Partial word

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A partial word is a string that may contain a number of "do not know" or "do not care" symbols. More formally, it is a partial function $u: \{ 0, \ldots, n-1 \} \rightarrow A$ where $A$ is some finite alphabet. If $i \in \{ 0, \ldots, n-1 \}$ but u(i) is not defined then it is called a "hole." For example, aab^ab^b is a partial word of length 8 over the alphabet A ={a,b} with two holes.^[1]