Reachable Vectors

As depicted in the Matrix Implementation, we can see that each word $w = a_{1} a_{2} \dots a_{n}$ can be represented as the vector $I \cdot μ_{a_{1}} \cdot μ_{a_{2}} \dots μ_{a_{n}}$ .

We call the set of all vectors that can be produced in this manner as Reachable Vectors.

If the semi-ring of the weighted automata is a sub-semi-ring of a field, then we can think of the above set of Reachable Vectors sitting in a vector space, called the Reachable Subspace. This, in general will be a strict super-set of the set of Reachable Vectors.

The basis for the Reachable Subspace is called the Reachable Basis and is useful in many algorithms.

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Reachable Vectors

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