Module tropical 

Tropical Algebra (Max-Plus Semiring)

Tropical algebra replaces (×, +) with (max, +) or (min, +). Applications:

• Neural network analysis (piecewise linear functions)
• Shortest path algorithms
• Dynamic programming
• Linear programming duality

Mathematical Background

The tropical semiring (ℝ ∪ {-∞}, ⊕, ⊗) where:

• a ⊕ b = max(a, b)
• a ⊗ b = a + b
• Zero element: -∞
• Unit element: 0

Key Results

• Tropical polynomials are piecewise linear
• Neural networks with ReLU = tropical rational functions
• Tropical geometry provides bounds on linear regions

Structs

LinearRegionCounter

Counter for linear regions of piecewise linear functions

MinPlusMatrix

Min-plus matrix for shortest paths

Tropical

Tropical number in the max-plus semiring

TropicalEigen

Tropical eigenvalue and eigenvector

TropicalMatrix

Tropical matrix (max-plus)

TropicalMonomial

A monomial in tropical arithmetic: a ⊗ x^k = a + k*x

TropicalNeuralAnalysis

Analyzes ReLU neural networks using tropical geometry

TropicalPolynomial

Tropical polynomial: max_i(a_i + i*x)

Traits

TropicalSemiring

Trait for tropical semiring operations