Partial words are finite sequences over a finite alphabet *A* that may contain a number of "do not know" symbols,
denoted by ◊'s. Setting *A*_{◊}=*A* ∪ {◊}, *A*_{◊}^{*}
denotes the set of all partial words over *A*. In this paper, we investigate the border correlation function
β: *A*_{◊}^{*} → {*a, b*}^{*} that specifies which conjugates (cyclic shifts) of a
given word *w* of length *n* are bordered, that is,
β(*w*)=*c*_{0}c_{1}
. . . c_{n-1} where *c*_{i}
=a or *c*_{i}=b according to whether the *i*th cyclic shift
σ^{i}(*w*) of *w* is unbordered or bordered. A partial word *w* is bordered if a proper prefix
*x*_{1} of *w* is compatible with a proper suffix *x*_{2} of *w*,
in which case any partial word *x* containing both *x*_{1} and *x*_{2}
is called a border of *w*. In addition to β, we investigate an extension β':{*a, b*}_{◊}^{*} →
**N**^{*} that maps a partial word *w* of length *n* to *m*_{0}
m_{1} . . . m_{n-1} where *m
*_{i} is the length of a shortest border of σ^{i}(*w*). Our results extend
those of Harju and Nowotka.

*Keywords*: Combinatorics on words; Border correlations; Partial words.