Third order matching is decidable

The problem of determining whether a term is an instance of another in the simply typed λ-calculus, i.e. of solving the equation a=b where a and b are simply typed λ-terms and b is ground, is addressed. An algorithm that decides whether a matching problem in which all the variables are at most third order has a solution is given. The main idea is that if the problem a=b has a solution, then it also has a solution whose depth is bounded by some integer s depending only on the problem a=b, so a simple enumeration of the substitutions whose depth is bounded by s gives a decision algorithm. This result can also be used to bound the depth of the search tree in Huet´s semi-decision algorithm and thus to turn it into an always-terminating algorithm. The problems that occur in trying to generalize the proof given to higher-order matching are discussed