--- a/Paper.thy Thu Jan 17 11:51:00 2013 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,684 +0,0 @@
-(*<*)
-theory Paper
-imports UTM
-begin
-
-hide_const (open) s
-
-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)
-
-abbreviation
- "run p inp out == \<exists>n. steps (1, inp) p n = (0, out)"
-
-lemma haltP_def2:
- "haltP p n = (\<exists>k l m.
- run p ([], exponent Oc n) (exponent Bk k, exponent Oc l @ exponent Bk m))"
-unfolding haltP_def
-apply(auto)
-done
-
-lemma tape_of_nat_list_def2:
- shows "tape_of_nat_list [] = []"
- and "tape_of_nat_list [n] = exponent Oc (n+1)"
- and "ns\<noteq> [] ==> tape_of_nat_list (n#ns) = (exponent Oc (n+1)) @ [Bk] @ (tape_of_nat_list ns)"
-apply(auto simp add: tape_of_nat_list.simps)
-apply(case_tac ns)
-apply(auto simp add: tape_of_nat_list.simps)
-done
-
-lemma tshift_def2:
- fixes p::"tprog"
- shows "tshift p n == (map (\<lambda> (a, s). (a, (if s = 0 then 0 else s + n))) p)"
-apply(rule eq_reflection)
-apply(auto simp add: tshift.simps)
-done
-
-lemma change_termi_state_def2:
- "change_termi_state p ==
- (map (\<lambda> (a, s). (a, if s = 0 then ((length p) div 2) + 1 else s)) p)"
-apply(rule eq_reflection)
-apply(auto simp add: change_termi_state.simps)
-done
-
-
-
-consts DUMMY::'a
-
-notation (latex output)
- Cons ("_::_" [78,77] 73) and
- set ("") and
- W0 ("W\<^bsub>\<^raw:\hspace{-2pt}>Bk\<^esub>") and
- W1 ("W\<^bsub>\<^raw:\hspace{-2pt}>Oc\<^esub>") and
- t_correct ("twf") and
- tstep ("step") and
- steps ("nsteps") and
- abc_lm_v ("lookup") and
- abc_lm_s ("set") and
- haltP ("stdhalt") and
- tshift ("shift") and
- tcopy ("copy") and
- change_termi_state ("adjust") and
- tape_of_nat_list ("\<ulcorner>_\<urcorner>") and
- t_add ("_ \<oplus> _") 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 take 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 definitions are sets of states together with
-transition functions, all 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 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 \emph{not} ``sufficiently accurate to be directly usable 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
-generalise 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, is 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 are 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. In order
-to construct a universal Turing machine we therefore do not follow
-\cite{AspertiRicciotti12}, instead 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:} We formalised in Isabelle/HOL Turing machines following the
-description of Boolos et al \cite{Boolos87} where tapes only have blank or
-occupied cells. We mechanise the undecidability of the halting problem and
-prove the correctness of concrete Turing machines that are needed
-in this proof; such correctness proofs are left out in the informal literature.
-We construct the universal Turing machine from \cite{Boolos87} by
-relating recursive functions to abacus machines and abacus machines to
-Turing machines. Since we have set up in Isabelle/HOL a very general computability
-model and undecidability result, we are able to formalise the
-undecidability of Wang's tiling problem. We are not aware of any other
-formalisation of a substantial undecidability problem.
-*}
-
-section {* Turing Machines *}
-
-text {* \noindent
- Turing machines can be thought of as having a read-write-unit, also
- referred to as \emph{head},
- ``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
- head and @{term r} for the tape on the right-hand side. We have the
- convention that the head, abbreviated @{term hd}, of the right-list is
- the cell on which the head of the Turing machine 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 or occupied cell
- whenever the head 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 halting computations and also universal Turing
- machines. Given a tape and an action, we can define the
- following tape 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 a new @{term Bk} or @{term Oc}, respectively. To see that
- these two clauses make sense in case where @{text r} is the empty
- list, one has to know that the tail function, @{term tl}, is defined in
- Isabelle/HOL
- such that @{term "tl [] == []"} holds. The third clause
- implements the move of the head 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 fourth clause for a right move action. The @{term Nop} operation
- leaves the the tape unchanged (last clause).
