[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

*To*: categories@mta.ca*Subject*: categories: Re: Reality check*From*: Martin Escardo <mhe@dcs.st-and.ac.uk>*Date*: Fri, 4 Aug 2000 11:23:36 +0100 (BST)*In-Reply-To*: <vkaaeeuytpw.fsf@gs2.sp.cs.cmu.edu>*References*: <200007311544.e6VFijI18118@saul.cis.upenn.edu><vkaaeeuytpw.fsf@gs2.sp.cs.cmu.edu>*Sender*: cat-dist@mta.ca

I can't answer Andrej's questions, but I can make a few observations. > It would be interesting to simplify this presentation even further by > using a signed representation with digits -1 and 1 only, in base B > strictly between 1 and 2. For example, the golden ratio base B = (1 + > sqrt(5))/2 seems to be very popular among exact real arithmetic > people. But it's unclear how to make a finite state automaton for > negation of === in this case. In base Golden Ratio with digits 0 and 1 (proposed by Pietro Di Gianantonio), the family of identities that generates === is ... 100 ... === ... 011 ... It corresponds to the fact that the Golden Ratio is the positive solution of the equation x^2 = x + 1. (i.e. 1 x^2 + 0 x^1 + 0 x^0 = 0 x^2 + 1 x^1 + 1 x^0) = = = = = = What I have reported about signed-digit binary notation has also been developed for Golden-Ratio notation by David McGaw in his Honours project ( http://www.dcs.st-and.ac.uk/~mhe/macgaw.ps.gz ). You may be able to get a finite automaton from his algorithm for solving the word problem. It should be even simpler than the one for signed binary, because there are fewer cases to consider. > [Discussion about intuitionistic versions of Freyd's construction > deleted.] > This [discussion] leads to the idea that we should think of > the closed interval I as being glued like this: > > I > |------------|--R--| > |--L--|------------| > I This is precisely what the Golden-Ratio notation achieves. The interval I is now [0,phi], where phi is the Golden Ratio. Then the "digit maps" are l(x)=(x+0)/phi and r(x)=(x+1)/phi. The intersection of the images of l and r is a closed interval with non-empty interior, as in your picture. The above family of identities is equivalent to the single equation l o r o r = r o l o l. One could try to consider algebras with two operations and this equation in order to get the interval in a more constructive way.

**Follow-Ups**:**Re: categories: Reality check***From:*Alex Simpson <als@dcs.ed.ac.uk>

**References**:**categories: Reality check***From:*Peter Freyd <pjf@saul.cis.upenn.edu>

**categories: Re: Reality check***From:*Andrej.Bauer@cs.cmu.edu

- Prev by Date:
**categories: Re: Reality check** - Next by Date:
**categories: Re: Reality check** - Prev by thread:
**categories: Re: Reality check** - Next by thread:
**Re: categories: Reality check** - Index(es):