De Bruijn index

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

1. find the instances of the variables n₁, n₂, ..., n_k in M that are bound by the λ in λ M,
2. decrement the free variables of M to match the removal of the outer λ-binder, and
3. replace n₁, n₂, ..., n_k with N, suitably incrementing the free variables occurring in N each time, to match the number of λ-binders, under which the corresponding variable occurs when N substitutes for one of the n_i.

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 destined for substitution are replaced with boxes. Step 2 decrements the free variables, giving λ 3 □ (λ 1 □). Finally, in step 3, we replace the boxes with the argument, namely λ 5 1; 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 terms, written M₁.M₂..., where M_i 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, and is defined by

${\displaystyle \uparrow ^{k}=(k+1).(k+2)....}$

For example, ↑⁰ is the identity substitution, leaving a term unchanged. A finite list of terms M₁.M₂...M_n abbreviates the substitution M₁.M₂...M_n.(n+1).(n+2)... leaving all variables greater than n 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 is defined by

(M₁.M₂...) s = M₁[s].M₂[s]...

satisfying the property

M [s₁ s₂] = (M [s₁]) [s₂],

and substitution is defined on terms as follows:

${\displaystyle {\begin{aligned}n[N_{1}\ldots N_{n}\ldots ]=&N_{n}\\(M_{1}\;M_{2})[s]=&(M_{1}[s])(M_{2}[s])\\(\lambda \;M)[s]=&\lambda \;(M[1.s'])\\&{\text{where }}s'=s\uparrow ^{1}\end{aligned}}}$

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

(λ M) N →_β M [N].

When using the standard "named" representation of λ-terms, where variables are treated as labels or strings, one must explicitly handle α-conversion when defining any operation on the terms. 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.

An alternative way to view De Bruijn indices is as De Bruijn levels, which indexes variables from the bottom of the stack rather than from the top. This eliminates the need to reindex free variables, for example when weakening the context, whereas De Bruijn indices eliminate the need to reindex bound variables, for example when substituting a closed expression in another context.[3]

De Bruijn indices are not the only representation of λ-terms that obviates the problem of α-conversion. Among named representations, the nominal techniques 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 name or rename 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]

• The De Bruijn notation for λ-terms.
• Combinatory logic, a more essential way to eliminate variable names.