Date: 3 August 1980 02:20-EDT From: Alan Bawden Subject: more on metrics To: McKeeman at PARC-MAXC cc: CUBE-HACKERS at MIT-MC Yes, it is true that the four conditions I gave for twist measures don't guarentee that the function will behave anything like the kind of complexity measure we are looking for. I was only trying to show how some of the properties you might expect of a twist measure could be used to generate a metric, so I didn't actually need strong enough properties to ensure reasonable twist measures. The additional property I have been using to assure reasonability is the following: 5) For all M, if T(M) > 1, then there exists an N such that 0 < T(N) < T(M) and T(N) + T(N'M) = T(M). Note that N'M has the property that 0 < T(N'M) < T(M) (easy to show) so the two manipulations N and N'M are both "simpler" than M. We can thus easily show that any manipulation M can be expressed as the product of T(M) twists (where a twist is defined as a manipulation such that T(N) = 1).