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zadania.tex
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\documentclass[leqno]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{xstring}
\usepackage{hyperref}
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\begin{document}
\tableofcontents
\newpage
\begin{center}
W każdym zadaniu w którym trzeba posługiwać się alfabetem unarnym używam alfabetu \textbf{unarnego z zerem}
\end{center}
\section{Maszyny Turinga obliczające funkcje w alfabecie unarnym}
\subsection{Zadanie 1.1}
Zaprojektuj maszynę Turinga, które oblicza funkcję odejmowania
ograniczonego f dla liczb naturalnych m i n w reprezentacji unarnej, czyli
\begin{equation}
f(m, n) = m - n =
\begin{cases}
m-n, & \text{jeżeli } m \ge n, \\
0, & \text{jeżeli } m < n
\end{cases}
\end{equation}
Narysuj diagram przejść. Dla zaprojektowanej maszyny wykonaj dwa obliczenia
(wykonaj rysunki taśmy i zapisz konfiguracje).
\begin{equation}
M = (Q, \Sigma, \Gamma, \delta, q_0, \nabla, F) = (
\{q_0, q_1, q_2, q_3, q_4, q_5, q_6, q_7\},
\{1, 0\},
\{1, 0, \nabla\},
\delta,
q_0,
\nabla,
\{q_7\}
)
\end{equation}
\begin{center}
\begin{tikzpicture}
\node[state, initial] (q0) {$q_0$};
\node[state, right of=q0] (q1) {$q_1$};
\node[state, below of=q1] (q5) {$q_5$};
\node[state, right of=q5] (q4) {$q_4$};
\node[state, right of=q4] (q3) {$q_3$};
\node[state, above of=q3] (q2) {$q_2$};
\node[state, below of=q5] (q6) {$q_6$};
\node[state, below of=q3, accepting] (q7) {$q_7$};
\draw {}
(q0) edge[above] node{ \TRH { 1 } { B } { R } } (q1)
(q1) edge[above] node{ \TRH { B } { B } { L } } (q2)
(q2) edge[right] node{ \TRH { 1 } { B } { L } } (q3)
(q3) edge[above] node{ \TRH { 1 } { 1 } { L } } (q4)
(q4) edge[above] node{ \TRH { B } { B } { R } } (q5)
(q5) edge[left] node{ \TRH { 0 } { 0 } { R } } (q6)
(q5) edge[left] node{ \TRH { 1 } { B } { R } } (q1)
(q6) edge[loop below] node{ \TRH { 1 } { B } { R } } (q6)
(q6) edge[below] node{ \TRH { B } { B } { L } } (q7)
(q1) edge[loop above] node[ align=center ] {
\TRH { 1 } { 1 } { R }\\
\TRH { 0 } { 0 } { R }} (q1)
(q3) edge[right] node[ align=center ] {
\TRH { 0 } { 1 } { N }\\
\TRH { B } { 0 } { N }} (q7)
(q4) edge[loop below] node[ align=center ] {
\TRH { 1 } { 1 } { L }\\
\TRH { 0 } { 0 } { L }} (q4);
\end{tikzpicture}
\end{center}
\subsubsection{f(3,2)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$11110111 \cont} {B,g1,1,1,1,0,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$1110111 \cont} {B,B,g1,1,1,0,1,1,1,B}
\item \TMT {$K_2$ \; 1$q_1$110111 \cont} {B,B,1,g1,1,0,1,1,1,B}
\item \TMT {$K_3$ \; 11$q_1$10111 \cont} {B,B,1,1,g1,0,1,1,1,B}
\item \TMT {$K_4$ \; 111$q_1$0111 \cont} {B,B,1,1,1,g0,1,1,1,B}
\item \TMT {$K_5$ \; 1110$q_1$111 \cont} {B,B,1,1,1,0,g1,1,1,B}
\item \TMT {$K_6$ \; 11101$q_1$11 \cont} {B,B,1,1,1,0,1,g1,1,B}
\item \TMT {$K_7$ \; 111011$q_1$1 \cont} {B,B,1,1,1,0,1,1,g1,B}
\item \TMT {$K_8$ \; 1110111$q_1$B \cont} {B,B,1,1,1,0,1,1,1,gB}
\item \TMT {$K_9$ \; 111011$q_2$1 \cont} {B,B,1,1,1,0,1,1,g1,B}
\item \TMT {$K_{10}$ \; 11101$q_3$1 \cont} {B,B,1,1,1,0,1,g1,B,B}
\item \TMT {$K_{11}$ \; 1110$q_4$11 \cont} {B,B,1,1,1,0,g1,1,B,B}
\item \TMT {$K_{12}$ \; 111$q_4$011 \cont} {B,B,1,1,1,g0,1,1,B,B}
\item \TMT {$K_{13}$ \; 11$q_4$1011 \cont} {B,B,1,1,g1,0,1,1,B,B}
\item \TMT {$K_{14}$ \; 1$q_4$11011 \cont} {B,B,1,g1,1,0,1,1,B,B}
\item \TMT {$K_{15}$ \; $q_4$111011 \cont} {B,B,g1,1,1,0,1,1,B,B}
\item \TMT {$K_{16}$ \; $q_4$B111011 \cont} {B,gB,1,1,1,0,1,1,B,B}
\item \TMT {$K_{17}$ \; $q_5$111011 \cont} {B,B,g1,1,1,0,1,1,B,B}
\item \TMT {$K_{18}$ \; $q_1$11011 \cont} {B,B,B,g1,1,0,1,1,B,B}
\item \TMT {$K_{19}$ \; 1$q_1$1011 \cont} {B,B,B,1,g1,0,1,1,B,B}
\item \TMT {$K_{20}$ \; 11$q_1$011 \cont} {B,B,B,1,1,g0,1,1,B,B}
\item \TMT {$K_{21}$ \; 110$q_1$11 \cont} {B,B,B,1,1,0,g1,1,B,B}
\item \TMT {$K_{22}$ \; 1101$q_1$1 \cont} {B,B,B,1,1,0,1,g1,B,B}
\item \TMT {$K_{23}$ \; 11011$q_1$B \cont} {B,B,B,1,1,0,1,1,gB,B}
\item \TMT {$K_{24}$ \; 1101$q_2$1 \cont} {B,B,B,1,1,0,1,g1,B,B}
