Some booly functions

(There's a list with all the lambda notes here.)

Bit of prelude first. Some stuff from an earlier numbery post:

0 ≜ λf.λx.x
1 ≜ λf.λx.f x
2 ≜ λf.λx.f (f x)
5 ≜ λf.λx.f (f (f (f (f x))))
+ ≜ λa.λb.λf.λx.a f (b f x)

(And in the earlier post we said something like: The number n is a function that, if given two arguments, applies the first argument n times to the second. The +-function takes the numbers a and b as arguments and gives back a new λf.λx.-function that applies f “a times” to the result of applying f “b times” to x.)

We can do multiplication as well before moving on. Multiplication is like addition, just more. We will make a function that, if given the numbers a and b as arguments will start with the number zero, and add a “b times” to it:

* ≜ λa.λb.b (+ a) 0

* 5 2

Soo. We have several numbers and also a couple of ways to make more numbers. So we pretty much have business: We can decide that one of the numbers is the number of monies and another one is the number of products. If we also have booleans we can do business logic.

Booleans are used for if-then-else. We have a stuff we maybe wanna do and another stuff we maybe wanna do instead, and we use a boolean to pick one of the stuffs. So, we would like to have a true-value and a false-value, and we wanna set things up so that something like if true stuff otherstuff will evaluate to stuff, and something like if false stuff otherstuff will evalute to otherstuff.

We will make the two boolean values be functions. (A reasonably easy choice, so long as we can only make functions in our language.) true will, if given two arguments, return the first one. false, if given two arguments, will return the second one.

true ≜ λa.λb.a
false ≜ λa.λb.b

true stuff otherstuff

false stuff otherstuff

Okay so it looks like true stuff otherstuff evaluates to stuff just fine on its own, and false stuff otherstuff evaluates to otherstuff, and we don’t really need if. But if we want an if we can have one. It can take a boolean as its first argument and then two more arguments, and then just hand those two last arguments over to the boolean.

if ≜ λb.λt.λe.b t e

if true stuff otherstuff

if false stuff otherstuff

Should get same results as with just the booleans. (If we want to, we can redefine if to be the identity function, λx.x. Will work fine.)

Okay, some boolean logic bits:

• not takes one boolean, b. If b is true, we return false. If b is false, we return true.
• and takes two booleans, a and b. If a is true, we return b. If a is false, we return false.
• or takes two booleans, a and b. If a is true, we return true. If a is false, we return b.

not ≜ λb.if b false true
and ≜ λa.λb.if a b false
or ≜ λa.λb.if a true b

if (and (or true false) (not false)) stuff otherstuff

We’ll do some boolean stuff with numbers. We can check if a number is zero by giving it two arguments. The first argument is a function that always returns false: λ_.false. If the number is not zero this function will be applied to something and we’ll get false back. The second argument is true. If the number is zero this true will be returned, without any applying of the λ_.false function.

zero? ≜ λn.n (λ_.false) true

zero? 0

zero? 2

Good. Now we can do everyday business logic things. Like you know when boss is like hey we need a program that checks if the number of monies times the number of products is zero and if it is zero we should add five to the number of monies and if it is not zero we should multiply the number of products by 2. No problem:

do-business ≜ λmonies.λproducts.if (zero? (* monies products)) (+ 2 monies) (* 2 products)

do-business 2 5

do-business 2 0

That’s a lot of business.

(Oh. If we cared about side effects or performance or something, we could worry about stuff like: Will the else-stuff be evaluated even if the boolean is true and we really only want the then-stuff? We don’t really care, since it’s going to correctly return the stuff we want either way. But if we cared: We’re evaluating things in normal order, and will give the then-stuff and the else-stuff to the boolean first, before we “step into” any of the then-stuff or else-stuff. So in the case of true the else-stuff will disappear before we do anything with it.)