-
- Note that our treatment of the tape is rather ``unsymmetric''---we
- have the convention that the head of the right-list is where the
- head is currently positioned. Asperti and Ricciotti
- \cite{AspertiRicciotti12} also considered such a representation, but
- dismiss it as it complicates their definition for \emph{tape
- equality}. The reason is that moving the head 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 head
- 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 uniformly all cases including 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
- presentations, it might be surprising that in a theorem prover we
- have to select carefully 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: we would need to combine two finite sets of states,
- possibly renaming states apart whenever both machines share
- states.\footnote{The usual disjoint union operation in Isabelle/HOL
- cannot be used as it does not preserve types.} 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 the number of states of the
- machine. In doing so we can compose two Turing machine by
- shifting the states of one by an appropriate amount to a higher
- segment and adjusting some ``next states'' in the other.
-
- An \emph{instruction} @{term i} of a Turing machine is a pair consisting of
- an action and a natural number (the next state). A \emph{program} @{term p} of a Turing
- machine is then a list of such pairs. Using as an example the following Turing machine
- program, which consists of four instructions
-
- \begin{equation}
- \begin{tikzpicture}
- \node [anchor=base] at (0,0) {@{thm dither_def}};
- \node [anchor=west] at (-1.5,-0.42) {$\underbrace{\hspace{21mm}}_{\text{1st state}}$};
- \node [anchor=west] at ( 1.1,-0.42) {$\underbrace{\hspace{17mm}}_{\text{2nd state}}$};
- \node [anchor=west] at (-1.5,0.65) {$\overbrace{\hspace{10mm}}^{\text{@{term Bk}-case}}$};
- \node [anchor=west] at (-0.1,0.65) {$\overbrace{\hspace{6mm}}^{\text{@{term Oc}-case}}$};
- \end{tikzpicture}
- \label{dither}
- \end{equation}
-
- \noindent
- the reader can see we have organised our Turing machine programs so
- that segments of two belong to a state. The first component of the
- segment determines what action should be taken and which next state
- should be transitioned to in case the head reads a @{term Bk};
- similarly the second component determines what should be done in
- case of reading @{term Oc}. We have the convention that the first
- state is always the \emph{starting state} of the Turing machine.
- The zeroth state is special in that it will be used as the
- ``halting state''. There are no instructions for the @{text
- 0}-state, but it will always perform a @{term Nop}-operation and
- remain in the @{text 0}-state. Unlike Asperti and Riccioti
- \cite{AspertiRicciotti12}, we have chosen a very concrete
- representation for programs, because when constructing a universal
- Turing machine, we need to define a coding function for programs.
- This can be easily done for our programs-as-lists, but is more
- difficult for the functions used by Asperti and Ricciotti.
-
- Given a program @{term p}, a state
- and the cell being read by the head, 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}
-
- \noindent
- In this definition the function @{term nth_of} returns the @{text n}th element
- from a list, provided it exists (@{term Some}-case), or if it does not, it
- returns the default action @{term Nop} and the default state @{text 0}
- (@{term None}-case). In doing so we slightly deviate from the description
- in \cite{Boolos87}: if their Turing machines transition to a non-existing
- state, then the computation is halted. We will transition in such cases
- to the @{text 0}-state. However, with introducing the
- notion of \emph{well-formed} Turing machine programs we will later exclude such
- cases and make the @{text 0}-state the only ``halting state''. A program
- @{term p} is said to be well-formed if it satisfies
- the following three properties:
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
- @{term "t_correct p"} & @{text "\<equiv>"} & @{term "2 <= length p"}\\
- & @{text "\<and>"} & @{term "iseven (length p)"}\\
- & @{text "\<and>"} & @{term "\<forall> (a, s) \<in> set p. s <= length p div 2"}
- \end{tabular}
- \end{center}
-
- \noindent
- The first says that @{text p} must have at least an instruction for the starting
- state; the second that @{text p} has a @{term Bk} and @{term Oc} instruction for every
- state, and the third that every next-state is one of the states mentioned in
- the program or being the @{text 0}-state.
