Date: Nov 5, 2013 10:38 PM
Author: Ben Bacarisse
Subject: Re: AN ERROR IN PEANO ARITHMETIC!     x X [ s Y ] [ + X Z ]   <-  x X Y Z

grahamcooper7@gmail.com writes:
<snip>
>> > But all of them require unlimited memory to perform unlimited size
>> > number multiplication.

>>
>> It would be nice to see a proof. I don't have one.

>
> There is a proof but I don't recall it.
>
> The number of states of the machine is less than the
> intermediate sums to perform multiplication.
>
> Starting with Least Significant Bit
>
> 010101010101
> X
> 010101
> ____________
>
>
> Now you have X carry states, LEN(TERM2)
> and only Y internal states.


That's an argument that one representation (I'm guessing -- you don't
actually give the input representation) and method fails. Whilst I am
sure they all do, any argument for that must be more general.

To all intents a purposes, it probably suffices to say that neither

{ "x*y=z" | v(x) * v(y) = v(z) }
{ interleave(x, y, z) | v(x) * v(y) = v(z) }

is regular.

<snip>
--
Ben.