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Inhabitation in simply typed lambda-calculus through a lambda-calculus for proof search

Published online by Cambridge University Press:  17 April 2018

JOSÉ ESPÍRITO SANTO
Affiliation:
Centro de Matemática, Universidade do Minho, Braga, Portugal Email: jes@math.uminho.pt, luis@math.uminho.pt
RALPH MATTHES
Affiliation:
Institut de Recherche en Informatique de Toulouse (IRIT), CNRS and University of Toulouse, Toulouse, France Email: Ralph.Matthes@irit.fr
LUÍS PINTO
Affiliation:
Centro de Matemática, Universidade do Minho, Braga, Portugal Email: jes@math.uminho.pt, luis@math.uminho.pt

Abstract

A new approach to inhabitation problems in simply typed lambda-calculus is shown, dealing with both decision and counting problems. This approach works by exploiting a representation of the search space generated by a given inhabitation problem, which is in terms of a lambda-calculus for proof search that the authors developed recently. The representation may be seen as extending the Curry–Howard representation of proofs by lambda terms. Our methodology reveals inductive descriptions of the decision problems, driven by the syntax of the proof-search expressions, and produces simple, recursive decision procedures and counting functions. These allow to predict the number of inhabitants by testing the given type for syntactic criteria. This new approach is comprehensive and robust: based on the same syntactic representation, we also derive the state-of-the-art coherence theorems ensuring uniqueness of inhabitants.

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© Cambridge University Press 2018 

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