HOL-lambda-sigma: an intentional first-order expression of higher-order logic

Gilles Dowek Thérèse Hardin 1 Claude Kirchner
1 SPI - Sémantiques, preuves et implantation
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.
Document type :
Journal articles
Complete list of metadatas

Contributor : Lip6 Publications <>
Submitted on : Tuesday, September 15, 2015 - 3:13:48 PM
Last modification on : Friday, May 24, 2019 - 5:31:54 PM


  • HAL Id : hal-01199524, version 1


Gilles Dowek, Thérèse Hardin, Claude Kirchner. HOL-lambda-sigma: an intentional first-order expression of higher-order logic. Mathematical Structures in Computer Science, Cambridge University Press (CUP), 2001, 11, pp.21-45. ⟨hal-01199524⟩



Record views