From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1592 Path: news.gmane.org!not-for-mail From: Martin Escardo Newsgroups: gmane.science.mathematics.categories Subject: Re: Reality check Date: Fri, 4 Aug 2000 11:23:36 +0100 (BST) Message-ID: <14730.39336.321672.460059@mosstowie.dcs.st-and.ac.uk> References: <200007311544.e6VFijI18118@saul.cis.upenn.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017953 31986 80.91.229.2 (29 Apr 2009 15:12:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:33 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Aug 4 12:50:10 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id MAA25001 for categories-list; Fri, 4 Aug 2000 12:49:46 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: X-Mailer: VM 6.71 under 21.1 (patch 3) "Acadia" XEmacs Lucid Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 4 Original-Lines: 53 Xref: news.gmane.org gmane.science.mathematics.categories:1592 Archived-At: 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.