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%% section 8 From the TicToc Lab to the Digital Lab [slacpub7205008 in slacpub7205008: slacpub7205009]
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\section{\usemenu{slacpub7205::context::slacpub7205008}{From the TicToc Lab to the Digital Lab}}\label{section::slacpub7205008}
Our model for an observatory is a number of {\it input} devices which,
relying on our general model, produce bitstrings of arbitrary length
which we can segment, compare, duplicate and operate on using the
three operations ${\bf a}\oplus {\bf b}$, ${\bf a}\Vert {\bf b}$,
and ${\bf a}\cdot {\bf b}$, and record the results. Our model for
a laboratory adds to these observatory facilities, {\it output} devices
which convert bitstrings into signals we can {\it calibrate} in the
laboratory by disconnecting our input devices from the (unknown)
signals coming from outside the laboratory and connecting them to our
locally constructed output devices. We require that the results
correspond to the predictions of the theory which led to their construction.
We then take the critical step from being observers
to being {\it participant observers} by connecting our output devices
to the outside of the laboratory and seeing if the input signals
from the outside into the laboratory
change in a correlated way. Just how we construct input and output
devices is a matter of {\it experimental protocol}, which we must
test by having other observerparticipants construct similar devices
and assuring ourselves that they achieve comparable results to our own.
{\it All} of this ``background'' is presupposed in what I mean by
the phrase ``the practice of physics'' which Gefwert, McGoveran and
I employed in our discussion of methodology at ANPA
9\cite{32,49,61}. It will be seen
that from the point of view of {\it theoretical physics} I am
claiming that all of our operations can, in principle, be reduced to bitstring
operations looking at input tapes to the laboratory,
preparing output tapes connected to the outside world, and
comparing the new inputs which result from these outputs.
It will sometimes be convenient to refer to these tapes by more
familiar names, such as ``clocks'', ``accumulating counters'', etc.,
without going through the detailed translation
into laboratory protocol that is required for the actual practice
of physics.
The most important device with which to start either an astronomical
observatory or a physical laboratory is a reliable clock.
For us a {\it standard clock} will simply consist of
an input device which produces a tape with
an alternating sequence of ``1'' 's and ``0'' 's, which
we will also respectively refer to as {\it tic}'s and {\it toc}'s.
It may be a device we construct ourselves or something that occurs
without our intervention other than what is involved in producing the
tape. Note that the fact that we have constructed it still
leaves the physical clock {\it outside} our (theoretical) bitstring
world; it remains essentially just as mysterious as, for example, the pulsations
of a pulsar, as recorded in our observatory.
We have two types of clock with period $2W$: a {\it tictoc} clock in which the
sequence of $2W$ alternating symbols starts with a ``1'', and a {\it toctic} clock
in which the sequence starts with a ``0''. We will represent the first
by a bitstring which we call ${\bf L}(W;2W)$ and the second by
a bitstring ${\bf R}(W;2W)$. The arbitrariness of the designation
R or L corresponds the fact that our choice of the symbols on
the bitstrings is also arbitrary, reflecting what we called Amson
invariance in our discussion of program universe above. Independent
of the specific symbolization, these two bitstrings have the following
properties in comparison with each other and with the (unique) antinull string
${\bf I}(2W)$ of length $2W$ (which could be called a ``tictic clock''):
\begin{eqnarray}
{\bf R}\cdot {\bf R} =&W& = {\bf L}\cdot {\bf L}\nonumber\\
{\bf R}\cdot {\bf I} =&W& = {\bf L}\cdot {\bf I}\nonumber\\
{\bf R}\cdot {\bf L} =&0&\\
{\bf I}\cdot {\bf I} =&2W&\nonumber\\
{\bf R}\oplus {\bf L}=&{\bf I}&\nonumber
\end{eqnarray}
We now have two calibrated clocks one of which we can use to make
measurements, and the second to obtain the {\it redundant} data
which is so useful in checking for experimental error  a matter
of laboratory protocol which I could expound on at some length,
but will refrain from so doing. We now consider an {\it arbitrary}
signal ${\bf a}(a;2W)$, and compare it with the antinull string.
We must have either that (1) ${\bf a}\cdot {\bf I} < W$ or that
(2) ${\bf a}\cdot {\bf I} = W$ or that (3) ${\bf a}\cdot {\bf I} > W$.
Actually, we need consider only the first two cases, because if
we define ${\bf \bar a}$ by the equation ${\bf \bar a}\equiv
{\bf a}\oplus {\bf I}$, we can reduce the third case to the first
simply by replacing ${\bf a}$ by ${\bf \bar a}$.
If we know the Hamming measure $a$, for instance by running
the string through an accumulating counter which simply records
the number of ``l'' 's in the string, we do not have to make this test
because, independent of the order of the bits in the string
the definitions of the inner product, the antinull string ${\bf I}$
and the conjugate string ${\bf \bar a}$ guarantee that
\begin{eqnarray}
{\bf a}\cdot {\bf a} &=& a = {\bf a}\cdot {\bf I}\nonumber\\
{\bf \bar a}\cdot {\bf \bar a} &=& 2Wa = {\bf \bar a}\cdot {\bf I}\\
{\bf a}\cdot {\bf \bar a} &=& 0\nonumber
\end{eqnarray}
Hence an accumulating counter, which throws away most of the
information contained in the bitstring ${\bf a}$, still gives us useful
structural information if we know the context in which
it is employed to produce its single integer result $a$.
If we compare the bitstring ${\bf a}$ with our standard clock {\it before} throwing away the
string, we get two additional integers, only one of which is
independent. Define these by
\begin{equation}
a_R \equiv {\bf a}\cdot {\bf R}; \ \ \ \
a_L \equiv {\bf a}\cdot {\bf L}
\end{equation}
Without actually constructing them, we now know that there exist
in our space of length $2W$ two bitstrings ${\bf a}_R$
${\bf a}_L$ with the following properties
\begin{eqnarray}
{\bf a}_R \oplus {\bf a}_L &=& {\bf a}\nonumber\\
{\bf a}_R \cdot {\bf a}_R &= a_R =& {\bf a}\cdot {\bf R}\nonumber\\
{\bf a}_L \cdot {\bf a}_L &= a_L =& {\bf a}\cdot {\bf L}\\
{\bf a}_R\cdot {\bf L} = 0 = &{\bf a}_R \cdot {\bf a}_L&
= 0 = {\bf a}_L \cdot {\bf R}\nonumber\\
a_R+a_L &=& a\nonumber
\end{eqnarray}
Suppose we have a second arbitrary string ${\bf b}$ coming from
some independent input device. Clearly we can get some structural
information in the same way as before, succinctly summarized
by the three integers $b, b_R$ and $b_L$ and the
constraint $b=b_L+b_R$. If the two sources
are {\it uncorrelated}, these amount to a pair of {\it classical
measurements}, which we can, given enough data of the same type,
analyze statistically by the methods developed in classical statistical
mechanics. But if the two sources are correlated {\it and} we
construct the string ${\bf a} \oplus {\bf b}$ and take its
inner product both with itself and with our standard clock before
throwing it away, we will have the starting point for a model
of {\it quantum measurement}. This is a deep subject, on which
much light has been shed by Etter's recent papers on Link Theory
\cite{22,23,24,25,26}.
We have started to investigate the connection
to bitstring physics\cite{70}, but have only scratched the
surface. We trust that the more systematic analysis started in
this paper will, eventually, help in bringing the two together.
Here, we will, instead, show how our tictoc laboratory can
give us useful information about the world in which it is embedded.
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