[FOM] Feasible and Utterable Numbers

[scerir]

Mirco Mannucci writes: 
What matters, is that I seek a theory of feasible 
computations that is not a SERVANT of current scientific 
theories, but that can at the same time accomodate
ANY constraints and resources (physical, biological, 
sociological, psychological, or whatever) one can 
possibly imagine.

# Not sure I understand the meaning of 'feasible
computations'. I would say there are, at the present 
time, and according to the current theories, different
limits.

The quantum geometric limit puts a bound on the number
of 'ops' of a computation proceeding in a four volume
of space-time, of covariant radius R, and spatial
extent T. QGL = R x T /(pi x l_Planck x t_Planck).

The holographic bound states that the max number 
of degrees of freedom (or bits) that can be stored
in a space-like region is bounded by the area
of that region divided by the square of the Planck 
lenght and the covariant entropy bound.

>From the Margolus-Levitin theorem (the time it takes
for a quantum state to evolve into an orthogonal
state is dt = pi / 2 E, where E is the expectation
value for the energy) a system with energy E can
carry a number of operations per second = 2 E / pi
(making Planck constant = 1). So, an 'ultimate' 
laptop (Planckian laptop?) should perform circa 
10^51 operations per second. Introducing 
quantum-relativistic refinements that figure would 
be smaller, something like !0^48, or so.

Note that we do not know how to make such an ultimate
laptop and that - as far as I know - problems like
heat production, during computation, are not
taken into account. (This problem is perhaps a minor
problem in the so called counterfactual quantum
computation, based on the interaction-free measurements
and on the quantum interrogation protocol, but this is
a completely different topic).

Saluti,
serafino cerulli-irelli
scerir at libero.it