Es una tupla $(Q,\Sigma,q_0,A,\delta)$
Estrategías diferentes entre AF y ER
Un cobro de una máquina chicles de 5 pesos que sólo acepta monedas de 1, 2 y 5 pesos
Reorganizando
Juntando
Es una tupla $(Q,\Sigma,q_0,A,\delta)$
Con el conjunto $C$, $2^C$ es el conjunto de todos los conjuntos posibles con elementos de $C$
$\delta:Q \times A \rightarrow 2^Q$
$\delta$ regresa un conjunto de estados
Ambos, son una tupla $(Q,\Sigma,q_0,A,\delta)$
AF | AFND |
---|---|
$Q$ | $Q$ |
$\Sigma$ | $\Sigma$ |
$q_0 \in Q$ | $q_0 \in Q$ |
$A \subseteq Q$ | $A \subseteq Q$ |
$\delta: Q \times A \rightarrow Q$ | $\delta: Q \times A \rightarrow 2^Q$ |
De nuevo
Función de transición
Q | 1 | 2 | 5 |
---|---|---|---|
$q_0$ | $\{q_{11/1},q_{11/2},q_{11/3},q_{11/4},q_{11/5}\}$ | $\{q_{12/6},q_{12/7},q_{12/8}\}$ | $\{q_5\}$ |
$q_{11/1}$ | $\{q_{21/1}\}$ | $\emptyset$ | $\emptyset$ |
$q_{21/1}$ | $\{q_{31/1}\}$ | $\emptyset$ | $\emptyset$ |
$q_{31/1}$ | $\{q_{41/1}\}$ | $\emptyset$ | $\emptyset$ |
$q_{41/1}$ | $\{q_5\}$ | $\emptyset$ | $\emptyset$ |
$q_{11/2}$ | $\emptyset$ | $\{q_{11-12/2}\}$ | $\emptyset$ |
$q_{11-12/2}$ | $\emptyset$ | $\{q_5\}$ | $\emptyset$ |
Función de transición (cont.)
Q | 1 | 2 | 5 |
---|---|---|---|
$q_{11/3}$ | $\emptyset$ | $\{q_{11-12/3}\}$ | $\emptyset$ |
$q_{11-12/3}$ | $\{q_{21-12/3}\}$ | $\emptyset$ | $\emptyset$ |
$q_{21-12/3}$ | $\{q_5\}$ | $\emptyset$ | $\emptyset$ |
$q_{11/4}$ | $\{q_{21/4}\}$ | $\emptyset$ | $\emptyset$ |
$q_{21/4}$ | $\emptyset$ | $\{q_{21-12/4}\}$ | $\emptyset$ |
$q_{21-12/4}$ | $\{q_5\}$ | $\emptyset$ | $\emptyset$ |
$q_{11/5}$ | $\{q_{21/5}\}$ | $\emptyset$ | $\emptyset$ |
$q_{21/5}$ | $\{q_{31/5}\}$ | $\emptyset$ | $\emptyset$ |
$q_{31/5}$ | $\emptyset$ | $\{q_5\}$ | $\emptyset$ |
Función de transición (cont. 2)
Q | 1 | 2 | 5 |
---|---|---|---|
$q_{12/6}$ | $\emptyset$ | $\{q_{22/6}\}$ | $\emptyset$ |
$q_{22/6}$ | $\{q_5\}$ | $\emptyset$ | $\emptyset$ |
$q_{12/7}$ | $\{q_{11-12/7}\}$ | $\emptyset$ | $\emptyset$ |
$q_{11-12/7}$ | $\emptyset$ | $\{q_5\}$ | $\emptyset$ |
$q_{12/8}$ | $\{q_{11-12/8}\}$ | $\emptyset$ | $\emptyset$ |
$q_{11-12/8}$ | $\{q_{21-12/8}\}$ | $\emptyset$ | $\emptyset$ |
$q_{21-12/8}$ | $\{q_5\}$ | $\emptyset$ | $\emptyset$ |
$\delta^*=\begin{cases} \delta^*(q,\epsilon)=\{q\} & q \in Q \\ \delta^*(q,wa)= \bigcup\limits_{r \in \delta^*(q,w)}\delta(r,a) & q,r \subseteq Q, w \subseteq \Sigma^*, a \subseteq \Sigma \\ \end{cases}$
Acepta el siguiente pago: $122$
Para el AFND $M=(Q,\Sigma,q_0,A,\delta)$
La cadena $w\in \Sigma^*$ se acepta si:
$L(M)$ es el lenguaje conformado por cadenas aceptadas por $M$
La cadena se acepta
Comenzar por codificar los estados de forma binaria