\item \TMT {$K_{25}$ \; 110$q_3$1 \cont} {B,B,B,1,1,0,g1,B,B,B}
\item \TMT {$K_{26}$ \; 11$q_4$01 \cont} {B,B,B,1,1,g0,1,B,B,B}
\item \TMT {$K_{27}$ \; 1$q_4$101 \cont} {B,B,B,1,g1,0,1,B,B,B}
\item \TMT {$K_{28}$ \; $q_4$1101 \cont} {B,B,B,g1,1,0,1,B,B,B}
\item \TMT {$K_{29}$ \; $q_4$B1101 \cont} {B,B,gB,1,1,0,1,B,B,B}
\item \TMT {$K_{30}$ \; $q_5$1101 \cont} {B,B,B,g1,1,0,1,B,B,B}
\item \TMT {$K_{31}$ \; $q_1$101 \cont} {B,B,B,B,g1,0,1,B,B,B}
\item \TMT {$K_{32}$ \; 1$q_1$01 \cont} {B,B,B,B,1,g0,1,B,B,B}
\item \TMT {$K_{33}$ \; 10$q_1$1 \cont} {B,B,B,B,1,0,g1,B,B,B}
\item \TMT {$K_{34}$ \; 101$q_1$B \cont} {B,B,B,B,1,0,1,gB,B,B}
\item \TMT {$K_{35}$ \; 10$q_2$1 \cont} {B,B,B,B,1,0,g1,B,B,B}
\item \TMT {$K_{36}$ \; 1$q_3$0 \cont} {B,B,B,B,1,g0,B,B,B,B}
\item \TMT {$K_{37}$ \; 1$q_7$1} {B,B,B,B,1,g1,B,B,B,B}
\end{enumerate}
\subsubsection{f(2, 3)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$11101111 \cont} {B,g1,1,1,0,1,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$1101111 \cont} {B,B,g1,1,0,1,1,1,1,B}
\item \TMT {$K_2$ \; 1$q_1$101111 \cont} {B,B,1,g1,0,1,1,1,1,B}
\item \TMT {$K_3$ \; 11$q_1$01111 \cont} {B,B,1,1,g0,1,1,1,1,B}
\item \TMT {$K_4$ \; 110$q_1$1111 \cont} {B,B,1,1,0,g1,1,1,1,B}
\item \TMT {$K_5$ \; 1101$q_1$111 \cont} {B,B,1,1,0,1,g1,1,1,B}
\item \TMT {$K_6$ \; 11011$q_1$11 \cont} {B,B,1,1,0,1,1,g1,1,B}
\item \TMT {$K_7$ \; 110111$q_1$1 \cont} {B,B,1,1,0,1,1,1,g1,B}
\item \TMT {$K_8$ \; 1101111$q_1$B \cont} {B,B,1,1,0,1,1,1,1,gB}
\item \TMT {$K_9$ \; 110111$q_2$1 \cont} {B,B,1,1,0,1,1,1,g1,B}
\item \TMT {$K_{10}$ \; 11011$q_3$1 \cont} {B,B,1,1,0,1,1,g1,B,B}
\item \TMT {$K_{11}$ \; 1101$q_4$11 \cont} {B,B,1,1,0,1,g1,1,B,B}
\item \TMT {$K_{12}$ \; 110$q_4$111 \cont} {B,B,1,1,0,g1,1,1,B,B}
\item \TMT {$K_{13}$ \; 11$q_4$0111 \cont} {B,B,1,1,g0,1,1,1,B,B}
\item \TMT {$K_{14}$ \; 1$q_4$10111 \cont} {B,B,1,g1,0,1,1,1,B,B}
\item \TMT {$K_{15}$ \; $q_4$110111 \cont} {B,B,g1,1,0,1,1,1,B,B}
\item \TMT {$K_{16}$ \; $q_4$B110111 \cont} {B,gB,1,1,0,1,1,1,B,B}
\item \TMT {$K_{17}$ \; $q_5$110111 \cont} {B,B,g1,1,0,1,1,1,B,B}
\item \TMT {$K_{18}$ \; $q_1$10111 \cont} {B,B,B,g1,0,1,1,1,B,B}
\item \TMT {$K_{19}$ \; 1$q_1$0111 \cont} {B,B,B,1,g0,1,1,1,B,B}
\item \TMT {$K_{20}$ \; 10$q_1$111 \cont} {B,B,B,1,0,g1,1,1,B,B}
\item \TMT {$K_{21}$ \; 101$q_1$11 \cont} {B,B,B,1,0,1,g1,1,B,B}
\item \TMT {$K_{22}$ \; 1011$q_1$1 \cont} {B,B,B,1,0,1,1,g1,B,B}
\item \TMT {$K_{23}$ \; 10111$q_1$B \cont} {B,B,B,1,0,1,1,1,gB,B}
\item \TMT {$K_{24}$ \; 1011$q_2$1 \cont} {B,B,B,1,0,1,1,g1,B,B}
\item \TMT {$K_{25}$ \; 101$q_3$1 \cont} {B,B,B,1,0,1,g1,B,B,B}
\item \TMT {$K_{26}$ \; 10$q_4$11 \cont} {B,B,B,1,0,g1,1,B,B,B}
\item \TMT {$K_{27}$ \; 1$q_4$011 \cont} {B,B,B,1,g0,1,1,B,B,B}
\item \TMT {$K_{28}$ \; $q_4$1011 \cont} {B,B,B,g1,0,1,1,B,B,B}
\item \TMT {$K_{29}$ \; $q_4$B1011 \cont} {B,B,gB,1,0,1,1,B,B,B}
\item \TMT {$K_{30}$ \; $q_5$1011 \cont} {B,B,B,g1,0,1,1,B,B,B}
\item \TMT {$K_{31}$ \; $q_1$011 \cont} {B,B,B,B,g0,1,1,B,B,B}
\item \TMT {$K_{32}$ \; 0$q_1$11 \cont} {B,B,B,B,0,g1,1,B,B,B}
\item \TMT {$K_{33}$ \; 01$q_1$1 \cont} {B,B,B,B,0,1,g1,B,B,B}
\item \TMT {$K_{34}$ \; 011$q_1$B \cont} {B,B,B,B,0,1,1,gB,B,B}
\item \TMT {$K_{35}$ \; 01$q_2$1 \cont} {B,B,B,B,0,1,g1,B,B,B}
\item \TMT {$K_{36}$ \; 0$q_3$1 \cont} {B,B,B,B,0,g1,B,B,B,B}
\item \TMT {$K_{37}$ \; $q_4$01 \cont} {B,B,B,B,g0,1,B,B,B,B}
\item \TMT {$K_{38}$ \; $q_4$B01 \cont} {B,B,B,gB,0,1,B,B,B,B}
\item \TMT {$K_{39}$ \; $q_5$01 \cont} {B,B,B,B,g0,1,B,B,B,B}
\item \TMT {$K_{40}$ \; 0$q_6$1 \cont} {B,B,B,B,0,g1,B,B,B,B}
\item \TMT {$K_{41}$ \; 0$q_6$B \cont} {B,B,B,B,0,gB,B,B,B,B}
\item \TMT {$K_{42}$ \; $q_7$0} {B,B,B,B,g0,B,B,B,B,B}
\end{enumerate}
\newpage
\subsection{Zadanie 1.2}
Zaprojektuj maszynę Turinga, które oblicza funkcję mnożenia f dla
liczb naturalnych m i n w reprezentacji unarnej, czyli
\begin{equation}
f(m, n) = m \cdot n
\end{equation}
Narysuj diagram przejść. Dla zaprojektowanej maszyny wykonaj dwa
obliczenia, w tym pomnóż 3·2 lub 2·3 (wykonaj rysunki taśmy i zapisz konfiguracje).
