The grammar we gave for the lambda calculus leads to some verbosity. There are several informal conventions in widespread use, which enable the language to be written more compactly. (If you like, you could instead articulate a formal grammar which incorporates these additional conventions. Instead of showing it to you, we'll leave it as an exercise for those so inclined.)
-Dot notation: dot means "put a left paren here, and put the right
+**Dot notation** Dot means "put a left paren here, and put the right
paren as far the right as possible without creating unbalanced
parentheses". So:
- (\x (\y (xy)))
+ (\x (\y (x y)))
can be abbreviated as:
and:
- (\x \y. (z y) z)
+ (\x (\y. (z y) z))
would abbreviate:
- (\x \y ((z y) z))
+ (\x (\y ((z y) z)))
This on the other hand:
- ((\x \y. (z y) z)
+ (\x (\y. z y) z)
would abbreviate:
- ((\x (\y (z y))) z)
+ (\x (\y (z y)) z)
-Parentheses: outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:
+**Parentheses** Outermost parentheses around applications can be dropped. Moreover, applications will associate to the left, so `M N P` will be understood as `((M N) P)`. Finally, you can drop parentheses around abstracts, but not when they're part of an application. So you can abbreviate:
- (\x x y)
+ (\x. x y)
as:
z (\x. x y)
-Merging lambdas: an expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
+**Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
(\x y. M)
All (recursively computable) functions can be represented by lambda
terms (the untyped lambda calculus is Turing complete). For some lambda terms, it is easy to see what function they represent:
-(\x x) represents the identity function: given any argument M, this function
-simply returns M: ((\x x) M) ~~> M.
+> `(\x x)` represents the identity function: given any argument `M`, this function
+simply returns `M`: `((\x x) M) ~~> M`.
-(\x (x x)) duplicates its argument:
-((\x (x x)) M) ~~> (M M)
+> `(\x (x x))` duplicates its argument:
+`((\x (x x)) M) ~~> (M M)`
-(\x (\y x)) throws away its second argument:
-(((\x (\y x)) M) N) ~~> M
+> `(\x (\y x))` throws away its second argument:
+`(((\x (\y x)) M) N) ~~> M`
and so on.