$|q_0,q_{11/1},q_{11/2},q_{11/3},q_{11/4},q_{11/5},q_{12/6},q_{12/7},q_{12/8},$ $q_{21/1},q_{11-12/2},q_{11-12/3},q_{21/4},q_{21/5},q_{22/6},q_{11-21/7},q_{11-21/8},$ $q_{31/1},q_{21-12/3},q_{21-12/4},q_{31/5},q_{21-12/8},$ $q_{41/1},q_{5}|=24$
0-00000000-00000000-00000-0-0
$2^Q$ | 1 | 2 | 5 |
---|---|---|---|
$1:00000000$ $00000000$ $00000:0:0$ | $0:11111000:$ $00000000$ $00000:0:0$* | $0:00000111$ $00000000$ $00000:0:0$* | $0:00000000$ $00000000$ $00000:0:1$* |
$2^Q$ | 1 | 2 | 5 |
---|---|---|---|
$1:00000000$ $00000000$ $00000:0:0$ | $0:11111000:$ $00000000$ $00000:0:0$ | $0:00000111$ $00000000$ $00000:0:0$ | $0:00000000$ $00000000$ $00000:0:1$ |
$0:11111000$ $00000000$ $00000:0:0$ | $0:00000000$ $10011000$ $00000:0:0$* | $0:00000000$ $01100000$ $00000:0:0$* | $\emptyset$ |
$0:00000111$ $00000000$ $00000:0:0$ | $0:00000000$ $00000011$ $00000:0:0$* | $0:00000000$ $00000100$ $00000:0:0$* | $\emptyset$ |
$0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ | $\emptyset$ | $\emptyset$ |
$2^Q$ | 1 | 2 | 5 |
---|---|---|---|
$0:00000000$ $10011000$ $00000:0:0$ | $0:00000000$ $00000000$ $10010:0:0$* | $0:00000000$ $00000000$ $00100:0:0$* | $\emptyset$ |
$0:00000000$ $01100000$ $00000:0:0$ | $0:00000000$ $00000000$ $01000:0:0$* | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ |
$0:00000000$ $00000011$ $00000:0:0$ | $0:00000111$ $00000000$ $00001:0:0$* | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ |
$0:00000000$ $00000100$ $00000:0:0$ | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ | $\emptyset$ |
$2^Q$ | 1 | 2 | 5 |
---|---|---|---|
$0:00000000$ $00000000$ $10010:0:0$ | $0:00000000$ $00000000$ $00000:1:0$* | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ |
$0:00000000$ $00000000$ $00100:0:0$ | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ | $\emptyset$ |
$0:00000000$ $00000000$ $01000:0:0$ | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ | $\emptyset$ |
$0:00000111$ $00000000$ $00001:0:0$ | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ | $\emptyset$ |
$2^Q$ | 1 | 2 | 5 |
---|---|---|---|
$0:00000000$ $00000000$ $00000:1:0$ | $0:00000000$ $00000000$ $00000:0:1$ | $\emptyset$ | $\emptyset$ |
Renombrando los estados
$Q'$ | 1 | 2 | 5 |
---|---|---|---|
$q_0$ | $q_1$ | $q_2$ | $q_3$ |
$q_1$ | $q_4$ | $q_5$ | $q_e$ |
$q_2$ | $q_6$ | $q_7$ | $q_e$ |
$q_3$ | $q_e$ | $q_e$ | $q_e$ |
$q_4$ | $q_8$ | $q_9$ | $q_e$ |
$q_5$ | $q_{10}$ | $q_3$ | $q_e$ |
$q_6$ | $q_{11}$ | $q_3$ | $q_e$ |
$q_7$ | $q_3$ | $q_e$ | $q_e$ |
$q_8$ | $q_{12}$ | $q_3$ | $q_e$ |
$q_9$ | $q_3$ | $q_e$ | $q_e$ |
$q_{10}$ | $q_3$ | $q_e$ | $q_e$ |
$q_{11}$ | $q_3$ | $q_e$ | $q_e$ |
$q_{12}$ | $q_3$ | $q_e$ | $q_e$ |
Todos los cobros de una máquina chicles de 5 pesos que sólo acepta monedas de 1, 2 y 5 pesos
Es una tupla $(Q,\Sigma,q_0,A,\delta)$
Ambos, son una tupla $(Q,\Sigma,q_0,A,\delta)$
AF | AFND | AFND-$\epsilon$ |
---|---|---|
