Partial word

Two partial words are said to be compatible when they have the same length and when every position that is a non-wildcard in both of them has the same character in both. If one forms an undirected graph with a vertex for each partial word in a collection of partial words, and an edge for each compatible pair, then the cliques of this graph come from sets of partial words that all match at least one common string. This graph-theoretical interpretation of compatibility of partial words plays a key role in the proof of hardness of approximation of the clique problem, in which a collection of partial words representing successful runs of a probabilistically checkable proof verifier has a large clique if and only if there exists a valid proof of an underlying NP-complete problem.^[5]

The faces (subcubes) of an ${\displaystyle n}$-dimensional hypercube can be described by partial words of length ${\displaystyle n}$ over a binary alphabet, whose symbols are the Cartesian coordinates of the hypercube vertices (e.g., 0 or 1 for a unit cube). The dimension of a subcube, in this representation, equals the number of don't-care symbols it contains. The same representation may also be used to describe the implicants of Boolean functions.^[6]

Partial words may be generalized to parameter words, in which some of the "do not know" symbols are marked as being equal to each other. A partial word is a special case of a parameter word in which each do not know symbol may be substituted by a character independently of all of the other ones.^[7]