On Jun 26, 8:48 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > Charlie-Boo <shymath...@gmail.com> writes: > >> Example 1: The TM that never halts and never changes the tape does not > >> represent a computable real. > > > It depends on the system of representation. An empty tape typically > > represents 0. > > If the machine does not halt, you can't say that the tape is empty for > certain.
Likewise if it does halt - until it halts. But who said that you have to say anything?
> The usual definitions of computable real require that the machine accept > input n, halts and outputs the first n digits, or something like that.
Well, you could do that too if you want, but in that case not all TM define a number. You are simply choosing a sparse representation instead of a dense one. You know how when they're defining a system they either have the set of wffs be represented by a sequential (dense) number or a number where only certain (r.e. set) numbers represent a wff? You're doing it the second way. If you do it the first way you can have a total map from TM onto the computable real numbers.
> >> Example 2: The TM that repeatedly changes the value in one cell, never > >> halting, does not represent a computable real. > > > The sequence of values put into a given cell defines the decimal > > expansion of a real number. Every machine must distinguish between > > scratch values and actual output. Having a certain cell represent > > output is also common. > > > Google "Turing Machines". > > So, what real number do you think example 2 computes? And what > convention of computable real number do you have in mind?
It represents the real number whose expansion is the sequence of non- blank values set into that cell.
> Show me a reference or web page, rather than telling me to google Turing > Machines. > > In other words, stop bluffing.
I'm not bluffing - you really can Google "Turing Machines".
Or better yet, Google "Quine Atom" and click I'm Feeling Lucky.
(Google's team of experts carefully ranks all sites of a scholarly nature.)
> -- > Jesse F. Hughes > "It's easy folks. Just talk about my approach to your favorite > mathematician. If they can't be interested in it, they've > demonstrated a lack of mathematical skill." -- James Harris