(*<*)
theory Paper
imports UTM
begin

hide_const (open) s 
hide_const (open) R

abbreviation 
  "update p a == new_tape a p"


lemma fetch_def2: 
  shows "fetch p 0 b == (Nop, 0)"
  and "fetch p (Suc s) Bk == 
     (case nth_of p (2 * s) of
        Some i \<Rightarrow> i
      | None \<Rightarrow> (Nop, 0))"
  and "fetch p (Suc s) Oc == 
     (case nth_of p ((2 * s) + 1) of
         Some i \<Rightarrow> i
       | None \<Rightarrow> (Nop, 0))"
apply -
apply(rule_tac [!] eq_reflection)
by (auto split: block.splits simp add: fetch.simps)

lemma new_tape_def2: 
  shows "new_tape W0 (l, r) == (l, Bk#(tl r))" 
  and "new_tape W1 (l, r) == (l, Oc#(tl r))"
  and "new_tape L (l, r) == 
     (if l = [] then ([], Bk#r) else (tl l, (hd l) # r))" 
  and "new_tape R (l, r) ==
     (if r = [] then (Bk#l,[]) else ((hd r)#l, tl r))" 
  and "new_tape Nop (l, r) == (l, r)"
apply -
apply(rule_tac [!] eq_reflection)
apply(auto split: taction.splits simp add: new_tape.simps)
done


abbreviation 
  "read r == if (r = []) then Bk else hd r"

lemma tstep_def2:
  shows "tstep (s, l, r) p == (let (a, s') = fetch p s (read r) in (s', new_tape a (l, r)))"
apply -
apply(rule_tac [!] eq_reflection)
by (auto split: if_splits prod.split list.split simp add: tstep.simps)

consts DUMMY::'a

notation (latex output)
  Cons ("_::_" [78,77] 73) and
  W0 ("W\<^bsub>\<^raw:\hspace{-2pt}>Bk\<^esub>") and
  W1 ("W\<^bsub>\<^raw:\hspace{-2pt}>Oc\<^esub>") and
  tstep ("step") and
  steps ("nsteps") and
  DUMMY  ("\<^raw:\mbox{$\_$}>")

declare [[show_question_marks = false]]
(*>*)

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's tiling problem---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 useable as a
guideline for formalization'' \cite[Page 2]{AspertiRicciotti12}. For
our formalisation we followed mainly 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 \emph{blank} or \emph{occupied} 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 equivalent,
the more restrictive notion of Turing machines in \cite{Boolos87} makes
the reasoning more uniform. In addition some proofs \emph{about} Turing
machines will be simpler.  The reason is that one often needs to encode
Turing machines---consequently if the Turing machines are simpler, then the coding
functions are simpler too. Unfortunately, the restrictiveness also makes
it harder to design programs for these 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. The universal Turing
machine can then be constructed as a recursive function.

\smallskip
\noindent
{\bf Contributions:} 

*}

section {* Turing Machines *}

text {* \noindent
  Turing machines can be thought of as having a read-write-unit
  ``gliding'' over a potentially infinite tape. Boolos et
  al~\cite{Boolos87} only consider tapes with cells being either blank
  or occupied, which we represent by a datatype having two
  constructors, namely @{text Bk} and @{text Oc}.  One way to
  represent such tapes is to use a pair of lists, written @{term "(l,
  r)"}, where @{term l} stands for the tape on the left-hand side of the
  read-write-unit and @{term r} for the tape on the right-hand side. We have the
  convention that the head, written @{term hd}, of the right-list is
  the cell on which the read-write-unit currently operates. This can
  be pictured as follows:

