On 08 Feb 2011, at 09:03, Andrew Soltau wrote:
On 06/02/11 22:06, Russell Standish wrote:
Fascinating. At every turn we seem to find additional complexity and
holistic phenomena in the processes of life giving rise to
biological computation. It reminds me of Bonnie Bassler's quorum
function which enables bacteria in the body to act in concert as a
Neurobiologists Find that Weak Electrical Fields in the Brain Help
Neurons Fire Together
Reminds me of what Colin says he is doing...
Ah ah ! I <3 Bonnie. I <3 bacteria. It is always a pleasure to listen
Bacteria are 100% Turing universal.
Give me one thousand molecular biologists, and I will build the most
powerful parallel computers. The difficulty: mixing a big number of
phages and bacteria. Ethical difficulties too.
Are bacteria Löbian? I doubt this, but who knows, really. I mean, in
I think that the eucaryote cells is a bacteria (+ virus, for the
nucleus) construction, like the choroplast is a descendent of the
We are bacteria colonies, bacteria swarm.
All my interest in mechanism stemmed from my interest for bacteria and
cells, notably at the molecular level. I discovered the computer
science "IF ... THEN" in the Lactose Operon (Jacob and Monod, and then
in Gödel's paper).
I consider Kleene second recursion theorem (AxEeAz phi_e(z) = phi_x(e,
z)) as being the most fundamental theorem in "abstract biology". I
apply it in the long version of the thesis to program finite and
infinite 'planaria' (my favorite worm). The program, when cutted in
part, is such that each part generate the whole program, like the bio
Planarias who are the champion of animal regeneration. I used an
operator form of the theorem due to John Case.
I think recursion theory contains an abstract biology, an abstract
psychology and an abstract theology, including the theory of matter.
And recursion theory is easily embedded in the theory of diophantine
polynomial. You don't even need more than a polynomial of degree 4, by
the work of Matiyazevitch and Jones.
This is hardly believable. You can verify the truth on any sigma_1
true sentence by less than 100 additions and multiplication. Of course
to emulate the collision between the Milky Way and Andromeda with a
low degree universal diophantine polynomial, you will have to encode a
lot of information in individual numbers. But no matter the complexity
of the task, you can verify it in less than 100 hundred operation
(addition and multiplication).
You might as well code for the quantum vacuum.
The simple counting algorithm 0, 1, 2, 3, ... is not turing universal,
but that was close! Just one 100 operations for testing arbitrary
Diophantine polynomials are Turing universal. That would have pleased
Hypatia who was teaching Plotinus and Diophantus in Alexandria, some
time ago. I am pretty sure.
Of degree four!
The question of the existence of a universal diophantine polynomial of
degree three remains open. We know that there are no universal
diophantine polynomial of degree two. (diophantine means that the
variables variate on the integers).
On the reals, you don't get the Turing universality with the
polynomials,. You need the sine or the cosine, to reintroduce the
natural numbers, or the complex numbers.
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