Date: Jan 18, 2013 7:13 PM
Author: Graham Cooper
Subject: Re: The MYTH of UNCOMPUTABLE FUNCTIONS

On Jan 19, 5:40 am, George Greene <gree...@email.unc.edu> wrote:
> On Jan 12, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > Turing's Halt Proof clearly proves that
>
> > HALT( program, input )  --> [yes/no]
>
> > is not a PURE FUNCTION!
>
> It's not OUR fault if you don't understand the proof.
> What the proof actually "clearly" proves ("clearly"
> in scare-quotes because it is unfortunately NOT clear to YOU)
> is that HALT( , ) is not A TURING MACHINE.
>
> >> How do you define "pure function"?
> > the output is determined by the input alone
>
> In the case of HALT ( , ), the output IS determined by the input
> alone.  For every TMprogram, it either halts or it doesn't.
> ON EVERY input.  Given the program,  THE INPUT ALONE determines
> that truth-value.  This is just a fact about TMs.  Given this
> fact, it will follow that Halt( , ) is not a TM.  But that does
> not stop it from being a pure function -- it's DEFINED as a pure
> function, so that is not even open to dispute.
>

Assume a process exists that runs any other process and ADDS 1.

Run 2 of these processes and cross the inputs.

Each process has it's one required argument.

P_1(P_2)

P_2(P_1)

Neither can terminate - DEADLOCK

According to you, this proves the  SUCCESSOR FUNCTION  is un-
computable!

STEP 1:  ASSUME P HALTS GIVEN AN INPUT PROCESS THAT HALTS
STEP 2:  P_1 HALTS  (from 1)
STEP 3:  P_2 HALTS  (from 1)
STEP 4:  P_1(P_2)  should HALT  (from 2 & 3)
CONTRADICTON
THEREFORE  ADDING 1 TO A PROCESS' RESULT IN UN-COMPUTABLE

S: if stops(S) gosub S
G. GREENE:  this proves stops() must be un-computable!

G. Cooper  (BInfTech)
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