Define the notion of "pairs" using higher-order logic

Question

Revising for a course on automated reasoning and I don't quite understand how to answer this question:

Show how the notion of pairs (x, y) can be defined in higher-order logic using a lambda abstraction. Define a function π1 that returns the first element of such a pair. Finally, show that π1(x, y) = x.

I've found similar questions on stackoverflow, but they're all to do with scheme, which I've never used. An explanation in English/relevant mathematical notation would be appreciated

Here is some runnable code for your favorite browser (I use the `Number` type for sake of simplicity): `const Pair = (x, y) => f => f(x, y); const fst = pair => pair((x, y) => x); fst(Pair(2, 3)) === 2;` yields `true` — , May 19 '17 at 14:23

score 3 · Answer 1 · answered May 19 '17 at 19:14

3

Here you go

PAIR := λx. λy. λp. p x y

π1 := λp. p (λx. λy. x)

π2 := λp. p (λx. λy. y)

π1 (PAIR a b) => a

π2 (PAIR a b) => b

Check the wiki entry on Church encoding for some good examples, too

answered May 19 '17 at 19:14

Mulan

129,518
31
228
259

score 2 · Answer 2 · answered May 19 '17 at 17:22

The main topic of this question is to understand how data can be represented as functions. When you're working with other paradigms , the normal way of thinking is "data = something that's stored in a variable" (could be an array, object, whatever structure you want).

But when we're in functional programming, we can also represent data as functions. So let's say you want a function pair(x,y)

This is "pseudo" lisp language:

(function pair x y = 
   lambda(pick) 
      if pick = 1 return x 
      else return y  )

That example, is showing a function that returns a lambda function which expects a parameter.

(function pi this-is-pair = this-is-pair 1)

this-is-pair should be constructed with a pair function, therefore, the parameter is a function which expects other parameter (pick).

And now, you can test what you need

(pi (pair x y ) ) should return x

I would highly recommend you to see this video about compound data. Most of the examples are made on lisp, but it's great to understand a concept like that.

score 0 · Answer 3 · answered May 19 '17 at 13:47

Pairs or tuples describes Products Domain, is the union of all elements of the set A and all elements of the set B:

A × B = { (a, b) | a ∈ A, b ∈ B }

Here, A and B are diferent types, so if you for example are in a laguage program like C, Java, you can have pair like (String, Integer), (Char, Boolean), (Double, Double)

Now, the function π1, is just a function that takes a pair and returns the first element, this function is called in usually first, and that's how it looks like π1(x, y) = x, on the other hand you have second, doing the same thing but returning the second element:

fst(a, b) = a
snd(a, b) = b

When I studied the signature "Characteristics of the programming languages" in college our professor recommended this book, see the chapter Product Domain to understand well all this concepts.

Define the notion of "pairs" using higher-order logic

3 Answers3