De Bruijn index

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The correct title of this article is de Bruijn index. It appears incorrectly here because of technical restrictions.

In mathematical logic, the de Bruijn index is a notation invented by the Dutch mathematician Nicolaas Govert de Bruijn (IPA: [dɪ bʁœyn]) for representing terms in the λ calculus.^[1] Terms written using these indexes 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 numbers 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 $\lambda x.\ \lambda y.\ x$ , sometimes called the K combinator, is written as $\lambda\;\lambda\;2$ with de Bruijn indexes. The binder for the occurrence $x$ is the second $λ$ in scope.
The term $\lambda x.\ \lambda y.\ \lambda z.\ x\;z\;(y\;z)$ (the S combinator), with de Bruijn indexes, is $\lambda\;\lambda\;\lambda\;3\;1\;(2\;1)$ .
The term $\lambda z.\ (\lambda y.\ y\;(\lambda x.\ x))\;(\lambda x.\ z\;x)$ is $\lambda\;(\lambda\;1\;(\lambda\;1))\;(\lambda\;2\;1)$ .

De Bruijn indexes are commonly used in higher-order reasoning systems such as automated theorem provers and logic programming systems.^{[citation needed]}

1 Formal definition
2 Alternatives to de Bruijn indexes
3 See also
4 References

[edit] Formal definition

Formally, λ-terms ( $M, N, \ldots$ ) written using de Bruijn indexes have the following syntax (parentheses allowed freely):

$M, N, \ldots\ ::=\ n\ |\ M\;N\ |\ \lambda\;M$

where $n$ — natural numbers greater than 0 — are the variables. A variable $n$ is bound if it is in the scope of at least $n$ binders ( $λ$ ); otherwise it is free. The binding site for a variable $n$ is the $n$ th binder it is in the scope of, starting from the innermost binder.

The most primitive operation on λ-terms is substitution: replacing free variables in a term with other terms. In the β-reduction $(\lambda M)\;N$ , for example, we must:

find the variables $n_1, n_2, \ldots, n_k$ in $M$ that are bound by the $λ$ in $λ M$ ,
decrease the free variables of $M$ to match the removal of the outer $λ$ -binder, and
replace $n_1, \ldots, n_k$ with $B$ , suitably increasing its free variables each time to match the number of $λ$ -binders the correponding variable occurs under.

To illustrate, consider the application

$(\lambda\;\lambda\;4\;2\;(\lambda\;1\;3))\;(\lambda\;5\;1)$

which might correspond to the following term written in the usual notation

$(\lambda x.\ \lambda y.\ z\;x\;(\lambda u.\ u\;x))\;(\lambda x.\ w\;x)$ .

After step 1, we obtain the term $\lambda 4\;\square\;(\lambda 1\;\square)$ , where the variables that are substituted for are replaced with boxes. Step 2 lowers the free variables, giving $\lambda 3\;\square\;(\lambda 1\;\square)$ . Finally, in step 3 we replace the boxes with the argument; the first box is under one binder, so we replace it with $\lambda 6\; 1$ (which is $\lambda 5\; 1$ with the free variables increased by 1); the second is under two binders, so we replace it with $\lambda 7\;1$ . The final result is $\lambda 3\;(\lambda 6\;1)\;(\lambda 1\;(\lambda 7\;1))$ .

Formally, a substitution is an unbounded list of term replacements for the free variables, written $M_1.M_2.\ldots$ , where $M i$ is the replacement for the $i$ th free variable. The increasing operation in step 3 is sometimes called shift and written $\uparrow^k$ where $k$ is a natural number indicating the amount to increase the variables; $\uparrow^k$ is thus a shorthand for the substitution $k+1.k+2\ldots$ . Ths identity substitution that leaves a term unchanged is $\uparrow^0$ , i.e., $1.2.\ldots$ .

The application of a substitution $s$ to a term $M$ is written $M [s]$ . The sequencing of two substitions $s 1$ and $s 2$ is written simply $s_1\ s_2$ , with

$M [s_1\ s_2] = (M [s_1]) [s_2]$ .

The rules for application are as follows:

$\begin{align} n [N_1\ldots N_n\ldots] =& N_n \\ (M_1\;M_2) [s] =&\ (M_1[s])\;(M_2[s]) \\ (\lambda M) [s] =&\ \lambda (M [1.1[s'].2[s'].3[s']\ldots]) \\ &\ \hbox{where}\ s' = s \uparrow^1 \end{align}$

The steps outlined for the β-reduction above are thus more concisely expressed as:

$(\lambda\;M)\;N\ \ \longrightarrow_\beta\ \ M [N.1.2.3.\ldots]$ .

[edit] Alternatives to de Bruijn indexes

When using the standard "named" representation of λ-terms, where variables are treated as labels or strings, one has to explicitly handle ά-conversion when defining any operation on the terms. The standard Variable Convention^[2] of Barendregt is one such approach where ά-conversion is applied as needed to ensure that:

bound variables are distinct from free variables, and
all binders bind variables not already in scope.

In practice this is cumbersome, inefficient, and often error-prone. It has therefore led to the search for different representations of such terms. On the other hand, the named representation of λ-terms is more pervasive and can be more immediately understandable by others because the variables can be given descriptive names. Thus, even if a system uses de Bruijn indexes internally, it will present a user interface with names.

De Bruijn indexes are not the only representation of λ-terms that obviates the problem of ά-conversion. Among named representations, the nominal logic of Pitts is one approach, where the representation of a λ-term is treated as an equivalence class of all terms rewritable to it using variable permutations.^[3] This approach is taken by the Nominal Datatype Package of Isabelle/HOL.^[4]

Another common alternative is an appeal to higher-order representations where the $λ$ -binder is treated as a true function. In such representations, the issues of ά-equivalence, substitution, etc. are identified with the same operations in a meta-logic.

[edit] See also

The de Bruijn notation for λ-terms. This notation has little to do with de Bruijn indexes, but the name "de Bruijn notation" is often (erroneously) used to stand for it.

[edit] References

^ De Bruijn, Nicolaas Govert (1972). "Lambda Calculus Notation with Nameless Dummies: A Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem". Indagationes Mathematicae 34: 381–392. ISSN 0019-3577.
^ Barendregt, Henk P. (1984). The Lambda Calculus: Its Syntax and Semantics. North Holland. ISBN 0-444-87508-5.
^ Pitts, Andy M. (2003). "Nominal Logic: A First Order Theory of Names and Binding". Information and Computation 186: 165–193. ISSN 0890-5401.
^ Nominal Isabelle web-site. Retrieved on 2007-03-28.