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theory Paper
imports UTM
begin

declare [[show_question_marks = false]]
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section {* Introduction *}

text {*

\noindent
We formalised in earlier work the correctness proofs for two
algorithms in Isabelle/HOL---one about type-checking in
LF~\cite{UrbanCheneyBerghofer11} and another about deciding requests
in access control~\cite{WuZhangUrban12}.  The formalisations
uncovered a gap in the informal correctness proof of the former and
made us realise that important details were left out in the informal
model for the latter. However, in both cases we were unable to
formalise in Isabelle/HOL computability arguments about the
algorithms. The reason is that both algorithms are formulated in terms
of inductive predicates. Suppose @{text "P"} stands for one such
predicate.  Decidability of @{text P} usually amounts to showing
whether \mbox{@{term "P \<or> \<not>P"}} holds. But this does \emph{not} work
in Isabelle/HOL, since it is a theorem prover based on classical logic
where the law of excluded middle ensures that \mbox{@{term "P \<or> \<not>P"}}
is always provable no matter whether @{text P} is constructed by
computable means. The same problem would arise if we had formulated
the algorithms as recursive functions, because internally in
Isabelle/HOL, like in all HOL-based theorem provers, functions are
represented as inductively defined predicates too.

The only satisfying way out of this problem in a theorem prover based on classical
logic is to formalise a theory of computability. Norrish provided such
a formalisation for the HOL4 theorem prover. He choose the
$\lambda$-calculus as the starting point for his formalisation
of computability theory,
because of its ``simplicity'' \cite[Page 297]{Norrish11}.  Part of his
formalisation is a clever infrastructure for reducing
$\lambda$-terms. He also established the computational equivalence
between the $\lambda$-calculus and recursive functions.  Nevertheless he
concluded that it would be ``appealing'' to have formalisations for more
operational models of computations, such as Turing machines or register
machines.  One reason is that many proofs in the literature use 
them.  He noted however that in the context of theorem provers
\cite[Page 310]{Norrish11}:

\begin{quote}
\it``If register machines are unappealing because of their 
general fiddliness, Turing machines are an even more 
daunting prospect.''
\end{quote}

\noindent
In this paper we took on this daunting prospect and provide a
formalisation of Turing machines, as well as abacus machines (a kind
of register machines) and recursive functions. To see the difficulties
involved with this work, one has to understand that interactive
theorem provers, like Isabelle/HOL, are at their best when the
data-structures at hand are ``structurally'' defined, like lists,
natural numbers, regular expressions, etc. Such data-structures come
with convenient reasoning infrastructures (for example induction
principles, recursion combinators and so on).  But this is \emph{not}
the case with Turing machines (and also not with register machines):
underlying their definition is a set of states together with a
transition function, both of which are not structurally defined.  This
means we have to implement our own reasoning infrastructure in order
to prove properties about them. This leads to annoyingly fiddly
formalisations.  We noticed first the difference between both,
structural and non-structural, ``worlds'' when formalising the
Myhill-Nerode theorem, where regular expressions fared much better
than automata \cite{WuZhangUrban11}.  However, with Turing machines
there seems to be no alternative if one wants to formalise the great
many proofs from the literature that use them.  We will analyse one
example---undecidability of Wang tilings---in Section~\ref{Wang}. The
standard proof of this property uses the notion of \emph{universal
Turing machines}.

We are not the first who formalised Turing machines in a theorem
prover: we are aware of the preliminary work by Asperti and Ricciotti
\cite{AspertiRicciotti12}. They describe a complete formalisation of
Turing machines in the Matita theorem prover, including a universal
Turing machine. They report that the informal proofs from which they
started are not ``sufficiently accurate to be directly used as a
guideline for formalization'' \cite[Page 2]{AspertiRicciotti12}. For
our formalisation we followed the proofs from the textbook
\cite{Boolos87} and found that the description there is quite
detailed. Some details are left out however: for example, it is only
shown how the universal Turing machine is constructed for Turing
machines computing unary functions. We had to figure out a way to
generalize this result to $n$-ary functions. Similarly, when compiling
recursive functions to abacus machines, the textbook again only shows
how it can be done for 2- and 3-ary functions, but in the
formalisation we need arbitrary functions. But the general ideas for
how to do this are clear enough in \cite{Boolos87}. However, one
aspect that is completely left out from the informal description in
\cite{Boolos87}, and similar ones we are aware of, are arguments why certain Turing
machines are correct. We will introduce Hoare-style proof rules
which help us with such correctness arguments of Turing machines.

The main difference between our formalisation and the one by Asperti
and Ricciotti is that their universal Turing machine uses a different
alphabet than the machines it simulates. They write \cite[Page
23]{AspertiRicciotti12}:

\begin{quote}\it
``In particular, the fact that the universal machine operates with a
different alphabet with respect to the machines it simulates is
annoying.'' 
\end{quote}

\noindent
In this paper we follow the approach by Boolos et al \cite{Boolos87},
which goes back to Post \cite{Post36}, where all Turing machines
operate on tapes that contain only blank or filled cells (represented 
by @{term Bk} and @{term Oc}, respectively, in our
formalisation). Traditionally the content of a cell can be any
character from a finite alphabet. Although computationally
equivalennt, the more restrictive notion of Turing machines make
the reasoning more uniform. Unfortunately, it also makes it
harder to design programs for Turing machines. Therefore
in order to construct a \emph{universal Turing machine} we follow
the proof in \cite{Boolos87} by relating abacus machines to
turing machines and in turn recursive functions to abacus machines. 

\medskip
\noindent
{\bf Contributions:} 

*}

section {* Turing Machines *}

text {*

  Tapes

  %\begin{center}
  %\begin{tikzpicture}
  %%
  %\end{tikzpicture}
  %\end{center}
  
  An action is defined as 

  \begin{center}
  \begin{tabular}{rcll}
  @{text "a"} & $::=$  & @{term "W0"} & write blank (@{term Bk})\\
  & $\mid$ & @{term "W1"} & write occupied (@{term Oc})\\
  & $\mid$ & @{term L} & move left\\
  & $\mid$ & @{term R} & move right\\
  & $\mid$ & @{term Nop} & do nothing\\
  \end{tabular}
  \end{center}

  For showing the undecidability of the halting problem, we need to consider
  two specific Turing machines.
  
*}

section {* Abacus Machines *}

section {* Recursive Functions *}

section {* Wang Tiles\label{Wang} *}

text {*
  Used in texture mapings - graphics
*}


section {* Related Work *}

text {*
  The most closely related work is by Norrish \cite{Norrish11}, and Asperti and 
  Ricciotti \cite{AspertiRicciotti12}. Norrish bases his approach on 
  lambda-terms. For this he introduced a clever rewriting technology
  based on combinators and de-Bruijn indices for
  rewriting modulo $\beta$-equivalence (to keep it manageable)
*}


(*
Questions:

Can this be done: Ackerman function is not primitive 
recursive (Nora Szasz)

Tape is represented as two lists (finite - usually infinite tape)?

*)


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end
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