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An introduction to deciding higher-order matching

Colin Stirling ( Laboratory for Foundations of Computer Science, University of Edinburgh )
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 uses a game-theoretic analysis of beta-reduction which is essentially game semantics. Besides the use of games to understand beta-reduction, I also emphasize how tree automata can recognize terms of simply typed lambda calculus.

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