$Q$ | $Q$ | $Q$ |
$\Sigma$ | $\Sigma$ | $\Sigma$ |
$q_0 \in Q$ | $q_0 \in Q$ | $q_0 \in Q$ |
$A \subseteq Q$ | $A \subseteq Q$ | $A \subseteq Q$ |
$\delta: Q \times \Sigma \rightarrow Q$ | $\delta: Q \times \Sigma \rightarrow 2^Q$ | $\delta: Q \times (\Sigma \cup \{\epsilon\}) \rightarrow 2^Q$ |
$\delta^*=\begin{cases} \delta^*(q,\epsilon)= exp_{\epsilon}(\{q\}) & q \in Q \\ \delta^*(q,wa)= exp_{\epsilon} (\bigcup\limits_{r \in \delta^*(q,w)}\delta(r,a)) & q,r \subseteq Q, w \subseteq \Sigma^*, a \subseteq \Sigma \\ \end{cases}$
Acepta los pagos: $5212$
Pero $\{q_0,q_3\} \cap \{ q_3 \} \not = \emptyset$, se aceptan los pagos
$Q'$ | $\epsilon$ | 1 | 2 | 5 | $\delta^*(q,1)$ | $\delta^*(q,2)$ | $\delta^*(q,5)$ | |
---|---|---|---|---|---|---|---|---|
$q_0$ | $\{q_e\}$ | $\{q_1\}$ | $\{q_2\}$ | $\{q_3\}$ | | | $\{q_1\}$ | $\{q_2\}$ | $\{q_3,q_0\}$ |
$q_1$ | $\{q_e\}$ | $\{q_4\}$ | $\{q_5\}$ | $\{q_e\}$ | | | $\{q_4\}$ | $\{q_5\}$ | $\{q_e\}$ |
$\{q_2\}$ | $\{q_e\}$ | $\{q_6\}$ | $\{q_7\}$ | $\{q_e\}$ | | | $\{q_6\}$ | $\{q_7\}$ | $\{q_e\}$ |
$\{q_3\}$ | $\{q_0\}$ | $\{q_e\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q_e\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\{q_4\}$ | $\{q_e\}$ | $\{q_8\}$ | $\{q_9\}$ | $\{q_e\}$ | | | $\{q_8\}$ | $\{q_9\}$ | $\{q_e\}$ |
$\{q_5\}$ | $\{q_e\}$ | $\{q_{10}\}$ | $\{q_3\}$ | $\{q_e\}$ | | | $\{q_{10}\}$ | $\{q_0,q_3\}$ | $\{q_e\}$ |
$\{q_6\}$ | $\{q_e\}$ | $\{q_{11}\}$ | $\{q_0,q_3\}$ | $\{q_e\}$ | | | $\{q_{11}\}$ | $\{q_0,q_3\}$ | $\{q_e\}$ |
$\{q_7\}$ | $\{q_e\}$ | $\{q_3\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q0,q_3\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\{q_8\}$ | $\{q_e\}$ | $\{q_{12}\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q_{12}\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\{q_9\}$ | $\{q_e\}$ | $\{q_3\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q_0,q_3\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\{q_{10}\}$ | $\{q_e\}$ | $\{q_3\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q_0,q_3\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\{q_{11}\}$ | $\{q_e\}$ | $\{q_3\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q_0,q_3\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\{q_{12}\}$ | $\{q_e\}$ | $\{q_3\}$ | $\{q_e\}$ | $\{q_e\}$ | | | $\{q_0,q_3\}$ | $\{q_e\}$ | $\{q_e\}$ |
$\delta'(q,\epsilon)=\emptyset$
$\delta'(q,a)=\{\delta(q,a)\}$
$Q'$ | a | b | $\epsilon$ | $a$ | $b$ | |
---|---|---|---|---|---|---|
$q_0$ | $q_0$ | $q_1$ | | | $\emptyset$ | $\{q_0\}$ | $\{q_1\}$ |
$q_1$ | $q_1$ | $q_0$ | | | $\emptyset$ | $\{q_1\}$ | $\{q_0\}$ |
Con los siguientes lenguajes:
$L_1:10^*$ y $L_2:1^*0$
$10^*+1^*0$
$10^*1^*0$
$(10^*)^*$
Tenemos unión, concatenación y cerradura ¡podemos hacer cualquier expresión regular!