\begin{equation}
M = (Q, \Sigma, \Gamma, \delta, q_0, \nabla, F) = (
\{q_0, q_1, q_2, q_3, q_4, q_5, q_6, q_7, q_8, q_9, q_{10}, q_{11}, q_{12}, q_{13}\},
\{1, 0\},
\{1, 0, \nabla\},
\delta,
q_0,
\nabla,
\{q_{13}\}
)
\end{equation}
\begin{center}
\begin{tikzpicture}
\node[state, initial left] (q0) {$q_0$};
\node[state, above of=q0] (q1) {$q_1$};
\node[state, right of=q1] (q2) {$q_2$};
\node[state, below of=q2] (q3) {$q_3$};
\node[state, below of=q3] (q11) {$q_{11}$};
\node[state, below of=q11] (q4) {$q_4$};
\node[state, right of=q11] (q10) {$q_{10}$};
\node[state, right of=q10] (q6) {$q_6$};
\node[state, below of=q6] (q5) {$q_5$};
\node[state, right of=q6] (q7) {$q_7$};
\node[state, above of=q7] (q8) {$q_8$};
\node[state, above of=q10] (q9) {$q_9$};
\node[state, left of=q4] (q12) {$q_{12}$};
\node[state, above of=q12, accepting] (q13) {$q_{13}$};
\draw {}
(q0) edge[left] node{ \TRH{ 1 } { B } { R } } (q1)
(q1) edge[loop above] node{ \TRH{ 1 } { 1 } { R } } (q1)
(q1) edge[above] node{ \TRH{ 0 } { 0 } { R } } (q2)
(q2) edge[right] node{ \TRH{ B } { B } { L } } (q3)
(q2) edge[loop above] node{ \TRH{ 1 } { 1 } { R } } (q2)
(q3) edge[left] node{ \TRH{ 1 } { B } { L } } (q11)
(q4) edge[above] node{ \TRH{ 0 } { B } { R } } (q12)
(q4) edge[above] node{ \TRH{ 1 } { B } { R } } (q5)
(q5) edge[right] node{ \TRH{ 0 } { 0 } { R } } (q6)
(q5) edge[loop below] node{ \TRH{ 1 } { 1 } { R } } (q5)
(q6) edge[right] node{ \TRH{ B } { B } { L } } (q9)
(q6) edge[above] node{ \TRH{ 1 } { B } { R } } (q7)
(q7) edge[right] node{ \TRH{ B } { B } { R } } (q8)
(q7) edge[loop below] node{ \TRH{ 1 } { 1 } { R } } (q7)
(q8) edge[loop above] node{ \TRH{ 1 } { 1 } { R } } (q8)
(q8) edge[above] node{ \TRH{ B } { 1 } { L } } (q9)
(q9) edge[left] node{ \TRH{ 0 } { 0 } { L } } (q11)
(q9) edge[loop above] node{ \TRH{ 1 } { 1 } { L } } (q9)
(q10) edge[above] node{ \TRH{ B } { 1 } { R } } (q6)
(q10) edge[above] node{ \TRH{ 0 } { 0 } { L } } (q11)
(q10) edge[loop below] node{ \TRH{ 1 } { 1 } { L } } (q10)
(q11) edge[right] node{ \TRH{ B } { B } { R } } (q4)
(q12) edge[left] node{ \TRH{ B } { 1 } { N } } (q13)
(q12) edge[loop below] node{ \TRH{ 1 } { B } { R } } (q12)
(q9) edge[above] node[midarr]{ \TRH{ B } { B } { L } } (q10)
(q11) edge[loop left] node[ align=center ]{
\TRH{ 0 } { 0 } { L }\\
\TRH{ 1 } { 1 } { L }} (q11);
\end{tikzpicture}
\end{center}
\subsubsection{f(3, 2)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$11110111 \cont} {B,g1,1,1,1,0,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$1110111 \cont} {B,B,g1,1,1,0,1,1,1,B}
\item \TMT {$K_2$ \; 1$q_1$110111 \cont} {B,B,1,g1,1,0,1,1,1,B}
\item \TMT {$K_3$ \; 11$q_1$10111 \cont} {B,B,1,1,g1,0,1,1,1,B}
\item \TMT {$K_4$ \; 111$q_1$0111 \cont} {B,B,1,1,1,g0,1,1,1,B}
\item \TMT {$K_5$ \; 1110$q_2$111 \cont} {B,B,1,1,1,0,g1,1,1,B}
\item \TMT {$K_6$ \; 11101$q_2$11 \cont} {B,B,1,1,1,0,1,g1,1,B}
\item \TMT {$K_7$ \; 111011$q_2$1 \cont} {B,B,1,1,1,0,1,1,g1,B}
\item \TMT {$K_8$ \; 1110111$q_2$B \cont} {B,B,1,1,1,0,1,1,1,gB}
\item \TMT {$K_9$ \; 111011$q_3$1 \cont} {B,B,1,1,1,0,1,1,g1,B}
\item \TMT {$K_{10}$ \; 11101$q_{11}$1 \cont} {B,B,1,1,1,0,1,g1,B,B}
\item \TMT {$K_{11}$ \; 1110$q_{11}$11 \cont} {B,B,1,1,1,0,g1,1,B,B}
\item \TMT {$K_{12}$ \; 111$q_{11}$011 \cont} {B,B,1,1,1,g0,1,1,B,B}
\item \TMT {$K_{13}$ \; 11$q_{11}$1011 \cont} {B,B,1,1,g1,0,1,1,B,B}
\item \TMT {$K_{14}$ \; 1$q_{11}$11011 \cont} {B,B,1,g1,1,0,1,1,B,B}
\item \TMT {$K_{15}$ \; $q_{11}$111011 \cont} {B,B,g1,1,1,0,1,1,B,B}
\item \TMT {$K_{16}$ \; $q_{11}$B111011 \cont} {B,gB,1,1,1,0,1,1,B,B}
\item \TMT {$K_{17}$ \; $q_4$111011 \cont} {B,B,g1,1,1,0,1,1,B,B}
\item \TMT {$K_{18}$ \; $q_5$11011 \cont} {B,B,B,g1,1,0,1,1,B,B}
\item \TMT {$K_{19}$ \; 1$q_5$1011 \cont} {B,B,B,1,g1,0,1,1,B,B}
\item \TMT {$K_{20}$ \; 11$q_5$011 \cont} {B,B,B,1,1,g0,1,1,B,B}
\item \TMT {$K_{21}$ \; 110$q_6$11 \cont} {B,B,B,1,1,0,g1,1,B,B}
\item \TMT {$K_{22}$ \; 110B$q_7$1 \cont} {B,B,B,1,1,0,B,g1,B,B}
\item \TMT {$K_{23}$ \; 110B1$q_7$B \cont} {B,B,B,1,1,0,B,1,gB,B}
\item \TMT {$K_{24}$ \; 110B1B$q_8$B \cont} {B,B,B,1,1,0,B,1,B,gB}
\item \TMT {$K_{25}$ \; 110B1$q_9$B1 \cont} {B,B,B,1,1,0,B,1,gB,1}
\item \TMT {$K_{26}$ \; 110B$q_{10}$1B1 \cont} {B,B,B,1,1,0,B,g1,B,1}
\item \TMT {$K_{27}$ \; 110$q_{10}$B1B1 \cont} {B,B,B,1,1,0,gB,1,B,1}
\item \TMT {$K_{28}$ \; 1101$q_6$1B1 \cont} {B,B,B,1,1,0,1,g1,B,1}
\item \TMT {$K_{29}$ \; 1101B$q_7$B1 \cont} {B,B,B,1,1,0,1,B,gB,1}
