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A new, comprehensive approach to inhabitation problems in simply-typedlambda-calculus is shown, dealing with both decision and counting problems.This approach works by exploiting a representation of the search spacegenerated by a given inhabitation problem, which is in terms of alambda-calculus for proof search that the authors developed recently. Therepresentation may be seen as extending the Curry-Howard representation ofproofs by lambda-terms, staying within the methods of lambda-calculus and typesystems. Our methodology reveals inductive descriptions of the decisionproblems, driven by the syntax of the proof-search expressions, and the endproducts are simple, recursive decision procedures and counting functions.

Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search