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In [[mathematical logic]], the '''De Bruijn index''' is a notation invented by the [[Netherlands|Dutch]] [[mathematician]] [[Nicolaas Govert de Bruijn]] for representing terms in the [[λ calculus]] with the purpose of eliminating the names of the variable from the notation.<ref name="de bruijn 72">{{cite journal|author=De Bruijn, Nicolaas Govert|title=Lambda Calculus Notation with Nameless Dummies: A Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem| url=http://alexandria.tue.nl/repository/freearticles/597619.pdf|journal=[[Indagationes Mathematicae]]|volume=34|pages=381–392|publisher=[[Elsevier]]|issn=0019-3577|year=1972}}</ref> Terms written using these indices are invariant with respect to [[α conversion]], so the check for [[Lambda_calculus#Alpha_equivalence|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 (mathematics)|variable]] in a λ-term, and denotes the number of binders that are in [[scope (programming)|scope]] between that occurrence and its corresponding binder. The following are some examples: | |||
* The term λ''x''. λ''y''. ''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 λ''x''. λ''y''. λ''z''. ''x'' ''z'' (''y'' ''z'') (the [[S combinator]]), with De Bruijn indices, is λ λ λ 3 1 (2 1). | |||
* The term λ''z''. (λ''y''. ''y'' (λ''x''. ''x'')) (λ''x''. ''z'' ''x'') is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are coloured and the references are shown with arrows.<br />[[Image:De Bruijn index illustration 1.svg|Pictorial depiction of example]] | |||
De Bruijn indices are commonly used in [[higher-order logic|higher-order]] reasoning systems such as [[automated theorem proving|automated theorem prover]]s and [[logic programming]] systems.<ref name="gabbay 03">{{cite conference|last=Gabbay|first=Murdoch J.|coauthors=Pitts, Andy M.|title=A New Approach to Abstract Syntax Involving Binders|booktitle=14th Annual [[IEEE Symposium on Logic in Computer Science]]|year=1999|pages=214–224|doi=10.1109/LICS.1999.782617}}</ref> | |||
== Formal definition == | |||
Formally, [[λ calculus|λ-terms]] (''M'', ''N'', …) written using De Bruijn indices have the following syntax (parentheses allowed freely): | |||
:''M'', ''N'', … ::= ''n'' | ''M'' ''N'' | λ ''M'' | |||
where ''n'' — [[natural number]]s greater than 0 — are the variables. A variable ''n'' is [[bound variable|bound]] if it is in the scope of at least ''n'' binders (λ); otherwise it is [[free variable|free]]. The binding site for a variable ''n'' is the ''n''th binder it is in the [[scope (programming)|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]] (λ ''M'') ''N'', for example, we must: | |||
# find the variables ''n''<sub>1</sub>, ''n''<sub>2</sub>, …, ''n''<sub>k</sub> 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''<sub>1</sub>, ''n''<sub>2</sub>, …, ''n''<sub>k</sub> with ''N'', suitably increasing the free variables occurring in ''N'' each time, to match the number of λ-binders the corresponding variable occurs under when substituted. | |||
To illustrate, consider the application | |||
:(λ λ 4 2 (λ 1 3)) (λ 5 1) | |||
which might correspond to the following term written in the usual notation | |||
:(λ''x''. λ''y''. ''z'' ''x'' (λ''u''. ''u'' ''x'')) (λ''x''. ''w'' ''x''). | |||
After step 1, we obtain the term λ 4 □ (λ 1 □), where the variables that are substituted for are replaced with boxes. Step 2 lowers the free variables, giving λ 3 □ (λ 1 □). Finally, in step 3 we replace the boxes with the argument; the first box is under one binder, so we replace it with λ 6 1 (which is λ 5 1 with the free variables increased by 1); the second is under two binders, so we replace it with λ 7 1. The final result is λ 3 (λ 6 1) (λ 1 (λ 7 1)). | |||
Formally, a substitution is an unbounded list of term replacements for the free variables, written ''M''<sub>1</sub>.''M''<sub>2</sub>…, where ''M''<sub>''i''</sub> is the replacement for the ''i''th free variable. The increasing operation in step 3 is sometimes called ''shift'' and written ↑<sup>''k''</sup> where ''k'' is a natural number indicating the amount to increase the variables; For example, ↑<sup>0</sup> is the identity substitution, leaving a term unchanged. | |||