  \begin{center}
  \begin{tikzpicture}
  \draw[very thick] (-3.0,0)   -- ( 3.0,0);
  \draw[very thick] (-3.0,0.5) -- ( 3.0,0.5);
  \draw[very thick] (-0.25,0)   -- (-0.25,0.5);
  \draw[very thick] ( 0.25,0)   -- ( 0.25,0.5);
  \draw[very thick] (-0.75,0)   -- (-0.75,0.5);
  \draw[very thick] ( 0.75,0)   -- ( 0.75,0.5);
  \draw[very thick] (-1.25,0)   -- (-1.25,0.5);
  \draw[very thick] ( 1.25,0)   -- ( 1.25,0.5);
  \draw[very thick] (-1.75,0)   -- (-1.75,0.5);
  \draw[very thick] ( 1.75,0)   -- ( 1.75,0.5);
  \draw[rounded corners=1mm] (-0.35,-0.1) rectangle (0.35,0.6);
  \draw[fill]     (1.35,0.1) rectangle (1.65,0.4);
  \draw[fill]     (0.85,0.1) rectangle (1.15,0.4);
  \draw[fill]     (-0.35,0.1) rectangle (-0.65,0.4);
  \draw (-0.25,0.8) -- (-0.25,-0.8);
  \draw[<->] (-1.25,-0.7) -- (0.75,-0.7);
  \node [anchor=base] at (-0.8,-0.5) {\small left list};
  \node [anchor=base] at (0.35,-0.5) {\small right list};
  \node [anchor=base] at (0.1,0.7) {\small head};
  \node [anchor=base] at (-2.2,0.2) {\ldots};
  \node [anchor=base] at ( 2.3,0.2) {\ldots};
  \end{tikzpicture}
  \end{center}
  
  \noindent
  Note that by using lists each side of the tape is only finite. The
  potential infinity is achieved by adding an appropriate blank cell 
  whenever the read-write unit goes over the ``edge'' of the tape. To 
  make this formal we define five possible \emph{actions} 
  the Turing machine can perform:

  \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 operation\\
  \end{tabular}
  \end{center}

  \noindent
  We slightly deviate
  from the presentation in \cite{Boolos87} by using the @{term Nop} operation; however its use
  will become important when we formalise universal Turing 
  machines later. Given a tape and an action, we can define the
  following updating function:

  \begin{center}
  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  @{thm (lhs) new_tape_def2(1)} & @{text "\<equiv>"} & @{thm (rhs) new_tape_def2(1)}\\
  @{thm (lhs) new_tape_def2(2)} & @{text "\<equiv>"} & @{thm (rhs) new_tape_def2(2)}\\
  @{thm (lhs) new_tape_def2(3)} & @{text "\<equiv>"} & \\
  \multicolumn{3}{l}{\hspace{1cm}@{thm (rhs) new_tape_def2(3)}}\\
  @{thm (lhs) new_tape_def2(4)} & @{text "\<equiv>"} & \\
  \multicolumn{3}{l}{\hspace{1cm}@{thm (rhs) new_tape_def2(4)}}\\
  @{thm (lhs) new_tape_def2(5)} & @{text "\<equiv>"} & @{thm (rhs) new_tape_def2(5)}\\
  \end{tabular}
  \end{center}

  \noindent
  The first two clauses replace the head of the right-list
  with new a @{term Bk} or @{term Oc}, repsectively. To see that
  these tow clauses make sense in case where @{text r} is the empty
  list, one has to know that the tail function, @{term tl}, is defined
  such that @{term "tl [] == []"} holds. The third clause 
  implements the move of the read-write unit one step to the left: we need
  to test if the left-list @{term l} is empty; if yes, then we just prepend a 
  blank cell to the right-list; otherwise we have to remove the
  head from the left-list and prepend it to the right-list. Similarly
  in the clause for a right move action. The @{term Nop} operation
  leaves the the tape unchanged.

  Note that our treatment of the tape is rather ``unsymmetric''---we
  have the convention that the head of the right-list is where the
  read-write unit is currently positioned. Asperti and Ricciotti
  \cite{AspertiRicciotti12} also consider such a representation, but
  dismiss it as it complicates their definition for \emph{tape
  equality}. The reason is that moving the read-write unit one step to
  the left and then back to the right might change the tape (in case
  of going over the ``edge''). Therefore they distinguish four types
  of tapes: one where the tape is empty; another where the read-write
  unit is on the left edge, respectively right edge, and in the middle
  of the tape. The reading, writing and moving of the tape is then
  defined in terms of these four cases.  In this way they can keep the
  tape in a ``normalised'' form, and thus making a left-move followed
  by a right-move being the identity on tapes. Since we are not using
  the notion of tape equality, we can get away with the unsymmetric
  definition above and by using the @{term update} function
  cover uniformely all cases including the corner cases.

