Alphabet of Partial Sequence

Alphabet of Partial sequence is just an Alphabet. The - empty element is not counted as a unique element and is not included in the alphabet.

Mathematical Definition

Let \(X\) is Carrier Set

Let \(X_{-}\) is Partial Carrier Set

\[X \subset X_{-}\]

Let \(-\) is Empty element of Partial Carrier Set

Let \(S_p\) is Partial Sequence described as function \(S : \{1,...,l\} \longrightarrow X_{-}\)

Define Alphabet of partial sequence function

\[alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}\]

\[alphabet_p(S_p) = \big<S_p(i) \big| i \in \{1,...,l\}, \forall k < i, S_p(i) \notin \{-\} \land S_p(i) \neq S_p(k) \big>\]