$(a^*ba^*ba^*)^*+a^*$
$(a^*ba^*ba^*)^*+a^*$
Marcar mismos estados y no alcanzables
S | $q_0$ | $q_1$ | $q_2$ | $q_3$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✖ | ||||||
$q_1$ | ✕ | ✖ | ||||||
$q_2$ | ✕ | ✖ | ||||||
$q_3$ | ✖ | ✖ | ✖ | ✖ | ✖ | ✖ | ✖ | ✖ |
$q_4$ | ✖ | ✕ | ||||||
$q_5$ | ✖ | ✕ | ||||||
$q_6$ | ✖ | ✕ | ||||||
$q_7$ | ✖ | ✕ |
Marcar diferenciables: iniciales vs finales
S | $q_0$ | $q_1$ | $q_2$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✔ | |||||
$q_1$ | ✕ | ✔ | |||||
$q_2$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ |
$q_4$ | ✔ | ✕ | |||||
$q_5$ | ✔ | ✕ | |||||
$q_6$ | ✔ | ✕ | |||||
$q_7$ | ✔ | ✕ |
Marcar pares $q_i$ y $q_j$ si $\delta(q_i,a)$ y $\delta(q_j,a)$ están marcados: $q_0$ y $q_1$ a través de $1$
S | $q_0$ | $q_1$ | $q_2$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✔ | ✔ | ||||
$q_1$ | ✔ | ✕ | ✔ | ||||
$q_2$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ |
$q_4$ | ✔ | ✕ | |||||
$q_5$ | ✔ | ✕ | |||||
$q_6$ | ✔ | ✕ | |||||
$q_7$ | ✔ | ✕ |
Buscar un par que nos lleve a difenciables ya identificados
Otro par, a través de $1$
$q_7$ y $q_4$
S | $q_0$ | $q_1$ | $q_2$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✔ | ✔ | ||||
$q_1$ | ✔ | ✕ | ✔ | ||||
$q_2$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ |
$q_4$ | ✔ | ✕ | ✔ | ||||
$q_5$ | ✔ | ✕ | |||||
$q_6$ | ✔ | ✕ | |||||
$q_7$ | ✔ | ✔ | ✕ |
Otro par, a través de $1$
$q_4$ y $q_5$
S | $q_0$ | $q_1$ | $q_2$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✔ | ✔ | ||||
$q_1$ | ✔ | ✕ | ✔ | ||||
$q_2$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ |
$q_4$ | ✔ | ✕ | ✔ | ✔ | |||
$q_5$ | ✔ | ✔ | ✕ | ||||
$q_6$ | ✔ | ✕ | |||||
$q_7$ | ✔ | ✔ | ✕ |
Otro par, a través de $0$
$q_5$ y $q_6$
S | $q_0$ | $q_1$ | $q_2$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✔ | ✔ | ||||
$q_1$ | ✔ | ✕ | ✔ | ||||
$q_2$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ |
$q_4$ | ✔ | ✕ | ✔ | ✔ | |||
$q_5$ | ✔ | ✔ | ✕ | ✔ | |||
$q_6$ | ✔ | ✔ | ✕ | ||||
$q_7$ | ✔ | ✔ | ✕ |
Otro par, a través de $0$
Otro par, a través de $1$
Otro par, a través de $1$
Otro par, a través de $1$
Otro par, a través de $1$
Otro par, a través de $1$
Otro par, a través de $1$
Otro par, a través de $1$
Otro par, a través de $1$
Finalmente
S | $q_0$ | $q_1$ | $q_2$ | $q_4$ | $q_5$ | $q_6$ | $q_7$ |
---|---|---|---|---|---|---|---|
$q_0$ | ✕ | ✔ | ✔ | ✔ | ✔ | ✔ | |
$q_1$ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ | |
$q_2$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | ✔ |
$q_4$ | ✔ | ✔ | ✕ | ✔ | ✔ | ✔ | |
$q_5$ | ✔ | ✔ | ✔ | ✔ | ✕ | ✔ | ✔ |
$q_6$ | ✔ | ✔ | ✔ | ✔ | ✔ | ✕ | ✔ |
$q_7$ | ✔ | ✔ | ✔ | ✔ | ✔ | ✕ |