BLC 2010

An introduction to deciding higher order matching
Colin Stirling
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 belong to 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 in the 1970s. In the talk I will describe a proof of decidability that is based on a game theoretic analysis of beta-reduction which is essentially game semantics. Besides the use of games to understand beta-reduction, I also emphasise how tree automata can recognise terms of simply typed lambda calculus.