We regard a finite word u=u1​u2​⋯un​ up to word isomorphism as an
equivalence relation on {1,2,…,n} where i is equivalent to j if
and only if xi​=xj​. Some finite words (in particular all binary words) are
generated by "{\it palindromic}" relations of the form k∼j+i−k for some
choice of 1≤i≤j≤n and k∈{i,i+1,…,j}. That is to say,
some finite words u are uniquely determined up to word isomorphism by the
position and length of some of its palindromic factors. In this paper we study
the function μ(u) defined as the least number of palindromic relations
required to generate u. We show that every aperiodic infinite word must
contain a factor u with μ(u)≥3, and that some infinite words x have
the property that μ(u)≤3 for each factor u of x. We obtain a
complete classification of such words on a binary alphabet (which includes the
well known class of Sturmian words). In contrast for the Thue-Morse word, we
show that the function μ is unbounded