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Commit f971b1a1 authored by Michael Leuschel's avatar Michael Leuschel
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fix NAT symbols

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%% Cell type:markdown id: tags:
# Turing Maschinen und Berechenbarkeit
Bisher wurden Turingmaschinen als Akzeptoren definiert.
Nun wollen wir Turingmaschinen als Maschinen zur Berechnung von
Funktionen auffassen.
Eine Funktion $f: \nats^k \rightarrow \nats$ heißt
Eine Funktion $f: ℕ^k \rightarrow ℕ$ heißt
__Turing-berechenbar__, falls es eine deterministische
Turingmaschine $M=(\Sigma, \Gamma, Z, \delta ,z_0, \Box, F)$
mit $\Sigma = \{0,1\}$
gibt,
so dass für alle $n_1, n_2, \ldots , n_k, m \in \nats$:
* $f(n_1, n_2, \ldots , n_k) = m$ $\Leftrightarrow$
* $\exists z\in F . z_0 \mbox{bin}(n_1) \#
\mbox{bin}(n_2) \# \cdots \# \mbox{bin}(n_k) \vdash_{M}^{\ast} z
\mbox{bin}(m)$
Falls $f(n_1, n_2, \ldots , n_k)$ nicht definiert ist, läuft $M$ in
eine unendliche Schleife oder stoppt in einem Zustand $z\not\in F$.
%% Cell type:markdown id: tags:
Die (total definierte) \alert{\em Nachfolgerfunktion} $f: \nats \rightarrow \nats$ mit
Die (total definierte) __Nachfolgerfunktion__ $f: ℕ \rightarrow ℕ$ mit
* $f:n\rightarrow n+1$
ist Turing-berechenbar.
%% Cell type:code id: tags:
``` prob
::load
MACHINE TuringMachine_2
/* B Modell einer 1-band Turing Maschine */
/* by Michael Leuschel, 2012 */
/* The turing machine implements the +1 function on binary numbers */
/* See Slide 2 from folien-kapitel-7.pdf (Info 4) */
SETS
Alphabet ={O,I,Blank};
Z = {z0,z1,z2,ze};
Direction = {L,R,N}
DEFINITIONS
Σ == {O,I};
Γ == Σ;
CurSymbol == (Right<-Blank)(1);
ANIMATION_FUNCTION_DEFAULT == {(1,0,cur)};
ANIMATION_FUNCTION1 == {1} *( (%i.(i∈-size(Left)..-1|i+size(Left)+1) ; Left) \/ Right)
CONSTANTS Final, δ
PROPERTIES
Final ⊆ Z ∧
δ ∈ (Z * Alphabet) <-> (Z * Alphabet * Direction) ∧
δ = { (z0,O) ↦ (z0,O,R), (z0,I) ↦ (z0,I,R), (z0,Blank) ↦ (z1,Blank,L),
(z1,O) ↦ (z2,I,L), (z1,I) ↦ (z1,O,L), (z1,Blank) ↦ (ze,I,N),
(z2,O) ↦ (z2,O,L), (z2,I) ↦ (z2,I,L), (z2,Blank) ↦ (ze,Blank,R)
} ∧
Final = {ze}
VARIABLES Left,cur,Right
INVARIANT
cur ∈ Z ∧
Left ∈ seq(Alphabet) ∧ Right ∈ seq(Alphabet)
INITIALISATION Left,cur,Right := [],z0,[I,I,I,I,I] // [I,O,I]
OPERATIONS
Accept = PRE cur : Final THEN
Left,cur,Right := [],z0,Right /* restart machine with previous number */
END;
GoTo(z,S,znew,NewSymbol,Dir) =
PRE z=cur ∧ S=CurSymbol ∧
(z, S) ↦ (znew,NewSymbol,Dir) ∈ δ THEN
ANY tail_Right
WHERE (Right=[] ⇒ tail_Right=[]) ∧
(Right/=[] ⇒ tail_Right = tail(Right)) THEN
cur := znew ||
IF Dir = N THEN
Right := NewSymbol -> tail_Right
ELSIF Dir = R THEN
Left,Right := Left <- NewSymbol, tail_Right
ELSIF Left=[] THEN
Left,Right := [], Blank -> (NewSymbol -> tail_Right)
ELSE
Left,Right := front(Left), last(Left) -> (NewSymbol -> tail_Right)
END
END
END
END
```
%% Output
Loaded machine: TuringMachine_2
%% Cell type:code id: tags:
``` prob
:constants
```
%% Output
Machine constants set up using operation 0: $setup_constants()
%% Cell type:code id: tags:
``` prob
:init
```
%% Output
Machine initialised using operation 1: $initialise_machine()
%% Cell type:markdown id: tags:
Die Turing Maschine wird mit der Binärzahl 11111 (31 als Dezimalzahl) aufgerufen.
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z0</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,I,z0,I,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">z0</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,I,z0,I,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z0</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,I,z0,I,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z0</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,I,z0,I,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z0</td>
<td style="padding:10px">I</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,I,z0,I,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z0</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,Blank,z1,Blank,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z1</td>
<td style="padding:10px">I</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,I,z1,O,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z1</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,I,z1,O,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">I</td>
<td style="padding:10px">z1</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,I,z1,O,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">z1</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,I,z1,O,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z1</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,I,z1,O,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z1</td>
<td style="padding:10px">Blank</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,Blank,ze,I,N)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">ze</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec Accept
```
%% Output
Executed operation: Accept()
%% Cell type:markdown id: tags:
Die Turing Maschine hat 32 als Ergebnis ausgerechnet. Wir können die Maschine wieder mit 32 als Eingabe starten:
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z0</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,I,z0,I,R)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,O,z0,O,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">z0</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,O,z0,O,R)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,O,z0,O,R)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,O,z0,O,R)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,O,z0,O,R)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z0,Blank,z1,Blank,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">z1</td>
<td style="padding:10px">O</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z1,O,z2,I,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">z2</td>
<td style="padding:10px">O</td>
<td style="padding:10px">I</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z2,O,z2,O,L)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z2,O,z2,O,L)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z2,O,z2,O,L)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z2,O,z2,O,L)
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z2,I,z2,I,L)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z2</td>
<td style="padding:10px">Blank</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">I</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:exec GoTo
```
%% Output
Executed operation: GoTo(z2,Blank,ze,Blank,R)
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">Blank</td>
<td style="padding:10px">ze</td>
<td style="padding:10px">I</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">O</td>
<td style="padding:10px">I</td>
<td style="padding:10px">Blank</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:markdown id: tags:
Die Turing Maschine at 33 als Ergebnis zurückgegeben.
%% Cell type:code id: tags:
``` prob
:exec Accept
```
%% Output
Executed operation: Accept()
%% Cell type:code id: tags:
``` prob
```
......
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