The application of a substitution ''s'' to a term ''M'' is written ''M''[''s'']. The composition of two substitutions ''s''<sub>''1''</sub> and ''s''<sub>2</sub> is written ''s''<sub>1</sub> ''s''<sub>2</sub> and defined by | |||
:''M'' [''s''<sub>1</sub> ''s''<sub>2</sub>] = (''M'' [''s''<sub>1</sub>]) [''s''<sub>2</sub>]. | |||
The rules for application are as follows: | |||
<blockquote><math>\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]) \\ | |||
& \text{where } s' = s \uparrow^1 | |||
\end{align}</math></blockquote> | |||
The steps outlined for the β-reduction above are thus more concisely expressed as: | |||
:(λ ''M'') ''N'' →<sub>β</sub> ''M'' [''N''.1.2.3…]. | |||
== Alternatives to De Bruijn indices == | |||
<!-- This section is linked from redirect "[[Barendregt convention]]" --> | |||
When using the standard "named" representation of λ-terms, where variables are treated as labels or strings, one has to explicitly handle α-conversion when defining any operation on the terms. The standard ''Variable Convention''<ref name="barendregt 84">{{cite book|author=Barendregt, Henk P.|title=The Lambda Calculus: Its Syntax and Semantics|publisher=[[North Holland]]|year=1984|isbn=0-444-87508-5}}</ref> of [[Henk Barendregt|Barendregt]] is one such approach where α-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 indices internally, it will present a user interface with names. | |||
De Bruijn indices are not the only representation of λ-terms that obviates the problem of α-conversion. Among named representations, the [[Nominal techniques|nominal]] approaches of Pitts and Gabbay is one approach, where the representation of a λ-term is treated as an equivalence class of all terms [[rewriting|rewritable]] to it using variable permutations.<ref name="pitts 03">{{cite journal|author=Pitts, Andy M.|title=Nominal Logic: A First Order Theory of Names and Binding|journal=[[Information and Computation]]|volume=186|year=2003|pages=165–193|issn=0890-5401|doi=10.1016/S0890-5401(03)00138-X|issue=2}}</ref> This approach is taken by the Nominal Datatype Package of [[Isabelle theorem prover|Isabelle/HOL]].<ref name="nominal isabelle">{{cite web|url=http://isabelle.in.tum.de/nominal/|title=Nominal Isabelle web-site|accessdate=2007-03-28}}</ref> | |||
Another common alternative is an appeal to [[higher-order abstract syntax|higher-order representation]]s where the λ-binder is treated as a true function. In such representations, the issues of α-equivalence, substitution, etc. are identified with the same operations in a [[meta-logic]]. | |||
When reasoning about the meta-theoretic properties of a deductive system in a [[proof assistant]], it is sometimes desirable to limit oneself to first-order representations and to have the ability to (re)name assumptions. The ''locally nameless approach'' uses a mixed representation of variables—De Bruijn indices for bound variables and names for free variables—that is able to benefit from the α-canonical form of De Bruijn indexed terms when appropriate.<ref name="mckinna mcbride">{{cite conference|url=http://www.cs.ru.nl/~james/RESEARCH/haskell2004.pdf|title=I Am Not a Number; I Am a Free Variable|author=McBride, McKinna}}</ref><ref name="aydemir">{{cite conference|url=http://repository.upenn.edu/cgi/viewcontent.cgi?article=1394&context=cis_papers|title=Engineering Formal Metatheory|author=Aydemir, Chargueraud, Pierce, Pollack, Weirich}}</ref> <!-- these citations need to be cleaned up --> | |||
== See also == | |||
* The [[De Bruijn notation]] for λ-terms. This notation has little to do with De Bruijn indices, but the name "De Bruijn notation" is often (erroneously) used to stand for it. | |||
* [[Combinatory logic]], a more essential way to eliminate variable names. | |||
== References == | |||
{{reflist}} | |||
[[Category:Lambda calculus]] | |||
Latest revision as of 05:05, 29 January 2014
In mathematical logic, the De Bruijn index is a notation invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms in the λ calculus with the purpose of eliminating the names of the variable from the notation.[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. x z (y z) (the S combinator), with De Bruijn indices, is λ λ λ 3 1 (2 1).
