An Introduction to Functional Programming Through Lambda Calculus

- 31 -(λx.x λx.x) =>λx.xIt is possible, however, for bound variables in different functions to have the same name. Consider:(λf.(f λf.f) λs.(s s))This should give the same result as the previous expression. Here, the bound variable f should be replaced by:λs.(s s)Note that we should replace the first f in:(f λf.f)but not the f in the body of:λf.fThis is a new function with a new bound variable which just happens to have the same name as a previous boundvariable.To clarify this we need to be more specific about how bound variables relate to variables in function bodies. For anarbitrary function:λ.the bound variable may correspond to occurrences of in and nowhere else. Formally, thescope of the bound variable is .For example, in:λf.λs.(f (s s))the bound variable f is in scope in:In:λs.(f (s s))(λf.λg.λa.(f (g a)) λg.(g g))the leftmost bound variable f is in scope in:λg.λa.(f (g a))and nowhere else. Similarly, the rightmost bound variable g is in scope in:(g g)and nowhere else.Note that we have said may correspond. This is because the re-use of a name may alter a bound variable’s scope, aswe will see.Now we can introduce the idea of a variable being bound or free in an expression. A variable is said to be bound tooccurrences in the body of a function for which it is the bound variable provided no other functions within the body