> On 20 Apr 2018, at 19:04, John Clark <johnkcl...@gmail.com> wrote:
>
> I never got past the first line of Bruno’s post because he said:
>
> "Consider any Turing universal machinery, for example the programming
> language c++”
>
> C++ is Turing complete but is not a Turing machine because machines are
> physical objects made of atoms but C++ is not nor is any language.
>
That is not correct. There are many definition of Turing machine.
Let me give you the most standard definition of Turing machine. You need two
infinite set of symbols:
Q = {q_1, q_2, q_3, q_4, …}. (Called set of internal configurations}
S = {S_0, S_1, S_2, S_3, …}. (Called set of tape symbol}
Along with the symbols R and L. (Right and Left symbols)
A quadruplet is then define by a sequence of fours symbols having the shapes
q_i S_j S_k q_r
or q_i S_j R q_r
or q_i S_j L q_r
(Davis makes the general relativized theory at once, and so add q_i S_j q_k
q_l, which is used to handle Turing’s Oracle, but I will ignore this here).
Now a Turing machine is defined to be a finite set of quadruples. That’s all.
It is set, so the order of the quadruples is irrelevant.
If there is not two different quadruples with the same beginning q_i S_j, then
we say that the machine is deterministic. If not it is non deterministic.
The intuitive interpretation of q_i S_j S_k q_r is that if the machine is in
the internal configuration q_i, in front of the symbol S_j, and if some
quadruple in the machine contains that quadruplet, the next step of the
computation will be obtained by overwriting S_k (on S_j) and getting the state
q_r, or if q_i S_j R q_r was in the machine (which is a set of such
quadruplets) moving to the right of the" tape” and getting the internal
configuration q_r. (And similarly, for L = Left).
Then we can define computation by finite sequence of “instantaneous tape
description”, which are finite set of tape symbols, + a symbols of internal
configuration, like S_4 S_5 q_22 S_2 S_6, which means that the machine is in
front of S_2, in configuration q_22, and if q_22 S_2 begins one of its
quadruplet, the machine “acts” accordingly, and if not, the machine stops.
But the terms “tape”, “configuration” are just intuitive help, and not part of
the definition.
Note that most definition use only two tape symbols S_0 (called “blank”) and
S_1 (written 1).
Note that Turing’s original definition use quintuplet, and he allowed the
machine to “move” and overwrite symbols simultaneously. But it is better to
reason on the quadruplets (more general). As you can guess (and try to prove) a
the quadruplet-TM can emulate the quintuplet -TM.
Important remark: a universal Turing machine is a Turing machine, and thus a
FINITE set of quadruplets. There is nothing infinite in the universal machine.
It is an interpreter of Turing formalism coded in one (universal) program, that
is here a FINITE set of quadruplets. That machine, assuming only the two
symboles “.” (for blank, S_0), and 1 for S_1.
The universal machine is thus a finite set of quadruplet, and it starts on
q_1 1111111111111111111111111.11111111
And it interprets the first block “1111111111111111111111111” as some Gödel
number of a Turing machine (set of quadruplet) acting on the input represented
by the second bloc. Obviously it has q_1 1 as a beginning of some of its
quadruplet to avoid stopping at the start!
The key point is that the universal machine is a FINITE object.
Turing’s talk on an infinite tape is only an aid to the intuition. It is better
to consider the tape as being an environment, mental or physical, or even as a
special oracle.
> As for Löbian machines that is yet another term that Bruno made up and is
> seen on this list but nowhere else.
>
It is a slightly more precise version of the “enough rich”
machine/theory/set-of-beliefs notion, and it can be defined by for all p
sigma_1, the machine can prove “p -> []p”, with “[]p” being the usual tedious
arithmetical definition of provability.
As I explained recently, with p sigma_1,
The truth of all p -> []p is equivalent with “[]” (the machine’s arithmetical
provability predicate) being a universal machine, in this sense or equivalent.
Then a machine “[]" is Löbian if she can prove “p -> []p”
As “[]” is itself sigma_1, (this should be obvious!!! Please tell me if you
(anyone) don’t see this), we get
[]p -> [][]p, so Löbianity entails the axiom 4 (called self-awareness by
Smullyan).
A pure K4 reasoner has to go the Löb’s Island to become Löbian, but any enough
rich machine can prove Gödel’s diagonal lemma, making them “born in the Löb’s
Island” so to speak.
The typical example are any sound machine believing in enough axioms of
arithmetic, like PA, ZF, and many others (including us, as far as we are
correct and agree with PA, say).
> And Turing explained exactly precisely how to make one of his machines in the
> real physical world
>
He is even the first to build one. But his machine is still a mathematical
notion, as his notion of computations.
Amazingly Robinson Arithmetic is Turing complete, and so we have very quickly
that whatever you can do (mathematically or physically with Turing machine,
physically incarnated/implemented or not), number relations provable in RA can
mimic exactly any Turing machine.
A Turing machine is unable to detect by its personal introspective feeling (we
assume indexical computationalism) if she is incarnated relatively to an
arithmetical base, or a physical base, but the point is that she can test this
experimentally. I predicted Quantum Logic from this well before I knew about
quantum physics (except as tool to study enzyme kinetic.
> but Bruno has no idea how to even start to build one of his machines, which
> means he doesn’t understand how it works or even exactly what it is he’s
> talking about.
>
>
I give often this as a simple exercice to my student. It takes not much line
more than PA axiomatic if you use a language like Prolog.
Note that to extract physics from arithmetic, you need only to prove the
existence of the Löbian Machines’ computations in arithmetic, which can be done
already in RA, as it is well known that RA is Turing complete, and so mimic all
machines (even the Löbian one, which of coures assume more than RA).
There is no need to implement a Löbian machine physically to get the logical
point.
Your remark above is thus only an annoying ad hominem insulting assertion,
which does not help us to believe you ever tried to understand, and this means
you have some prejudices or that you repeat some lies I have often been
reported. It is pure direct or indirect bullying. You are either a liar, or a
victim of liars.
Bruno
>
> John K Clark
>
>
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