Director string

In mathematics, in the area of lambda calculus and computation, directors or director strings are a mechanism for keeping track of the free variables in a term.

Specifically, assume that a term takes the form

$f(t_{1},t_{2},\cdots ,t_{n})$

where f is a function, of arity n, with no free variables, and the $t_{i}$ are terms that may or may not contain free variables. Let V denote the set of free variables that may occur in the set of terms. The director is then the map

$\sigma _{t}:V\to P(\lbrace 1,2,\cdots ,n\rbrace )$

with $P(X)$ the power set of X.