Finite Acyclic Theories are Unitary

The main result in this paper is that the class of finite acyclic theories contains only unitary theories. As far as we are aware, this is the first class of unitary theories to be discovered. Interestingly, two of the known unitary theories (those of left and right distributivity) belong to this class. Moreover, it is possible to construct a universal unification procedure for this class of theories—the procedure is similar to the one developed by Kirchner for decomposable theories (generalizing the efficient free-unification algorithm of Martelli and Montanari). The decidability of unification in an arbitrary finite acyclic theory is open. The termination technique developed by Tiden and Arnborg for one-sided distributivity appears to be adaptable for many but not for all theories in the class.