name capture definition

<reduction> In beta reduction, when a term containing a free occurrence of a variable v is substituted into another term where v is bound the free v becomes spuriously bound or "captured". E.g.

(\ x . \ y . x y) y --> \ y . y y (WRONG)

This problem arises because two distinct variables have the same name. The most common solution is to rename the bound variable using alpha conversion:

(\ x . \ y' . x y') y --> \ y' . y y'

Another solution is to use de Bruijn notation.

Note that the argument expression, y, contained a free variable. The whole expression above must therefore be notionally contained within the body of some lambda abstraction which binds y. If we never reduce inside the body of a lambda abstraction (as in reduction to weak head normal form) then name capture cannot occur.

(01 Mar 1995)