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| # TropicalNumbers | ||
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| Tropical number algebra, still under development. | ||
| This package implements [tropical numbers](https://en.wikipedia.org/wiki/Tropical_geometry) and tropical algebras in Julia. Tropical algebra is also known as the [semiring](https://en.wikipedia.org/wiki/Semiring) algebra, which is a set $R$ equipped with two binary operations $\oplus$ and $\otimes$, called addition and multiplication, such that: | ||
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| * $(R, \oplus)$ is a monoid with identity element called $\mathbb{0}$; | ||
| * $(R, \otimes)$ is a monoid with identity element called $\mathbb{1}$; | ||
| * Addition is commutative; | ||
| * Multiplication by the additive identity $\mathbb{0}$ annihilates ; | ||
| * Multiplication left- and right-distributes over addition; | ||
| * Explicitly stated, $(R, \oplus)$ is a commutative monoid. | ||
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| [](https://travis-ci.com/TensorBFS/TropicalNumbers.jl) | ||
| [](https://codecov.io/gh/TensorBFS/TropicalNumbers.jl) | ||
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| ## To start | ||
| ## Installation | ||
| To install this package, press `]` in Julia REPL to enter package mode, then type | ||
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| ```julia | ||
| pkg> add TropicalNumbers | ||
| ``` | ||
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| ## Semiring and Tropical Algebras | ||
| ## Using | ||
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| A [`Semiring`](https://en.wikipedia.org/wiki/Semiring) is a set R equipped with two binary operations + and ⋅, called addition and multiplication, such that: | ||
| A Topical algebra can be described as a tuple $(R, \oplus, \otimes, \mathbb{0}, \mathbb{1})$, where $R$ is the set, $\oplus$ and $\otimes$ are the opeartions and $\mathbb{0}$, $\mathbb{1}$ are their identity element, respectively. In this package, the following tropical algebras are implemented: | ||
| * `TropicalAndOr`: $([T, F], \lor, \land, F, T)$; | ||
| * `Tropical` (`TropicalMaxPlus`): $(\mathbb{R}, \max, +, -\infty, 0)$; | ||
| * `TropicalMinPlus`: $(\mathbb{R}, \min, +, \infty, 0)$; | ||
| * `TropicalMaxMul`: $(\mathbb{R}^+, \max, \times, 0, 1)$. | ||
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| * (R, +) is a monoid with identity element called 0; | ||
| * (R, ⋅) is a monoid with identity element called 1; | ||
| * Addition is commutative; | ||
| * Multiplication by the additive identity 0 annihilates ; | ||
| * Multiplication left- and right-distributes over addition; | ||
| * Explicitly stated, (R, +) is a commutative monoid. | ||
| ```julia | ||
| julia> using TropicalNumbers | ||
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| [`Tropical number`](https://en.wikipedia.org/wiki/Tropical_geometry) are a set of semiring algebras, described as | ||
| * (R, +, ⋅, 0, 1). | ||
| where R is the set, + and ⋅ are the opeartions and 0, 1 are their identity element, respectively. | ||
| julia> Tropical(3) * Tropical(4) | ||
| Tropical(7) | ||
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| In this package, the following tropical algebras are implemented: | ||
| * TropicalAndOr, ([T, F], or, and, false, true); | ||
| * Tropical (TropicalMaxPlus), (ℝ, max, +, -Inf, 0); | ||
| * TropicalMinPlus, (ℝ, min, +, Inf, 0); | ||
| * TropicalMaxMul, (ℝ⁺, max, ⋅, 0, 1). | ||
| julia> TropicalMaxMul(3) * TropicalMaxMul(4) | ||
| TropicalMaxMul(12) | ||
| ``` | ||
| !!! note | ||
| 1. `TropicalMaxPlus` is an alias of `Tropical`. | ||
| 2. `TropicalMaxMul` should not contain negative numbers. However, this package does not check the data validity. Not only for performance reason, but also for future GPU support. | ||
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| ## Why another tropical number? | ||
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| Related packages includes | ||
| Related packages include | ||
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| * [SimpleTropical.jl](https://github.com/scheinerman/SimpleTropical.jl) | ||
| * [TropicalSemiring.jl](https://github.com/saschatimme/TropicalSemiring.jl) | ||
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| Tropical numbers in these packages contains an extra field `isinf`. Which is not nessesary because we have `Inf` and `-Inf` for floating point numbers already, we can just use them directly. It is memory more efficient and computational cheap. | ||
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| Most importantly, we are going to release a BLAS package for tropical numbers, which is two orders faster than naive Julia loops. If a tropical number is defined as a composite data structure, it is hard to utilize SIMD. | ||
| These packages include unnecessary fields in its tropical numbers, such as `isinf`. However, `Inf` and `-Inf` can be used directly for floating point numbers, which is more memory efficient and computationally cheap. `TropicalNumbers` is designed for high performance matrix multiplication on both CPU and GPU. | ||
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| ## Ecosystem | ||
| * [TropicalGEMM](https://github.com/TensorBFS/TropicalGEMM.jl), Tropical matrix multiplication with close to optimal speed. | ||
| * [TropicalGEMM](https://github.com/TensorBFS/TropicalGEMM.jl): The BLAS package for tropical numbers. | ||
| * [CuTropicalGEMM](https://github.com/ArrogantGao/CuTropicalGEMM.jl), The GPU version of TropicalGEMM. |