How to read the below expression in "plain" English

Question

I'm going through the book "Haskell programming from first principles" and although the authors try to explain in detail how to reduce expressions, I still have trouble understanding how to read some expressions in the examples provided.

Let's assume the below expression:

(λx.x) Then if I understand things correctly this can be read as: "A lambda function, that accepts an argument x and returns (basically translating . as "returns") the same value x.

Now supposing that that we have the below expression:

(λxy.xy), how do we read that expression in plain English? To make it even more complex the authors then expand (λxy.xy) to (λxy.xy)(λz.a) 1. How are these z and a appearing to the above expression? I was hoping that if I can properly transform the expression to plain English I'd be able to understand where z and a come from.

Answer 1

In lambda calculus, values that are next to each other imply that the first value is being called as a function with the second value as an argument. (λxy.xy) is shorthand for (λx.λy.xy), which means a function that takes an argument x then an argument y, and calls x with argument y. (if you're wondering why there's a function returning a function, it's due to currying). To figure out what (λxy.xy)(λz.a) 1 means, we have to take it step by step:

• Undo the shorthand: (λx. λy. xy) (λz. a) 1
• Apply first function (λx): (λy. (λz. a) y) 1
• Apply second function (λy): (λz. a) 1
• Apply third function (λz): a

So, we end up with whatever a is (probably an externally defined constant).