\item \TMT {$K_{30}$ \; 1101BB$q_8$1 \cont} {B,B,B,1,1,0,1,B,B,g1}
\item \TMT {$K_{31}$ \; 1101BB1$q_8$B \cont} {B,B,B,1,1,0,1,B,B,1,gB}
\item \TMT {$K_{32}$ \; 1101BB$q_9$11 \cont} {B,B,B,1,1,0,1,B,B,g1,1}
\item \TMT {$K_{33}$ \; 1101B$q_9$B11 \cont} {B,B,B,1,1,0,1,B,gB,1,1}
\item \TMT {$K_{34}$ \; 1101$q_{10}$BB11 \cont} {B,B,B,1,1,0,1,gB,B,1,1}
\item \TMT {$K_{35}$ \; 11011$q_6$B11 \cont} {B,B,B,1,1,0,1,1,gB,1,1}
\item \TMT {$K_{36}$ \; 1101$q_9$1B11 \cont} {B,B,B,1,1,0,1,g1,B,1,1}
\item \TMT {$K_{37}$ \; 110$q_9$11B11 \cont} {B,B,B,1,1,0,g1,1,B,1,1}
\item \TMT {$K_{38}$ \; 11$q_9$011B11 \cont} {B,B,B,1,1,g0,1,1,B,1,1}
\item \TMT {$K_{39}$ \; 1$q_{11}$1011B11 \cont} {B,B,B,1,g1,0,1,1,B,1,1}
\item \TMT {$K_{40}$ \; $q_{11}$11011B11 \cont} {B,B,B,g1,1,0,1,1,B,1,1}
\item \TMT {$K_{41}$ \; $q_{11}$B11011B11 \cont} {B,B,gB,1,1,0,1,1,B,1,1}
\item \TMT {$K_{42}$ \; $q_4$11011B11 \cont} {B,B,B,g1,1,0,1,1,B,1,1}
\item \TMT {$K_{43}$ \; $q_5$1011B11 \cont} {B,B,B,B,g1,0,1,1,B,1,1}
\item \TMT {$K_{44}$ \; 1$q_5$011B11 \cont} {B,B,B,B,1,g0,1,1,B,1,1}
\item \TMT {$K_{45}$ \; 10$q_6$11B11 \cont} {B,B,B,B,1,0,g1,1,B,1,1}
\item \TMT {$K_{46}$ \; 10B$q_7$1B11 \cont} {B,B,B,B,1,0,B,g1,B,1,1}
\item \TMT {$K_{47}$ \; 10B1$q_7$B11 \cont} {B,B,B,B,1,0,B,1,gB,1,1}
\item \TMT {$K_{48}$ \; 10B1B$q_8$11 \cont} {B,B,B,B,1,0,B,1,B,g1,1}
\item \TMT {$K_{49}$ \; 10B1B1$q_8$1 \cont} {B,B,B,B,1,0,B,1,B,1,g1}
\begin{center}
\textbf{Przesuwam taśmę o 3 miejsca w lewo bo by była zbyt długa}
\end{center}
\item \TMT {$K_{50}$ \; 10B1B11$q_8$B \cont} {B,1,0,B,1,B,1,1,gB}
\item \TMT {$K_{51}$ \; 10B1B1$q_9$11 \cont} {B,1,0,B,1,B,1,g1,1}
\item \TMT {$K_{52}$ \; 10B1B$q_9$111 \cont} {B,1,0,B,1,B,g1,1,1}
\item \TMT {$K_{53}$ \; 10B1$q_9$B111 \cont} {B,1,0,B,1,gB,1,1,1}
\item \TMT {$K_{54}$ \; 10B$q_{10}$1B111 \cont} {B,1,0,B,g1,B,1,1,1}
\item \TMT {$K_{55}$ \; 10$q_{10}$B1B111 \cont} {B,1,0,gB,1,B,1,1,1}
\item \TMT {$K_{56}$ \; 101$q_6$1B111 \cont} {B,1,0,1,g1,B,1,1,1}
\item \TMT {$K_{57}$ \; 101B$q_7$B111 \cont} {B,1,0,1,B,gB,1,1,1}
\item \TMT {$K_{58}$ \; 101BB$q_8$111 \cont} {B,1,0,1,B,B,g1,1,1}
\item \TMT {$K_{59}$ \; 101BB1$q_8$11 \cont} {B,1,0,1,B,B,1,g1,1}
\item \TMT {$K_{60}$ \; 101BB11$q_8$1 \cont} {B,1,0,1,B,B,1,1,g1}
\item \TMT {$K_{61}$ \; 101BB111$q_8$B \cont} {B,1,0,1,B,B,1,1,1,gB}
\item \TMT {$K_{62}$ \; 101BB11$q_9$11 \cont} {B,1,0,1,B,B,1,1,g1,1}
\item \TMT {$K_{63}$ \; 101BB1$q_9$111 \cont} {B,1,0,1,B,B,1,g1,1,1}
\item \TMT {$K_{64}$ \; 101BB$q_9$1111 \cont} {B,1,0,1,B,B,g1,1,1,1}
\item \TMT {$K_{65}$ \; 101B$q_9$B1111 \cont} {B,1,0,1,B,gB,1,1,1,1}
\item \TMT {$K_{66}$ \; 101$q_{10}$BB1111 \cont} {B,1,0,1,gB,B,1,1,1,1}
\item \TMT {$K_{67}$ \; 1011$q_6$B1111 \cont} {B,1,0,1,1,gB,1,1,1,1}
\item \TMT {$K_{68}$ \; 101$q_9$1B1111 \cont} {B,1,0,1,g1,B,1,1,1,1}
\item \TMT {$K_{69}$ \; 10$q_9$11B1111 \cont} {B,1,0,g1,1,B,1,1,1,1}
\item \TMT {$K_{70}$ \; 1$q_9$011B1111 \cont} {B,1,g0,1,1,B,1,1,1,1}
\item \TMT {$K_{71}$ \; $q_{11}$1011B1111 \cont} {B,g1,0,1,1,B,1,1,1,1}
\item \TMT {$K_{72}$ \; $q_{11}$B1011B1111 \cont} {gB,1,0,1,1,B,1,1,1,1}
\item \TMT {$K_{73}$ \; $q_4$1011B1111 \cont} {B,g1,0,1,1,B,1,1,1,1}
\item \TMT {$K_{74}$ \; $q_5$011B1111 \cont} {B,B,g0,1,1,B,1,1,1,1}
\item \TMT {$K_{75}$ \; 0$q_6$11B1111 \cont} {B,B,0,g1,1,B,1,1,1,1}
\item \TMT {$K_{76}$ \; 0B$q_7$1B1111 \cont} {B,B,0,B,g1,B,1,1,1,1}
\item \TMT {$K_{77}$ \; 0B1$q_7$B1111 \cont} {B,B,0,B,1,gB,1,1,1,1}
\item \TMT {$K_{78}$ \; 0B1B$q_8$1111 \cont} {B,B,0,B,1,B,g1,1,1,1}
\item \TMT {$K_{79}$ \; 0B1B1$q_8$111 \cont} {B,B,0,B,1,B,1,g1,1,1}
\item \TMT {$K_{80}$ \; 0B1B11$q_8$11 \cont} {B,B,0,B,1,B,1,1,g1,1}
\item \TMT {$K_{81}$ \; 0B1B111$q_8$1 \cont} {B,B,0,B,1,B,1,1,1,g1}
\item \TMT {$K_{82}$ \; 0B1B1111$q_8$B \cont} {B,B,0,B,1,B,1,1,1,1,gB}
\item \TMT {$K_{83}$ \; 0B1B111$q_9$11 \cont} {B,B,0,B,1,B,1,1,1,g1,1}
\item \TMT {$K_{84}$ \; 0B1B11$q_9$111 \cont} {B,B,0,B,1,B,1,1,g1,1,1}
\item \TMT {$K_{85}$ \; 0B1B1$q_9$1111 \cont} {B,B,0,B,1,B,1,g1,1,1,1}
\item \TMT {$K_{86}$ \; 0B1B$q_9$11111 \cont} {B,B,0,B,1,B,g1,1,1,1,1}
\item \TMT {$K_{87}$ \; 0B1$q_9$B11111 \cont} {B,B,0,B,1,gB,1,1,1,1,1}
\item \TMT {$K_{88}$ \; 0B$q_{10}$1B11111 \cont} {B,B,0,B,g1,B,1,1,1,1,1}
\item \TMT {$K_{89}$ \; 0$q_{10}$B1B11111 \cont} {B,B,0,gB,1,B,1,1,1,1,1}
\item \TMT {$K_{90}$ \; 01$q_6$1B11111 \cont} {B,B,0,1,g1,B,1,1,1,1,1}
\item \TMT {$K_{91}$ \; 01B$q_7$B11111 \cont} {B,B,0,1,B,gB,1,1,1,1,1}
