Unification in Permutative Equational Theories is Undecidable

An equational theory E is permutative if for all terms s, t : s =E t implies that the terms s and t contain the same symbols with the same number of occurrences. The class of permutative equational theories includes the theory of AC (associativity and commutativity). It is shown in this research note that there is no algorithm that decides E-unifiability of terms for all permutative theories. The proof technique is to provide for every Turing machine M, a permutative theory with a confluent term-rewriting system such that narrowing on certain terms simulates the Turing machine M.