  Next we need to define the \emph{states} of a Turing machine.  Given
  how little is usually said about how to represent them in informal
  presentaions, it might be surprising that in a theorem prover we have 
  to select carfully a representation. If we use the naive representation
  where a Turing machine consists of a finite set of states, then we
  will have difficulties composing two Turing machines. In this case we 
  would need to combine two finite sets of states, possibly requiring 
  renaming states apart whenever both machines share states. This 
  renaming can be quite cumbersome to reason about. Therefore we made 
  the choice of representing a state by a natural number and the states 
  of a Turing machine will always consist of the initial segment
  of natural numbers starting from @{text 0} up to number of states
  of the machine minus @{text 1}. In doing so we can compose
  two Turing machine by ``shifting'' the states of one by an appropriate
  amount to a higher segment.

  An \emph{instruction} @{term i} of a Turing machine is a pair consisting of a
  natural number (the next state) and an action. A \emph{program} @{term p} of a Turing
  machine is then a list of such pairs. Given a program @{term p}, a state
  and the cell being read by the read-write unit, we need to fetch
  the corresponding instruction from the program. For this we define 
  the function @{term fetch}
 
  \begin{center}
  \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
  \multicolumn{3}{l}{@{thm fetch_def2(1)[where b=DUMMY]}}\\
  @{thm (lhs) fetch_def2(2)} & @{text "\<equiv>"} & \\
  \multicolumn{3}{@ {\hspace{1cm}}l}{@{text "case nth_of p (2 * s) of"}}\\
  \multicolumn{3}{@ {\hspace{1.4cm}}l}{@{text "None \<Rightarrow> (Nop, 0) |"}}\\
  \multicolumn{3}{@ {\hspace{1.4cm}}l}{@{text "Some i \<Rightarrow> i"}}\\
  @{thm (lhs) fetch_def2(3)} & @{text "\<equiv>"} & \\
  \multicolumn{3}{@ {\hspace{1cm}}l}{@{text "case nth_of p (2 * s + 1) of"}}\\
  \multicolumn{3}{@ {\hspace{1.4cm}}l}{@{text "None \<Rightarrow> (Nop, 0) |"}}\\
  \multicolumn{3}{@ {\hspace{1.4cm}}l}{@{text "Some i \<Rightarrow> i"}}
  \end{tabular}
  \end{center}

  start state

  What is tinres? What is it used for?

  \begin{center}
  @{thm dither_def}
  \end{center}


  A \emph{configuration} @{term c} of a Turing machine is a state together with 
  a tape. This is written as the triple @{term "(s, l, r)"}. If we have a 
  configuration and a program, we can calculate
  what the next configuration is by fetching the appropriate next state
  and action from the program. Such a single step of execution can be defined as

  \begin{center}
  @{thm tstep_def2(1)}\\
  \end{center}

  No evaluator in HOL, because of totality.

  \begin{center}
  @{thm steps.simps(1)}\\
  @{thm steps.simps(2)}\\
  \end{center}

  \emph{well-formedness} of a Turing program

  programs halts

  shift and change of a p

  composition of two ps

  assertion holds for all tapes

  Hoare rule for composition

  For showing the undecidability of the halting problem, we need to consider
  two specific Turing machines. copying TM and dithering TM

  correctness of the copying TM

  measure for the copying TM, which we however omit.

  standard configuration

  halting problem
*}

section {* Abacus Machines *}

text {*
  \noindent
  Boolos et al \cite{Boolos87} use abacus machines as a 
  stepping stone for making it less laborious to write
  programs for Turing machines. Abacus machines operate
  over an unlimited number of registers $R_0$, $R_1$, \ldots
  each being able to hold an arbitrary large natural number.
  We use natural numbers to refer to registers and also
  to \emph{opcodes} of abacus machines. Obcodes are
  defined as the datatype

  \begin{center}
  \begin{tabular}{rcll}
  @{text "o"} & $::=$  & @{term "Inc R"} & increment register $R$ by one\\
  & $\mid$ & @{term "Dec R o"} & if content of $R$ is non-zero, then decrement it by one\\
  & & & otherwise jump to opcode $o$
  & $\mid$ & @{term "Goto o"} & jump to opcode $o$
  \end{tabular}
  \end{center}
*}


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)?

*)


(*<*)
end
(*>*)