\item \TMT {$K_{92}$ \; 01BB$q_8$11111 \cont} {B,B,0,1,B,B,g1,1,1,1,1}
\item \TMT {$K_{93}$ \; 01BB1$q_8$1111 \cont} {B,B,0,1,B,B,1,g1,1,1,1}
\item \TMT {$K_{94}$ \; 01BB11$q_8$111 \cont} {B,B,0,1,B,B,1,1,g1,1,1}
\item \TMT {$K_{95}$ \; 01BB111$q_8$11 \cont} {B,B,0,1,B,B,1,1,1,g1,1}
\item \TMT {$K_{96}$ \; 01BB1111$q_8$1 \cont} {B,B,0,1,B,B,1,1,1,1,g1}
\item \TMT {$K_{97}$ \; 01BB11111$q_8$B \cont} {B,B,0,1,B,B,1,1,1,1,1,gB}
\item \TMT {$K_{98}$ \; 01BB1111$q_9$11 \cont} {B,B,0,1,B,B,1,1,1,1,g1,1}
\item \TMT {$K_{99}$ \; 01BB111$q_9$111 \cont} {B,B,0,1,B,B,1,1,1,g1,1,1}
\item \TMT {$K_{100}$ \; 01BB11$q_9$1111 \cont} {B,B,0,1,B,B,1,1,g1,1,1,1}
\item \TMT {$K_{101}$ \; 01BB1$q_9$11111 \cont} {B,B,0,1,B,B,1,g1,1,1,1,1}
\item \TMT {$K_{102}$ \; 01BB$q_9$111111 \cont} {B,B,0,1,B,B,g1,1,1,1,1,1}
\item \TMT {$K_{103}$ \; 01B$q_9$B111111 \cont} {B,B,0,1,B,gB,1,1,1,1,1,1}
\item \TMT {$K_{104}$ \; 01$q_{10}$BB111111 \cont} {B,B,0,1,gB,B,1,1,1,1,1,1}
\item \TMT {$K_{105}$ \; 011$q_6$B111111 \cont} {B,B,0,1,1,gB,1,1,1,1,1,1}
\item \TMT {$K_{106}$ \; 01$q_9$1B111111 \cont} {B,B,0,1,g1,B,1,1,1,1,1,1}
\item \TMT {$K_{107}$ \; 0$q_9$11B111111 \cont} {B,B,0,g1,1,B,1,1,1,1,1,1}
\item \TMT {$K_{108}$ \; $q_9$011B111111 \cont} {B,B,g0,1,1,B,1,1,1,1,1,1}
\item \TMT {$K_{109}$ \; $q_{11}$B011B111111 \cont} {B,gB,0,1,1,B,1,1,1,1,1,1}
\item \TMT {$K_{110}$ \; $q_4$011B111111 \cont} {B,B,g0,1,1,B,1,1,1,1,1,1}
\item \TMT {$K_{111}$ \; $q_{12}$11B111111 \cont} {B,B,B,g1,1,B,1,1,1,1,1,1}
\item \TMT {$K_{112}$ \; $q_{12}$1B111111 \cont} {B,B,B,B,g1,B,1,1,1,1,1,1}
\item \TMT {$K_{113}$ \; $q_{12}$B111111 \cont} {B,B,B,B,B,gB,1,1,1,1,1,1}
\item \TMT {$K_{114}$ \; $q_{13}$1111111} {B,B,B,B,B,g1,1,1,1,1,1,1}
\end{enumerate}
\subsubsection{f(2, 2)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$1110111 \cont} {B,g1,1,1,0,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$110111 \cont} {B,B,g1,1,0,1,1,1,B}
\item \TMT {$K_2$ \; 1$q_1$10111 \cont} {B,B,1,g1,0,1,1,1,B}
\item \TMT {$K_3$ \; 11$q_1$0111 \cont} {B,B,1,1,g0,1,1,1,B}
\item \TMT {$K_4$ \; 110$q_2$111 \cont} {B,B,1,1,0,g1,1,1,B}
\item \TMT {$K_5$ \; 1101$q_2$11 \cont} {B,B,1,1,0,1,g1,1,B}
\item \TMT {$K_6$ \; 11011$q_2$1 \cont} {B,B,1,1,0,1,1,g1,B}
\item \TMT {$K_7$ \; 110111$q_2$B \cont} {B,B,1,1,0,1,1,1,gB}
\item \TMT {$K_8$ \; 11011$q_3$1 \cont} {B,B,1,1,0,1,1,g1,B}
\item \TMT {$K_9$ \; 1101$q_{11}$1 \cont} {B,B,1,1,0,1,g1,B,B}
\item \TMT {$K_{10}$ \; 110$q_{11}$11 \cont} {B,B,1,1,0,g1,1,B,B}
\item \TMT {$K_{11}$ \; 11$q_{11}$011 \cont} {B,B,1,1,g0,1,1,B,B}
\item \TMT {$K_{12}$ \; 1$q_{11}$1011 \cont} {B,B,1,g1,0,1,1,B,B}
\item \TMT {$K_{13}$ \; $q_{11}$11011 \cont} {B,B,g1,1,0,1,1,B,B}
\item \TMT {$K_{14}$ \; $q_{11}$B11011 \cont} {B,gB,1,1,0,1,1,B,B}
\item \TMT {$K_{15}$ \; $q_4$11011 \cont} {B,B,g1,1,0,1,1,B,B}
\item \TMT {$K_{16}$ \; $q_5$1011 \cont} {B,B,B,g1,0,1,1,B,B}
\item \TMT {$K_{17}$ \; 1$q_5$011 \cont} {B,B,B,1,g0,1,1,B,B}
\item \TMT {$K_{18}$ \; 10$q_6$11 \cont} {B,B,B,1,0,g1,1,B,B}
\item \TMT {$K_{19}$ \; 10B$q_7$1 \cont} {B,B,B,1,0,B,g1,B,B}
\item \TMT {$K_{20}$ \; 10B1$q_7$B \cont} {B,B,B,1,0,B,1,gB,B}
\item \TMT {$K_{21}$ \; 10B1B$q_8$B \cont} {B,B,B,1,0,B,1,B,gB}
\item \TMT {$K_{22}$ \; 10B1$q_9$B1 \cont} {B,B,B,1,0,B,1,gB,1}
\item \TMT {$K_{23}$ \; 10B$q_{10}$1B1 \cont} {B,B,B,1,0,B,g1,B,1}
\item \TMT {$K_{24}$ \; 10$q_{10}$B1B1 \cont} {B,B,B,1,0,gB,1,B,1}
\item \TMT {$K_{25}$ \; 101$q_6$1B1 \cont} {B,B,B,1,0,1,g1,B,1}
\item \TMT {$K_{26}$ \; 101B$q_7$B1 \cont} {B,B,B,1,0,1,B,gB,1}
\item \TMT {$K_{27}$ \; 101BB$q_8$1 \cont} {B,B,B,1,0,1,B,B,g1}
\item \TMT {$K_{28}$ \; 101BB1$q_8$B \cont} {B,B,B,1,0,1,B,B,1,gB}
\item \TMT {$K_{29}$ \; 101BB$q_9$11 \cont} {B,B,B,1,0,1,B,B,g1,1}
\item \TMT {$K_{30}$ \; 101B$q_9$B11 \cont} {B,B,B,1,0,1,B,gB,1,1}
\item \TMT {$K_{31}$ \; 101$q_{10}$BB11 \cont} {B,B,B,1,0,1,gB,B,1,1}
\item \TMT {$K_{32}$ \; 1011$q_6$B11 \cont} {B,B,B,1,0,1,1,gB,1,1}
\item \TMT {$K_{33}$ \; 101$q_9$1B11 \cont} {B,B,B,1,0,1,g1,B,1,1}
\item \TMT {$K_{34}$ \; 10$q_9$11B11 \cont} {B,B,B,1,0,g1,1,B,1,1}
\item \TMT {$K_{35}$ \; 1$q_9$011B11 \cont} {B,B,B,1,g0,1,1,B,1,1}
\item \TMT {$K_{36}$ \; $q_{11}$1011B11 \cont} {B,B,B,g1,0,1,1,B,1,1}
\item \TMT {$K_{37}$ \; $q_{11}$B1011B11 \cont} {B,B,gB,1,0,1,1,B,1,1}
\item \TMT {$K_{38}$ \; $q_4$1011B11 \cont} {B,B,B,g1,0,1,1,B,1,1}
\item \TMT {$K_{39}$ \; $q_5$011B11 \cont} {B,B,B,B,g0,1,1,B,1,1}
