This article is about a method for avoiding naming bound variables in lambda calculus. For an alternative syntax for lambda expressions, see De Bruijn notation.
In mathematical logic, the de Bruijn index is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables.[1] Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as that for syntactic equality. Each de Bruijn index is a natural number that represents an occurrence of a variable in a λ-term, and denotes the number of binders that are in scope between that occurrence and its corresponding binder. The following are some examples:
The term λx. λy. x, sometimes called the K combinator, is written as λ λ 2 with de Bruijn indices. The binder for the occurrence x is the second λ in scope.
The term λx. λy. λz. xz (yz) (the S combinator), with de Bruijn indices, is λ λ λ 3 1 (2 1).
The term λz. (λy. y (λx. x)) (λx. zx) is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are colored and the references are shown with arrows.
De Bruijn indices are commonly used in higher-order reasoning systems such as automated theorem provers and logic programming systems.[2]
^de Bruijn, Nicolaas Govert (1972). "Lambda Calculus Notation with Nameless Dummies: A Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem" (PDF). Indagationes Mathematicae. 34: 381–392. ISSN 0019-3577. Archived (PDF) from the original on 2011-05-20.
^Gabbay, Murdoch J.; Pitts, Andy M. (1999). "A New Approach to Abstract Syntax Involving Binders" (PDF). 14th Annual IEEE Symposium on Logic in Computer Science. pp. 214–224. doi:10.1109/LICS.1999.782617. Archived (PDF) from the original on 2004-07-27.
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