℨ] ≈ [ℨ·↑·2ᵇ]
= Equals Exact equality Test1ᵇ = Test2ᵇ [1ᵇ] = [2ᵇ] = [1ᵇ] = [2ᵇ]
⟷ Logical equivalence Bidirectional logical connection TestM ⟷ TestL [𝔪] ⟷ [𝕃] = [𝔪] ⟷ [𝕃]
Comparison Operators (Type: COMPARISON)
Symbol Name Usage Example Processing
> Greater than Value comparison Test♅ > Test𝔼 Sequential
< Less than Value comparison Test1ᵇ < Test2ᵇ Sequential
≥ Greater than or equal Value comparison Test𝔸³ ≥ Test𝔸 Sequential
≤ Less than or equal Value comparison TestZinf ≤ TestZinf² Sequential
≠ Not equal Inequality Test1ᵇ ≠ Test2ᵇ Sequential
Mathematical Operators (Type: MATHEMATICAL)
Symbol Name Precedence Example Result
× Multiplication 3 TestZinf × Test↑ [ℨ·↑]
÷ Division 3 Test♅ ÷ Test𝔼 [ℨ·↑·𝔼·𝔼⁻¹]
+ Addition 2 TestZinf + TestZinf [2ℨ]
- Subtraction 2 Test𝔸 - Test𝔸 [∅]

Processing Behavior Configuration

Sequential Processing

Used for comparison operators and chained equations. Processes expressions left-to-right in the order they appear.

Example:
Test1ᵇ ≤ Test2ᵇ ≥ Test1ᵇ ≠ Test∅
Processes as: ((Test1ᵇ ≤ Test2ᵇ) ≥ Test1ᵇ) ≠ Test∅

Binary Tree Processing

Used for simple equation operators and mathematical expressions. Uses precedence rules to build expression trees.

Example:
TestA × TestB + TestC
Processes as: (TestA × TestB) + TestC

Duplication Format

Shows both sides of equation operators in the final output, maintaining the original expression format.

Format:
[A] op [B] = [A] op [B]
Instead of just: [A] op [B]

Associativity Rules

Controls evaluation order for operators of the same precedence level.

Left: A - B - C = (A - B) - C
Right: A ^ B ^ C = A ^ (B ^ C)
None: Requires explicit parentheses

Dimensional Analysis Rules

Core Processing Rules

  • Multiplication: Add dimensional exponents (𝔼¹ × 𝔼² = 𝔼³)
  • Division: Subtract dimensional exponents (𝔼³ ÷ 𝔼¹ = 𝔼²)
  • Addition/Subtraction: Dimensions must match, coefficients combine
  • Cancellation: Same dimensions with opposite exponents cancel (𝔼¹ × 𝔼⁻¹ = ∅)
  • Ordering: Results follow BPT canonical dimension ordering
Canonical BPT Dimension Order:
ℨ (Zinf) • ↑ (Data) • 𝔸 (Mass) • 𝔼 (Energy) • 𝔪 (Matter) • 𝕃 (Length) • 𝕋 (Time) • nᵇ (Binary)

Implementation Architecture

Database Schema
CREATE TABLE pulse_core_variables ( symbol VARCHAR(10), name VARCHAR(255), property VARCHAR(50), -- 'Operator' or 'Variable' type VARCHAR(20), -- EQUATION, MATHEMATICAL, LOGICAL, etc. precedence INT, -- 1-4 (1=lowest, 4=highest) associativity VARCHAR(10), -- LEFT, RIGHT, NONE show_duplication_format BOOLEAN, show_intermediate_steps BOOLEAN, sequential_processing BOOLEAN, binary_tree_processing BOOLEAN );
Dynamic Processing Logic
// Database-driven operator detection const dbOperatorData = Object.values(variableCache) .filter(item => item.property === 'Operator'); // Type-based routing if (comparisonOps.length > 0) { return false; // Sequential processing } if (equationOps.length > 1) { return false; // Sequential processing } return true; // Binary tree processing

This operator system provides the foundation for BPT dimensional analysis, enabling precise mathematical operations within the Binary Pulse Theory framework while maintaining computational efficiency and dimensional consistency.