\item \TMT {$K_{40}$ \; 0$q_6$11B11 \cont} {B,B,B,B,0,g1,1,B,1,1}
\item \TMT {$K_{41}$ \; 0B$q_7$1B11 \cont} {B,B,B,B,0,B,g1,B,1,1}
\item \TMT {$K_{42}$ \; 0B1$q_7$B11 \cont} {B,B,B,B,0,B,1,gB,1,1}
\item \TMT {$K_{43}$ \; 0B1B$q_8$11 \cont} {B,B,B,B,0,B,1,B,g1,1}
\item \TMT {$K_{44}$ \; 0B1B1$q_8$1 \cont} {B,B,B,B,0,B,1,B,1,g1}
\item \TMT {$K_{45}$ \; 0B1B11$q_8$B \cont} {B,B,B,B,0,B,1,B,1,1,gB}
\item \TMT {$K_{46}$ \; 0B1B1$q_9$11 \cont} {B,B,B,B,0,B,1,B,1,g1,1}
\item \TMT {$K_{47}$ \; 0B1B$q_9$111 \cont} {B,B,B,B,0,B,1,B,g1,1,1}
\item \TMT {$K_{48}$ \; 0B1$q_9$B111 \cont} {B,B,B,B,0,B,1,gB,1,1,1}
\item \TMT {$K_{49}$ \; 0B$q_{10}$1B111 \cont} {B,B,B,B,0,B,g1,B,1,1,1}
\item \TMT {$K_{50}$ \; 0$q_{10}$B1B111 \cont} {B,B,B,B,0,gB,1,B,1,1,1}
\item \TMT {$K_{51}$ \; 01$q_6$1B111 \cont} {B,B,B,B,0,1,g1,B,1,1,1}
\item \TMT {$K_{52}$ \; 01B$q_7$B111 \cont} {B,B,B,B,0,1,B,gB,1,1,1}
\item \TMT {$K_{53}$ \; 01BB$q_8$111 \cont} {B,B,B,B,0,1,B,B,g1,1,1}
\item \TMT {$K_{54}$ \; 01BB1$q_8$11 \cont} {B,B,B,B,0,1,B,B,1,g1,1}
\item \TMT {$K_{55}$ \; 01BB11$q_8$1 \cont} {B,B,B,B,0,1,B,B,1,1,g1}
\item \TMT {$K_{56}$ \; 01BB111$q_8$B \cont} {B,B,B,B,0,1,B,B,1,1,1,gB}
\item \TMT {$K_{57}$ \; 01BB11$q_9$11 \cont} {B,B,B,B,0,1,B,B,1,1,g1,1}
\item \TMT {$K_{58}$ \; 01BB1$q_9$111 \cont} {B,B,B,B,0,1,B,B,1,g1,1,1}
\item \TMT {$K_{59}$ \; 01BB$q_9$1111 \cont} {B,B,B,B,0,1,B,B,g1,1,1,1}
\item \TMT {$K_{60}$ \; 01B$q_9$B1111 \cont} {B,B,B,B,0,1,B,gB,1,1,1,1}
\item \TMT {$K_{61}$ \; 01$q_{10}$BB1111 \cont} {B,B,B,B,0,1,gB,B,1,1,1,1}
\item \TMT {$K_{62}$ \; 011$q_6$B1111 \cont} {B,B,B,B,0,1,1,gB,1,1,1,1}
\item \TMT {$K_{63}$ \; 01$q_9$1B1111 \cont} {B,B,B,B,0,1,g1,B,1,1,1,1}
\item \TMT {$K_{64}$ \; 0$q_9$11B1111 \cont} {B,B,B,B,0,g1,1,B,1,1,1,1}
\item \TMT {$K_{65}$ \; $q_9$011B1111 \cont} {B,B,B,B,g0,1,1,B,1,1,1,1}
\item \TMT {$K_{66}$ \; $q_{11}$B011B1111 \cont} {B,B,B,gB,0,1,1,B,1,1,1,1}
\item \TMT {$K_{67}$ \; $q_4$011B1111 \cont} {B,B,B,B,g0,1,1,B,1,1,1,1}
\item \TMT {$K_{68}$ \; $q_{12}$11B1111 \cont} {B,B,B,B,B,g1,1,B,1,1,1,1}
\item \TMT {$K_{69}$ \; $q_{12}$1B1111 \cont} {B,B,B,B,B,B,g1,B,1,1,1,1}
\item \TMT {$K_{70}$ \; $q_{12}$B1111 \cont} {B,B,B,B,B,B,B,gB,1,1,1,1}
\item \TMT {$K_{71}$ \; $q_{13}$11111} {B,B,B,B,B,B,B,g1,1,1,1,1}
\end{enumerate}
\subsection{Zadanie 1.3}
Zaprojektuj maszynę Turinga, które oblicza funkcję f dla liczby
naturalnej n w reprezentacji unarnej, gdzie
\begin{equation}
f(n) =
\begin{cases}
\frac{n}{2}, & \text{jeżeli } n \text{ jest parzysta,} \\
\frac{n+1}{2}, & \text{jeżeli } n \text{ jest nieparzysta}
\end{cases}
\end{equation}
Narysuj diagram przejść. Dla zaprojektowanej maszyny wykonaj dwa
obliczenia (wykonaj rysunki taśmy i zapisz konfiguracje).
\begin{equation}
M = (Q, \Sigma, \Gamma, \delta, q_0, \nabla, F) = (
\{q_0, q_1, q_2, q_3, q_4, q_5, q_6\},
\{1\},
\{1, x, \nabla\},
\delta,
q_0,
\nabla,
\{q_6\}
)
\end{equation}
\begin{center}
\begin{tikzpicture}
\node[state, initial] (q0) {$q_0$};
\node[state, right of=q0] (q1) {$q_1$};
\node[state, below right of=q1, shift={(.75cm,-1cm)}] (q2) {$q_2$};
\node[state, below of=q1, yshift=-3cm] (q4) {$q_4$};
\node[state, right of=q4, xshift=3cm] (q3) {$q_3$};
\node[state, right of=q1, xshift=3cm] (q5) {$q_5$};
\node[state, right of=q5, accepting] (q6) {$q_6$};
\draw {}
(q0) edge[above] node{ \TRH{ 1 } { B } { R } } (q1)
(q1) edge[above] node{ \TRH{ B } { B } { L } } (q5)
(q1) edge[right] node{ \TRH{ 1 } { x } { R } } (q2)
(q2) edge[right] node{ \TRH{ B } { B } { L } } (q3)
(q2) edge[loop below] node{ \TRH{ 1 } { 1 } { R } } (q2)
(q3) edge[right] node{ \TRH{ x } { 1 } { L } } (q5)
(q3) edge[below] node{ \TRH{ 1 } { B } { L } } (q4)
(q4) edge[left] node{ \TRH{ x } { x } { R } } (q1)
(q4) edge[loop below] node{ \TRH{ 1 } { 1 } { L } } (q4)
(q5) edge[loop above] node{ \TRH{ x } { 1 } { N } } (q5)
(q5) edge[above] node{ \TRH{ B } { 1 } { L } } (q6);
\end{tikzpicture}
\end{center}
\subsubsection{f(3)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$1111 \cont} {B,g1,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$111 \cont} {B,B,g1,1,1,B}
\item \TMT {$K_2$ \; x$q_2$11 \cont} {B,B,x,g1,1,B}
\item \TMT {$K_3$ \; x1$q_2$1 \cont} {B,B,x,1,g1,B}
\item \TMT {$K_4$ \; x11$q_2$B \cont} {B,B,x,1,1,gB}
\item \TMT {$K_5$ \; x1$q_3$1 \cont} {B,B,x,1,g1,B}
\item \TMT {$K_6$ \; x$q_4$1 \cont} {B,B,x,g1,B,B}
\item \TMT {$K_7$ \; $q_4$x1 \cont} {B,B,gx,1,B,B}
\item \TMT {$K_8$ \; x$q_1$1 \cont} {B,B,x,g1,B,B}
