Tractable and Intractable Second-Order Matching Problems

The second-order matching problem is the problem of determining, for a finite set {〈ti, si〉 | i ∈ I} of pairs of a second-order term ti and a first-order closed term si, called a matching expression, whether or not there exists a substitution σ such that tiσ = si for each i ∈ I . It is well-known that the second-order matching problem is NP-complete. In this paper, we introduce the following restrictions of a matching expression: k-ary, k-fv , predicate, ground , and function-free. Then, we show that the second-order matching problem is NP-complete for a unary predicate, a unary ground, a ternary function-free predicate, a binary function-free ground, and an 1-fv predicate matching expressions, while it is solvable in polynomial time for a binary function-free predicate, a unary function-free, a k-fv function-free (k ≥ 0), and a ground predicate matching expressions.