Different reduction strategies

Normal order reduction means that whenever there is more than one reducible expression (redex) then the leftmost outermost reduction is chosen first.

Applicative order reduction means that whenever there is more than one reducible expression then the leftmost innermost reduction is chosen first.

Example:

leftmost outermost gives:

((x.y) ((x.x x)(x.x x))
=
((x.y) ((x.(x x))(x.(x x)))
->
y [((x.(x x))(x.(x x))/x]
=
y

whereas leftmost innermost gives:

((x.y) ((x.(x x))(x.(x x)))
->
(x.y)((x x)[(x.(x x))/x])
=
((x.y) ((x.(x x))(x.(x x)))