\item \TMT {$K_9$ \; xx$q_2$B \cont} {B,B,x,x,gB,B}
\item \TMT {$K_{10}$ \; x$q_3$x \cont} {B,B,x,gx,B,B}
\item \TMT {$K_{11}$ \; $q_5$x1 \cont} {B,B,gx,1,B,B}
\item \TMT {$K_{12}$ \; $q_5$B11 \cont} {B,gB,1,1,B,B}
\item \TMT {$K_{13}$ \; $q_6$B111} {gB,1,1,1,B,B}
\end{enumerate}
\subsubsection{f(6)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$1111111 \cont} {B,g1,1,1,1,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$111111 \cont} {B,B,g1,1,1,1,1,1,B}
\item \TMT {$K_2$ \; x$q_2$11111 \cont} {B,B,x,g1,1,1,1,1,B}
\item \TMT {$K_3$ \; x1$q_2$1111 \cont} {B,B,x,1,g1,1,1,1,B}
\item \TMT {$K_4$ \; x11$q_2$111 \cont} {B,B,x,1,1,g1,1,1,B}
\item \TMT {$K_5$ \; x111$q_2$11 \cont} {B,B,x,1,1,1,g1,1,B}
\item \TMT {$K_6$ \; x1111$q_2$1 \cont} {B,B,x,1,1,1,1,g1,B}
\item \TMT {$K_7$ \; x11111$q_2$B \cont} {B,B,x,1,1,1,1,1,gB}
\item \TMT {$K_8$ \; x1111$q_3$1 \cont} {B,B,x,1,1,1,1,g1,B}
\item \TMT {$K_9$ \; x111$q_4$1 \cont} {B,B,x,1,1,1,g1,B,B}
\item \TMT {$K_{10}$ \; x11$q_4$11 \cont} {B,B,x,1,1,g1,1,B,B}
\item \TMT {$K_{11}$ \; x1$q_4$111 \cont} {B,B,x,1,g1,1,1,B,B}
\item \TMT {$K_{12}$ \; x$q_4$1111 \cont} {B,B,x,g1,1,1,1,B,B}
\item \TMT {$K_{13}$ \; $q_4$x1111 \cont} {B,B,gx,1,1,1,1,B,B}
\item \TMT {$K_{14}$ \; x$q_1$1111 \cont} {B,B,x,g1,1,1,1,B,B}
\item \TMT {$K_{15}$ \; xx$q_2$111 \cont} {B,B,x,x,g1,1,1,B,B}
\item \TMT {$K_{16}$ \; xx1$q_2$11 \cont} {B,B,x,x,1,g1,1,B,B}
\item \TMT {$K_{17}$ \; xx11$q_2$1 \cont} {B,B,x,x,1,1,g1,B,B}
\item \TMT {$K_{18}$ \; xx111$q_2$B \cont} {B,B,x,x,1,1,1,gB,B}
\item \TMT {$K_{19}$ \; xx11$q_3$1 \cont} {B,B,x,x,1,1,g1,B,B}
\item \TMT {$K_{20}$ \; xx1$q_4$1 \cont} {B,B,x,x,1,g1,B,B,B}
\item \TMT {$K_{21}$ \; xx$q_4$11 \cont} {B,B,x,x,g1,1,B,B,B}
\item \TMT {$K_{22}$ \; x$q_4$x11 \cont} {B,B,x,gx,1,1,B,B,B}
\item \TMT {$K_{23}$ \; xx$q_1$11 \cont} {B,B,x,x,g1,1,B,B,B}
\item \TMT {$K_{24}$ \; xxx$q_2$1 \cont} {B,B,x,x,x,g1,B,B,B}
\item \TMT {$K_{25}$ \; xxx1$q_2$B \cont} {B,B,x,x,x,1,gB,B,B}
\item \TMT {$K_{26}$ \; xxx$q_3$1 \cont} {B,B,x,x,x,g1,B,B,B}
\item \TMT {$K_{27}$ \; xx$q_4$x \cont} {B,B,x,x,gx,B,B,B,B}
\item \TMT {$K_{28}$ \; xxx$q_1$B \cont} {B,B,x,x,x,gB,B,B,B}
\item \TMT {$K_{29}$ \; xx$q_5$x \cont} {B,B,x,x,gx,B,B,B,B}
\item \TMT {$K_{30}$ \; x$q_5$x1 \cont} {B,B,x,gx,1,B,B,B,B}
\item \TMT {$K_{31}$ \; $q_5$x11 \cont} {B,B,gx,1,1,B,B,B,B}
\item \TMT {$K_{32}$ \; $q_5$B111 \cont} {B,gB,1,1,1,B,B,B,B}
\item \TMT {$K_{33}$ \; $q_6$B1111} {gB,1,1,1,1,B,B,B,B}
\end{enumerate}
\newpage
\subsection{Zadanie 1.4}
Zaprojektuj maszynę Turinga, które oblicza funkcję f dla liczby
naturalnej n w reprezentacji unarnej, gdzie
\begin{equation}
f(n) = n \text{ mod } 5
\end{equation}
Narysuj diagram przejść. Dla zaprojektowanej maszyny wykonaj
dwa obliczenia (wykonaj rysunki taśmy i zapisz konfiguracje).
\begin{equation}
M = (Q, \Sigma, \Gamma, \delta, q_0, \nabla, F) = (
\{q_0, q_1, q_2, q_3, q_4, q_5, q_6, q_7, q_8, q_9, q_{10}, q_{11}\},
\{1\},
\{1, x, \nabla\},
\delta,
q_0,
\nabla,
\{q_{11}\}
)
\end{equation}
\begin{center}
\begin{tikzpicture}[node distance=3cm]
\node[state, initial] (q0) {$q_0$};
\node[state, below of=q0] (q1) {$q_1$};
\node[state, right of=q1] (q2) {$q_2$};
\node[state, right of=q2] (q3) {$q_3$};
\node[state, right of=q3] (q4) {$q_4$};
\node[state, right of=q4] (q5) {$q_5$};
\node[state, right of=q5] (q6) {$q_6$};
\node[state, below of=q2] (q7) {$q_7$};
\node[state, below of=q6] (q8) {$q_8$};
\node[state, above of=q6,yshift=3cm] (q9) {$q_9$};
\node[state, above of=q3] (q10) {$q_{10}$};
\node[state, accepting, above of=q10] (q11) {$q_{11}$};
\path (q10) edge [out=40,in=80,looseness=8] node[above right] { \TRH{ x } { 1 } { L } } (q10);
\draw {}
(q0) edge[left] node{ \TRH{ 1 } { B } { R } } (q1)
(q1) edge[below] node{ \TRH{ 1 } { x } { R } } (q2)
(q2) edge[below] node{ \TRH{ 1 } { x } { R } } (q3)
(q3) edge[below] node{ \TRH{ 1 } { x } { R } } (q4)
(q4) edge[below] node{ \TRH{ 1 } { x } { R } } (q5)
(q5) edge[below] node{ \TRH{ 1 } { x } { R } } (q6)
(q6) edge[left] node{ \TRH{ B } { B } { L } } (q9)
(q6) edge[left] node{ \TRH{ 1 } { 1 } { L } } (q8)
(q7) edge[loop below] node{ \TRH{ B } { B } { R } } (q7)
(q7) edge[left] node{ \TRH{ 1 } { x } { R } } (q2)
(q8) edge[loop below] node{ \TRH{ x } { B } { L } } (q8)
(q8) edge[above] node{ \TRH{ B } { B } { R } } (q7)
(q9) edge[loop above] node{ \TRH{ x } { B } { L } } (q9)
(q9) edge[above] node{ \TRH{ B } { 1 } { N } } (q11)
