On the Parameterized Complexity of Associative and Commutative Unification

Abstract: This paper studies the unification problem with associative, commutative, and associative-commutative functions mainly from a viewpoint of the parameterized complexity on the number of variables. It is shown that both associative and associative-commutative unification problems are $W[1]$-hard. A fixed-parameter algorithm and a polynomial-time algorithm are presented for special cases of commutative unification in which one input term is variable-free and the number of variables is bounded by a constant, respectively. Related results including those on the string and tree edit distance problems with variables are shown too.