-
- A \emph{configuration} @{term c} of a Turing machine is a state together with
- a tape. This is written as @{text "(s, (l, r))"}. If we have a
- configuration and a program, we can calculate
- what the next configuration is by fetching the appropriate action and next state
- from the program, and by updating the state and tape accordingly.
- This single step of execution is defined as the function @{term tstep}
-
- \begin{center}
- \begin{tabular}{l}
- @{text "step (s, (l, r)) p"} @{text "\<equiv>"}\\
- \hspace{10mm}@{text "let (a, s) = fetch p s (read r)"}\\
- \hspace{10mm}@{text "in (s', update (l, r) a)"}
- \end{tabular}
- \end{center}
-
- \noindent
- where @{term "read r"} returns the head of the list @{text r}, or if @{text r} is
- empty it returns @{term Bk}.
- It is impossible in Isabelle/HOL to lift the @{term step}-function realising
- a general evaluation function for Turing machines. The reason is that functions in HOL-based
- provers need to be terminating, and clearly there are Turing machine
- programs that are not. We can however define an evaluation
- function so that it performs exactly @{text n} steps:
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
- @{thm (lhs) steps.simps(1)} & @{text "\<equiv>"} & @{thm (rhs) steps.simps(1)}\\
- @{thm (lhs) steps.simps(2)} & @{text "\<equiv>"} & @{thm (rhs) steps.simps(2)}\\
- \end{tabular}
- \end{center}
-
- \noindent
- Recall our definition of @{term fetch} with the default value for
- the @{text 0}-state. In case a Turing program takes in \cite{Boolos87} less
- then @{text n} steps before it halts, then in our setting the @{term steps}-evaluation
- does not actually halt, but rather transitions to the @{text 0}-state and
- remains there performing @{text Nop}-actions until @{text n} is reached.
-
- Given some input tape @{text "(l\<^isub>i,r\<^isub>i)"}, we can define when a program
- @{term p} generates a specific output tape @{text "(l\<^isub>o,r\<^isub>o)"}
-
- \begin{center}
- \begin{tabular}{l}
- @{term "runs p (l\<^isub>i, r\<^isub>i) (l\<^isub>o,r\<^isub>o)"} @{text "\<equiv>"}\\
- \hspace{6mm}@{text "\<exists>n. nsteps (1, (l\<^isub>i,r\<^isub>i)) p n = (0, (l\<^isub>o,r\<^isub>o))"}
- \end{tabular}
- \end{center}
-
- \noindent
- where @{text 1} stands for the starting state and @{text 0} for our final state.
- A program @{text p} with input tape @{term "(l\<^isub>i, r\<^isub>i)"} \emph{halts} iff
-
- \begin{center}
- @{term "halts p (l\<^isub>i, r\<^isub>i) \<equiv>
- \<exists>l\<^isub>o r\<^isub>o. runs p (l\<^isub>i, r\<^isub>i) (l\<^isub>o,r\<^isub>o)"}
- \end{center}
-
- \noindent
- Later on we need to consider specific Turing machines that
- start with a tape in standard form and halt the computation
- in standard form. To define a tape in standard form, it is
- useful to have an operation @{term "tape_of_nat_list DUMMY"} that
- translates
- lists of natural numbers into tapes.
-
- \begin{center}
- \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
- @{thm (lhs) tape_of_nat_list_def2(1)} & @{text "\<equiv>"} & @{thm (rhs) tape_of_nat_list_def2(1)}\\
- @{thm (lhs) tape_of_nat_list_def2(2)} & @{text "\<equiv>"} & @{thm (rhs) tape_of_nat_list_def2(2)}\\
- @{thm (lhs) tape_of_nat_list_def2(3)} & @{text "\<equiv>"} & @{thm (rhs) tape_of_nat_list_def2(3)}\\
- \end{tabular}
- \end{center}
-
-
-
-
- By this we mean
-
- \begin{center}
- @{thm haltP_def2[where p="p" and n="n", THEN eq_reflection]}
- \end{center}
-
- \noindent
- This means the Turing machine starts with a tape containg @{text n} @{term Oc}s
- and the head pointing to the first one; the Turing machine
- halts with a tape consisting of some @{term Bk}s, followed by a
- ``cluster'' of @{term Oc}s and after that by some @{term Bk}s.
