Computation

@miloas

http://miloas.me

Outline

Histroy
Turing machine
Lambda calculus

David Hilbert

Kurt Friedrich Gödel

Alan Mathison Turing

Turing machine

(Q, \Sigma, \Gamma, \delta, q_0, q_{accept}, q_{reject})

The Halting Problem

bool God_algo(char *program, char *input) {
    if(<program> halts on <input>) return true;
    return false;
}


bool Satan_algo(char *program) {
    if(God_algo(program, program)) {
        while(1);
        return false;
    } else {
        return true;
    }
}

Satan_algo(Satan_algo);

Georg Ferdinand Ludwig Philipp Cantor

1
2
3
4
5

a_{10}a_{11}a_{12}a_{13} ...

a_{20}a_{21}a_{22}a_{23} ...

a_{30}a_{31}a_{32}a_{33} ...

...

	1	2	3	4	...
M1	N	1	N	N	...
M2	2	0	N	0	...
M3	0	1	2	0	...
M4	N	0	5	N	...
...

构造新的图灵机P(i)

当Mi(i) halts时,P(i) = Mi(i)+1

否则,P(i) = 0

必然存在k使得Mk = P, 就会有:

Mk(k) = P(k) =

1+Mk(k) if Mk(k) halts

0 if Mk(k) doesn't halt

不管哪种情况都矛盾

Lambda calculus

(Y F) = (F (Y F))

Alonzo Church

lambda calculus BNF:

<expr> ::= <identifier>
<expr> ::= lambda <identifier-list> . <expr>
<expr> ::= (<expr> <expr>)

Haskell Brooks Curry

Y Combinator

"Why does it exist? Why does it work the way it does?"

function f(n) {
    return n == 0 ? 1 : n * f(n - 1)
} // lambda n. If_Else n == 0 1 n * f(n - 1)

lambda n. If_Else n == 0 1 n * <self>(n - 1)

let F = lambda n. If_Else n == 0 1 n * F(n - 1)

let P = lambda self n. If_Else n == 0 1 n * self(self, n - 1) // P(P, 3)

let P = lambda self n. If_Else n == 0 1 n * self(n - 1) // P(f, 3)

P(f) ??? excuse me ???

----------->  P(f)(3) == P(f, 3)

P(f) == lambda n. If_Else n == 0 1 n * f(n - 1) == f

Y(F) = f = F(f) = F(Y(F))

let f_gen = lambda self. P(self(self))
f_gen(f_gen) = P(f_gen(f_gen))

let Y = lambda F. {
    let f_gen = lambda self. F(self(self))
    return f_gen(f_gen)
}

const Y = f => (x => f(v => x(x)(v)))(x => f(v => x(x)(v)))
(self => n => n ===0 ? 1 : n * self(n - 1))