On 2/14/2014 2:17 PM, LizR wrote:
On 15 February 2014 10:57, meekerdb <[email protected] <mailto:[email protected]>> wrote:On 2/14/2014 12:32 PM, LizR wrote:On 15 February 2014 09:12, meekerdb <[email protected] <mailto:[email protected]>> wrote: On 2/14/2014 8:14 AM, Bruno Marchal wrote:With some definition of the abacus, it is Turing universal. With others it is not. The slide rules is not Turing universal. You can add and multiply approximation of natural numbers only, or, if you want, you can analogically add and multiply the real numbers, and that is not Turing universal. (That is not entirely obvious).That's an interesting point to me (I own a collection of circular slide rules). Of course you can add and subtract on a slide rule as well as multiply, divide, exponentiate, and compute the value of other functions encoded on the rule (sin, tan), but the rule doesn't do it by itself; you provide the sequence of operations consisting of reading a cursor and moving the rule. So why would that not be Turing universal? I would guess because it isn't digital, but analogue? 'cause Turing machines use discrete symbols, while slide rules use a continuous scale?Yes, of course a real slide rule can't encode arbitrarily large integers because it only has finitely many distinguisable locations for the the cursor. But since a Turing machine is allowed an infinite tape, suppose my slide rule (Sliding Machine?) is allowed to expand the number of distinct positions arbitrarily?So you don't think the analogue/digital thing matters? I suppose a person using a slide rule could be trusted to correct for small errors....or could they?
I think it matters because the power of arithmetic to encode proofs depends on it having arbitrarily long strings of digits. But just as Turing idealized infinite tapes, I can idealize arbitrarily large slide rules to get arbitrarily high precision.
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