Tractable and intractable second-order matching problems

The second-order matching problem is to determine whether or not a first-order term without variables is an instance of a second-order term that is allowed to contain not only individual variables but also function variables. It is well known that the second-order matching problem is NP-complete in general. In this paper, we first introduce the several restrictions for the second-order matching problems, such as the bounded number, arity and occurrence of function variables, ground that contains no individual variables, flat that contains no function constants, and predicate that no function variable occurs in the terms of arguments of each function variable. By combining the above restrictions, we give the sharp separations of tractable second-order matching problems from intractable ones. Finally, we compare them with the separations of decidable second-order unification problems from undecidable ones.