I have a lambda expression: λx.λy.x(xy), and I'm supposed to infer the integer representation of it. I've read a lot about Church encodings and Church numerals specifically but I can't find what number is. Can you explain it to me in a way a 3 year old can understand or refer me to a resource better than wikipedia?
Church encoding of integers is the following:
"0" ≡ (λf.(λx.x)): Think of (λf.(λx.x)) as meaning: given a function f and an element x, the result is x: it's like applying the function f zero times to x."1" ≡ (λf.(λx.(fx))): Think of (λf.(λx.(fx))) as meaning: given a function f and an element x, the result is (fx): which should be thought of as apply f to x or, in more standard mathematical notation, like f(x)."2" ≡ (λf.(λx.(f(fx)))): Think of (λf.(λx.(f(fx)))) as meaning: given a function f and an element x, the result is (f(fx)): which should be thought of as apply f to x twice or, in more standard mathematical notation, like f(f(x))."3" ≡ (λf.(λx.(f(f(fx))))): Think of (λf.(λx.(f(f(fx))))) as meaning: given a function f and an element x, the result is (f(f(fx))): which should be thought of as apply f to x three times or, in more standard mathematical notation, like f(f(f(x))).I hope that you see the pattern (and the logic behind). In your case, (λx.(λy.(x(xy)))) is the Church encoding of the number 2 (using alpha-equivalence, of course).
The wikiped article is actually quite clear. What is it that you don't understand?