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\begin{document}
\title{The Decision Problem}
\author{\em Das Entscheidungsproblem}
\date{An Historical Project}
\maketitle
Alan Turing's 1936 paper ``On Computable Numbers with an Application
to the Entscheidungsproblem'' \cite{T} proved most influential not
only for mathematical logic, but also for the development of the
programmable computer, and together with work of Alonzo Church
(1903--1995) \cite{C1, C2} inaugurated a new field of study, known
today as computability. Recall that Turing's original motivation for
writing the paper was to answer the decision problem of David Hilbert
(1862--1943),
which asked whether there is a standard procedure that can be applied
to decide whether an arbitrary statement (within some system of logic)
is provable. A previous project examined the construction of Turing's
``universal computing machine,'' which accepts the instructions of any
other machine $M$ in standard form, and then outputs the same sequence
as $M$. The concept of a universal machine has evolved into what now
is known as a compiler or interpreter in computer science, and is
indispensable for the processing of any programming language. The
question then arises, does the universal computing machine provide a
solution to the decision problem? The universal machine is the
standard procedure for answering all questions that can in turn be
phrased in terms of a computer program.
First, study the following excerpts from Turing's paper \cite[p.\
232--233]{T}
\bigskip
{\sf
\centerline{\em Automatic machines.}
\smallskip
If at each stage the motion of a machine is {\em completely}
determined by the configuration, we shall call the machine an
``automatic machine'' (or $a$-machine). $\ldots$
\medskip
\centerline{\em Computing machines.}
\smallskip
If an $a$-machine prints two kinds of symbols, of which the first kind
(called figures) consists entirely of 0 and 1 (the others being called
symbols of the second kind), then the machine will be called a
computing machine. If the machine is supplied with a blank tape and
set in motion, starting from the correct initial $m$-configuration,
the subsequence of symbols printed by it which are of the first kind
will called the {\em sequence computed by the machine}. $\ldots$
\medskip
\centerline{\em Circular and circle-free machines.}
\smallskip
If a computing machine never writes down more than a finite number of
symbols of the first kind, it will be called {\em circular}.
Otherwise it is said to be {\em circle-free}. $\ldots$
A machine will be circular if it reaches a configuration from which
there is no possible move, or if it goes on moving and possibly
printing symbols of the second kind, but cannnot print any more
symbols of the first kind.
\medskip
\centerline{\em Computable sequences and numbers.}
\smallskip
A sequence is said to be computable if it can be computed by a
circle-free machine. A number is computable if it differs by an
integer from the number computed by a circle-free machine. $\ldots$}
\bigskip
\noindent
(a) Consider the following machine, $T_1$, which begins in
$m$-configuration $a$ with a blank tape, reading the blank at the far
left. Is $T_1$ circle-free? Justify your answer.
\medskip
$T_1: \ $\begin{tabular}{|c|c|c|c|}\hline
\multicolumn{2}{|c|}{Configuration} & \multicolumn{2}{c|}{Behavior} \\ \hline
m-config. & symbol & operation & final m-config. \\ \hline
$a$ & blank & $R, \ P(1)$ & $b$ \\
$a$ & $0$ & $R$ & $b$ \\ \hline
$b$ & $1$ & $R, \ R, \ P(0)$ & $a$ \\
$b$ & blank & (none) & $a$ \\ \hline
\end{tabular}
\bigskip
\noindent
(b) Consider the following machine, $T_2$, which begins in
$m$-configuration $a$ with a blank tape, reading the blank at the far
left. Is $T_2$ circle-free? Justify your answer.
\medskip
$T_2: \ $\begin{tabular}{|c|c|c|c|}\hline
\multicolumn{2}{|c|}{Configuration} & \multicolumn{2}{c|}{Behavior} \\ \hline
m-config. & symbol & operation & final m-config. \\ \hline
$a$ & blank & $R, \ P(1)$ & $b$ \\
$a$ & $0$ & $R$ & $b$ \\ \hline
$b$ & $1$ & $R, \ R, \ P(0)$ & $a$ \\
$b$ & $0$ & $R$ & $a$ \\ \hline
\end{tabular}
\bigskip
\noindent
(c) Describe in your own words the key feature which distinguishes a
circle-free machine from a circular machine.
\medskip
\noindent
(d) Is the sequence $101001000100001 \, \ldots$ computable? If so,
find a circle-free machine (with a finite number of
$m$-configurations) that computes this sequence on every other
square (the $F$-squares) of a tape which is originally blank. If not,
prove that there is no circle-free machine that computes the above
sequence.
\medskip
Turing's insight into the decision problem begins by listing all
computable sequences in some order:
$$ \phi_1, \ \phi_2, \ \phi_3, \ \ldots , \ \phi_n, \ \ldots , $$
where $\phi_n$ is the $n$-th computable sequence. Moreover, let
$\phi_n(k)$ denote the $k$-th figure (0 or 1) of $\phi_n$. For
example, if
$$ \phi_2 = 101010 \, \ldots , $$
then $\phi_2(1) = 1$, $\phi_2(2)=0$, $\phi_2(3)=1$, etc. Turing then
considers the sequence $\beta'$ defined by $\beta'(n) = \phi_n(n)$.
If the decision problem has a solution, then:
\smallskip
\noindent
{\sf we can invent a machine $D$ which, when supplied with the $S.D$
[standard description] of any computing machine $M$ will test this
$S.D$ and if $M$ is circular will mark the $S.D$ with the symbol ``u''
[unsatisfactory] and if it is circle-free will mark it with ``s''
[satisfactory]. By combining the machines $D$ and $U$ [the universal
computing machine] we could construct a machine $H$ to compute the
sequence $\beta'$} \cite[p.\ 247]{T}.
\medskip
\noindent
(e) Is the number of computable sequences finite or infinite?
If finite, list the computable sequences. If infinite, find a
one-to-one correspondence between the natural numbers, $\bn$, and a
subset of the computable sequences.
Use the result of this question to carefully explain why
$H$ must be circle-free.
\medskip
\noindent
(f) Since $H$ is circle-free, the sequence computed by $H$ must be
listed among the $\phi_n$'s. Suppose this occurs for $n=N_0$. In a
written paragraph, explain how $\beta'(N_0)$ should be computed. Is
it possible to construct a machine $H$ that computes $\beta'$? If so,
find the configuration table for $H$. If not, what part of $H$, i.e.,
$D$ or $U$, cannot be constructed? Justify your answer.
\medskip
\noindent
(g) Does the universal computing machine solve the decision problem?
Explain.
\medskip
\noindent
(h) By what name is the decision problem known today in computer
science? Support your answer with excerpts from outside sources.
\begin{thebibliography}{99}
\bibitem{C1} Church, A., ``An Unsolvable Problem of Elementary Number
Theory,'' {\em American Journal of Math.}, {\bf{58}} (1936), 345--363.
\bibitem{C2} Church, A., ``A Note on the Entscheidungsproblem,'' {\em
Journal of Symbolic Logic}, {\bf{1}} (1936), 40--41.
\bibitem{T} Turing, A., ``On Computable Numbers with an Application
to the Entscheidungsproblem,'' {\em Proceedings of the London
Mathematical Society}, {\bf{42}} (1936), 230--265.
\end{thebibliography}
\end{document}