Higher-Order Matching, Games and Automata

Higher-order matching is the problem given t = u where t, u are terms of simply typed lambda-calculus and u is closed, is there a substitution thetas such that tthetas and u have the same normal form with respect to betaeta-equality: can t be pattern matched to u? This paper considers the question: can we characterize the set of all solution terms to a matching problem? We provide an automata-theoretic account that is relative to resource: given a matching problem and a finite set of variables and constants, the (possibly infinite) set of terms that are built from those components and that solve the problem is regular. The characterization uses standard bottom-up tree automata.