Introduction to Decidability of Higher-Order Matching

Higher-order unification is the problem given an equation t = u containing free variables is there a solution substitution theta such that t theta and u theta have the same normal form? The terms t and u are from the simply typed lambda calculus and the same normal form is with respect to beta eta-equivalence. Higher-order matching is the particular instance when the term u is closed; can t be pattern matched to u? Although higher-order unification is undecidable, higher-order matching was conjectured to be decidable by Huet [2]. Decidability was shown in [7] via a game-theoretic analysis of beta-reduction when component terms are in eta-long normal form. In the talk we outline the proof of decidability. Besides the use of games to understand beta-reduction, we also emphasize how tree automata can recognize terms of simply typed lambda calculus as developed in [1, 3-6].