- The term λz. (λy. y (λx. x)) (λx. z x) is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are coloured 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]
Formal definition
Formally, λ-terms (M, N, …) written using De Bruijn indices have the following syntax (parentheses allowed freely):
- M, N, … ::= n | M N | λ 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 nth 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 (λ M) N, for example, we must:
- find the variables n1, n2, …, nk 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 n1, n2, …, nk with N, suitably increasing the free variables occurring in N each time, to match the number of λ-binders the corresponding variable occurs under when substituted.
To illustrate, consider the application
- (λ λ 4 2 (λ 1 3)) (λ 5 1)
which might correspond to the following term written in the usual notation
- (λx. λy. z x (λu. u x)) (λx. w x).
After step 1, we obtain the term λ 4 □ (λ 1 □), where the variables that are substituted for are replaced with boxes. Step 2 lowers the free variables, giving λ 3 □ (λ 1 □). Finally, in step 3 we replace the boxes with the argument; the first box is under one binder, so we replace it with λ 6 1 (which is λ 5 1 with the free variables increased by 1); the second is under two binders, so we replace it with λ 7 1. The final result is λ 3 (λ 6 1) (λ 1 (λ 7 1)).
Formally, a substitution is an unbounded list of term replacements for the free variables, written M1.M2…, where Mi is the replacement for the ith free variable. The increasing operation in step 3 is sometimes called shift and written ↑k where k is a natural number indicating the amount to increase the variables; For example, ↑0 is the identity substitution, leaving a term unchanged.
The application of a substitution s to a term M is written M[s]. The composition of two substitutions s1 and s2 is written s1 s2 and defined by
- M [s1 s2] = (M [s1]) [s2].
The rules for application are as follows:
The steps outlined for the β-reduction above are thus more concisely expressed as:
- (λ M) N →β M [N.1.2.3…].
Alternatives to De Bruijn indices
When using the standard "named" representation of λ-terms, where variables are treated as labels or strings, one has to explicitly handle α-conversion when defining any operation on the terms. The standard Variable Convention[3] of Barendregt is one such approach where α-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 indices internally, it will present a user interface with names.
De Bruijn indices are not the only representation of λ-terms that obviates the problem of α-conversion. Among named representations, the nominal approaches of Pitts and Gabbay 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.[4] This approach is taken by the Nominal Datatype Package of Isabelle/HOL.[5]
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 α-equivalence, substitution, etc. are identified with the same operations in a meta-logic.
When reasoning about the meta-theoretic properties of a deductive system in a proof assistant, it is sometimes desirable to limit oneself to first-order representations and to have the ability to (re)name assumptions. The locally nameless approach uses a mixed representation of variables—De Bruijn indices for bound variables and names for free variables—that is able to benefit from the α-canonical form of De Bruijn indexed terms when appropriate.[6][7]
See also
- The De Bruijn notation for λ-terms. This notation has little to do with De Bruijn indices, but the name "De Bruijn notation" is often (erroneously) used to stand for it.
- Combinatory logic, a more essential way to eliminate variable names.
References
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