(q10) edge[left] node{ \TRH{ B } { 1 } { N } } (q11)
(q1) edge[bend left] node[midarr] { \TRH{ B } { B } { L } } (q10)
(q2) edge[bend left] node[midarr] { \TRH{ B } { B } { L } } (q10)
(q3) edge[left] node[midarr] { \TRH{ B } { B } { L } } (q10)
(q4) edge[bend right] node[midarr] { \TRH{ B } { B } { L } } (q10)
(q5) edge[bend right] node[midarr] { \TRH{ B } { B } { L } } (q10);
\end{tikzpicture}
\end{center}
\subsubsection{f(5)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$111111 \cont} {B,g1,1,1,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$11111 \cont} {B,B,g1,1,1,1,1,B}
\item \TMT {$K_2$ \; x$q_2$1111 \cont} {B,B,x,g1,1,1,1,B}
\item \TMT {$K_3$ \; xx$q_3$111 \cont} {B,B,x,x,g1,1,1,B}
\item \TMT {$K_4$ \; xxx$q_4$11 \cont} {B,B,x,x,x,g1,1,B}
\item \TMT {$K_5$ \; xxxx$q_5$1 \cont} {B,B,x,x,x,x,g1,B}
\item \TMT {$K_6$ \; xxxxx$q_6$B \cont} {B,B,x,x,x,x,x,gB}
\item \TMT {$K_7$ \; xxxx$q_9$x \cont} {B,B,x,x,x,x,gx,B}
\item \TMT {$K_8$ \; xxx$q_9$x \cont} {B,B,x,x,x,gx,B,B}
\item \TMT {$K_9$ \; xx$q_9$x \cont} {B,B,x,x,gx,B,B,B}
\item \TMT {$K_{10}$ \; x$q_9$x \cont} {B,B,x,gx,B,B,B,B}
\item \TMT {$K_{11}$ \; $q_9$x \cont} {B,B,gx,B,B,B,B,B}
\item \TMT {$K_{12}$ \; $q_9$B \cont} {B,gB,B,B,B,B,B,B}
\item \TMT {$K_{13}$ \; $q_{11}$1} {B,g1,B,B,B,B,B,B}
\end{enumerate}
\subsubsection{f(7)}
\begin{enumerate}
\item \TMT {$K_0$ \; $q_0$11111111 \cont} {B,g1,1,1,1,1,1,1,1,B}
\item \TMT {$K_1$ \; $q_1$1111111 \cont} {B,B,g1,1,1,1,1,1,1,B}
\item \TMT {$K_2$ \; x$q_2$111111 \cont} {B,B,x,g1,1,1,1,1,1,B}
\item \TMT {$K_3$ \; xx$q_3$11111 \cont} {B,B,x,x,g1,1,1,1,1,B}
\item \TMT {$K_4$ \; xxx$q_4$1111 \cont} {B,B,x,x,x,g1,1,1,1,B}
\item \TMT {$K_5$ \; xxxx$q_5$111 \cont} {B,B,x,x,x,x,g1,1,1,B}
\item \TMT {$K_6$ \; xxxxx$q_6$11 \cont} {B,B,x,x,x,x,x,g1,1,B}
\item \TMT {$K_7$ \; xxxx$q_8$x11 \cont} {B,B,x,x,x,x,gx,1,1,B}
\item \TMT {$K_8$ \; xxx$q_8$xB11 \cont} {B,B,x,x,x,gx,B,1,1,B}
\item \TMT {$K_9$ \; xx$q_8$xBB11 \cont} {B,B,x,x,gx,B,B,1,1,B}
\item \TMT {$K_{10}$ \; x$q_8$xBBB11 \cont} {B,B,x,gx,B,B,B,1,1,B}
\item \TMT {$K_{11}$ \; $q_8$xBBBB11 \cont} {B,B,gx,B,B,B,B,1,1,B}
\item \TMT {$K_{12}$ \; $q_8$BBBBBB11 \cont} {B,gB,B,B,B,B,B,1,1,B}
\item \TMT {$K_{13}$ \; $q_7$BBBBB11 \cont} {B,B,gB,B,B,B,B,1,1,B}
\item \TMT {$K_{14}$ \; $q_7$BBBB11 \cont} {B,B,B,gB,B,B,B,1,1,B}
\item \TMT {$K_{15}$ \; $q_7$BBB11 \cont} {B,B,B,B,gB,B,B,1,1,B}
\item \TMT {$K_{16}$ \; $q_7$BB11 \cont} {B,B,B,B,B,gB,B,1,1,B}
\item \TMT {$K_{17}$ \; $q_7$B11 \cont} {B,B,B,B,B,B,gB,1,1,B}
\item \TMT {$K_{18}$ \; $q_7$11 \cont} {B,B,B,B,B,B,B,g1,1,B}
\item \TMT {$K_{19}$ \; x$q_2$1 \cont} {B,B,B,B,B,B,B,x,g1,B}
\item \TMT {$K_{20}$ \; xx$q_3$B \cont} {B,B,B,B,B,B,B,x,x,gB}
\item \TMT {$K_{21}$ \; x$q_{10}$x \cont} {B,B,B,B,B,B,B,x,gx,B}
\item \TMT {$K_{22}$ \; $q_{10}$x1 \cont} {B,B,B,B,B,B,B,gx,1,B}
\item \TMT {$K_{23}$ \; $q_{10}$B11 \cont} {B,B,B,B,B,B,gB,1,1,B}
\item \TMT {$K_{24}$ \; $q_{11}$111} {B,B,B,B,B,B,g1,1,1,B}
\end{enumerate}
\newpage
\subsection{Zadanie 1.5}
Zaprojektuj maszynę Turinga, które oblicza funkcję f dla liczby
naturalnej n w reprezentacji unarnej, gdzie
\begin{equation}
f(n) = \floor{\frac{n}{2}}
\end{equation}
Narysuj diagram przejść. Dla zaprojektowanej maszyny wykonaj
dwa obliczenia (wykonaj rysunki taśmy i zapisz konfiguracje).
\begin{equation}
M = (Q, \Sigma, \Gamma, \delta, q_0, \nabla, F) = (
\{q_0, q_1, q_2, q_3, q_4, q_5, q_6\},
\{1\},
\{1, x, \nabla\},
\delta,
q_0,
\nabla,
\{q_6\}
)
\end{equation}
\begin{center}
\begin{tikzpicture}
\node[state, initial] (q0) {$q_0$};
\node[state, right of=q0] (q1) {$q_1$};
\node[state, right of=q1] (q2) {$q_2$};
\node[state, right of=q2] (q3) {$q_3$};
\node[state, above of=q2] (q4) {$q_4$};
\node[state, below of=q2] (q5) {$q_5$};
\node[state, right of=q5, accepting] (q6) {$q_6$};
\draw {}
(q0) edge[above] node{ \TRH{ 1 } { B } { R } } (q1)
(q1) edge[left] node{ \TRH{ B } { B } { L } } (q5)
(q1) edge[above] node{ \TRH{ 1 } { x } { R } } (q2)
(q2) edge[above] node{ \TRH{ B } { B } { L } } (q3)
(q2) edge[loop above] node{ \TRH{ 1 } { 1 } { R } } (q2)
(q3) edge[right] node{ \TRH{ x } { B } { L } } (q5)
(q3) edge[right] node{ \TRH{ 1 } { B } { L } } (q4)
(q4) edge[left] node{ \TRH{ x } { x } { R } } (q1)
(q4) edge[loop above] node{ \TRH{ 1 } { 1 } { L } } (q4)
(q5) edge[loop above] node{ \TRH{ x } { 1 } { L } } (q5)
(q5) edge[above] node{ \TRH{ B } { 1 } { N } } (q6);
\end{tikzpicture}
\end{center}