SemiringFactorizations.jl

Semirings

A semiring is an algebraic structure that generalizes rings, dropping the requirement that each element must have an additive inverse. Examples include

$(\mathbb{R}, +, \times, 0, 1)$
$(\mathbb{Z}, +, \times, 0, 1)$
$(\mathbb{R}, \mathrm{max}, +, -\infty, 0)$
$(\mathbb{R}, \mathrm{max}, \mathrm{min}, -\infty, +\infty)$

Several semirings are implemented in the Julia library TropicalNumbers.jl.

Fixed-Point Equations

Let $A \in \mathbb{S}^{n \times n}$ and $B \in \mathbb{S}^{n \times m}$ be matrices over a semiring $\mathbb{S}$. The linear fixed-point equation

\[AX + B = X\]

is solved by the matrix $X = A^*B$, where $A^* \in \mathbb{S}^{n \times n}$ is a matrix called the quasi-inverse of $A$. With SemiringFactorizations.jl, we can solve linear fixed-point equations with the functions sinv(A), sldiv(A, B), and srdiv(B, A), which respectively compute

$A^*$
$A^*B$
$BA^*$

All three functions work by computing an LU factorization of $A$. A factorization can also be computed directly with the function slu and then reused. Both dense and sparse matrices are supported.

Examples

Linear System of Equations

Any linear system of equations

\[AX = B\]

can be reformulated as a linear fixed-point problem

\[(I - A)X + B = X.\]

This problem can be solved using the function sldiv.

julia> using LinearAlgebra, SemiringFactorizations

julia> A = Float64[
           1 1 1 1 1 1
           1 2 1 1 1 1
           1 1 2 1 1 1
           1 1 1 2 1 1
           1 1 1 1 2 1
           1 1 1 1 1 2
       ];

julia> b = Float64[
           1
           2
           3
           4
           5
           6 
       ];

julia> sldiv(I - A, b)
6-element Vector{Float64}:
 -14.0
   1.0
   2.0
   3.0
   4.0
   5.0

Path Counting

Directed graphs can be represented by 1-0 matrices called adjacency matrices.

The quasi-inverse of this matrix contains, for every pair of vertices, the number paths from one to the other.

julia> using SemiringFactorizations

julia> A = Int[
           0 1 1 0 0 0
           0 0 0 0 0 1
           0 0 0 1 1 0
           0 0 0 0 1 0
           0 0 0 0 0 1
           0 0 0 0 0 0
       ];

julia> sinv(A)
6×6 Matrix{Int64}:
 1  1  1  1  2  3
 0  1  0  0  0  1
 0  0  1  1  2  2
 0  0  0  1  1  1
 0  0  0  0  1  1
 0  0  0  0  0  1

Shortest Paths

A weighted graph is a directed graph with a weight assigned to each edge. Weighted graphs can be represented by adjacency matrices with entries in the min-plus semiring.

The quasi-inverse of this matrix contains, for every pair of vertices, the weight of the shortest path from one to the other.

julia> using SemiringFactorizations, TropicalNumbers

julia> A = TropicalMinPlusF64[
           Inf 9.0 1.0 Inf Inf Inf
           Inf Inf Inf Inf Inf 4.0
           Inf Inf Inf 2.0 6.0 Inf
           Inf Inf Inf Inf 3.0 Inf
           Inf Inf Inf Inf Inf 5.0
           Inf Inf Inf Inf Inf Inf
       ];

julia> sinv(A)
6×6 Matrix{TropicalMinPlusF64}:
 0.0ₛ  9.0ₛ  1.0ₛ  3.0ₛ  6.0ₛ  11.0ₛ
 Infₛ  0.0ₛ  Infₛ  Infₛ  Infₛ   4.0ₛ
 Infₛ  Infₛ  0.0ₛ  2.0ₛ  5.0ₛ  10.0ₛ
 Infₛ  Infₛ  Infₛ  0.0ₛ  3.0ₛ   8.0ₛ
 Infₛ  Infₛ  Infₛ  Infₛ  0.0ₛ   5.0ₛ
 Infₛ  Infₛ  Infₛ  Infₛ  Infₛ   0.0ₛ

Finite Automaton

A finite automaton is a graph with a string assigned to each edge. Finite automata can be represented by adjacency matrices with entries in the semiring of regular expressions.

The quasi-inverse of this matrix contains, for every pair of states, the regular expression that brings the automaton from one to the other.

julia> using SemiringFactorizations

julia> A = RE[
           "a^" "a"  "a"  "a^" "a^" "a^"
           "a^" "a^" "a^" "a^" "a^" "c"
           "a^" "a^" "a^" "c"  "b"  "a^"
           "a^" "a^" "a^" "a^" "b"  "a^"
           "a^" "a^" "a^" "a^" "a^" "a"
           "a^" "a^" "a^" "a^" "a^" "a^"
       ];

julia> sinv(A)
6×6 Matrix{RE}:
     a   a   ac  ab|acb  ac|(?:ab|acb)a
 a^      a^  a^  a^      c
 a^  a^      c   b|cb    (?:b|cb)a
 a^  a^  a^      b       ba
 a^  a^  a^  a^          a
 a^  a^  a^  a^  a^

Note that regular expression a^ matches nothing,