- The head in the output is pointing again at the first @{term Oc}.
- The intuitive meaning of this definition is to start the Turing machine with a
- tape corresponding to a value @{term n} and producing
- a new tape corresponding to the value @{term l} (the number of @{term Oc}s
- clustered on the output tape).
-
- Before we can prove the undecidability of the halting problem for Turing machines,
- we have to define how to compose two Turing machines. Given our setup, this is
- relatively straightforward, if slightly fiddly. We use the following two
- auxiliary functions:
-
- \begin{center}
- \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
- @{thm (lhs) tshift_def2} @{text "\<equiv>"}\\
- \hspace{4mm}@{thm (rhs) tshift_def2}\\
- @{thm (lhs) change_termi_state_def2} @{text "\<equiv>"}\\
- \hspace{4mm}@{text "map (\<lambda> (a, s)."}\\
- \hspace{14mm}@{text "(a, if s = 0 then length p div 2 + 1 else s)) p"}\\
- \end{tabular}
- \end{center}
-
- \noindent
- The first adds @{text n} to all states, exept the @{text 0}-state,
- thus moving all ``regular'' states to the segment starting at @{text
- n}; the second adds @{term "length p div 2 + 1"} to the @{text
- 0}-state, thus ridirecting all references to the ``halting state''
- to the first state after the program @{text p}. With these two
- functions in place, we can define the \emph{sequential composition}
- of two Turing machine programs @{text "p\<^isub>1"} and @{text "p\<^isub>2"}
-
- \begin{center}
- @{thm t_add.simps[where ?t1.0="p\<^isub>1" and ?t2.0="p\<^isub>2", THEN eq_reflection]}
- \end{center}
-
- \noindent
- This means @{text "p\<^isub>1"} is executed first. Whenever it originally
- transitioned to the @{text 0}-state, it will in the composed program transition to the starting
- state of @{text "p\<^isub>2"} instead. All the states of @{text "p\<^isub>2"}
- have been shifted in order to make sure that the states of the composed
- program @{text "p\<^isub>1 \<oplus> p\<^isub>2"} still only ``occupy''
- an initial segment of the natural numbers.
-
- \begin{center}
- \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}p{6.9cm}@ {}}
- @{thm (lhs) tcopy_def} & @{text "\<equiv>"} & @{thm (rhs) tcopy_def}
- \end{tabular}
- \end{center}
-
-
- 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.
-
- 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, but also
- to refer to \emph{opcodes} of abacus
- machines. Obcodes are given by the datatype
-
- \begin{center}
- \begin{tabular}{rcll}
- @{text "o"} & $::=$ & @{term "Inc R\<iota>"} & increment register $R$ by one\\
- & $\mid$ & @{term "Dec R\<iota> o\<iota>"} & if content of $R$ is non-zero,\\
- & & & then decrement it by one\\
- & & & otherwise jump to opcode $o$\\
- & $\mid$ & @{term "Goto o\<iota>"} & jump to opcode $o$
- \end{tabular}
- \end{center}
-
- \noindent
- A \emph{program} of an abacus machine is a list of such
- obcodes. For example the program clearing the register
- $R$ (setting it to 0) can be defined as follows:
-
- \begin{center}
- @{thm clear.simps[where n="R\<iota>" and e="o\<iota>", THEN eq_reflection]}
- \end{center}
-
- \noindent
- The second opcode @{term "Goto 0"} in this programm means we
- jump back to the first opcode, namely @{text "Dec R o"}.
- The \emph{memory} $m$ of an abacus machine holding the values
- of the registers is represented as a list of natural numbers.
- We have a lookup function for this memory, written @{term "abc_lm_v m R\<iota>"},
- which looks up the content of register $R$; if $R$
- is not in this list, then we return 0. Similarly we
- have a setting function, written @{term "abc_lm_s m R\<iota> n"}, which
- sets the value of $R$ to $n$, and if $R$ was not yet in $m$
- it pads it approriately with 0s.
-
-
- Abacus machine halts when it jumps out of range.
-*}
-
-
-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
-(*>*)
\ No newline at end of file