A word equation is a formal equality E := u = ⋅ v {\displaystyle E:=u{\overset {\cdot }{=}}v} between a pair of words u {\displaystyle u} and v {\displaystyle v} , each over an alphabet Σ ∪ Ξ {\displaystyle \Sigma \cup \Xi } comprising both constants (cf. Σ {\displaystyle \Sigma } ) and unknowns (cf. Ξ {\displaystyle \Xi } ). An assignment h {\displaystyle h} of constant words to the unknowns of E {\displaystyle E} is said to solve E {\displaystyle E} if it maps both sides of E {\displaystyle E} to identical words. Word equations are a central object in combinatorics on words; they play an analogous role in this area as do Diophantine equations in number theory. One stark difference is that Diophantine equations have an undecidable solubility problem, whereas the analogous problem for word equations is decidable. A classical example of a word equation is the commutation equation x w = ⋅ w x {\displaystyle xw{\overset {\cdot }{=}}wx} , in which x {\displaystyle x} is an unknown and w {\displaystyle w} is a primitive terminal word. It is well-known that the solutions of the commutation equation are exactly those assignments mapping x {\displaystyle x} to some power of w {\displaystyle w} . Another example is the conjugacy equation x z = ⋅ z y {\displaystyle xz{\overset {\cdot }{=}}zy} , in which x , y , {\displaystyle x,y,} and z {\displaystyle z} are all unknowns. The solutions of this equation are precisely those assignments h {\displaystyle h} sending x {\displaystyle x} and y {\displaystyle y} to conjugate words, with the image h ( z ) {\displaystyle h(z)} being filled in as appropriate. Many subclasses of word equations have been introduced, some of which include:
constant-free equations, which are those u = ⋅ v {\displaystyle u{\overset {\cdot }{=}}v} such that u , v {\displaystyle u,v} comprise unknowns only. Such equations have a trivial solution wherein all their unknowns are erased; as such, they are usually studied over free semigroups. quadratic equations, which are those containing each of their unknowns at most twice. This is exactly the class of word equations on which the Nielsen Transformations algorithm (cf. below) terminates. word equations in one unknown, which can be checked for their solubility in linear time.
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