Let $\Sigma=\{1,2,3,4\}$ For $x \in \Sigma^*$, let prod $(x)$ be the product of symbols in $x$ modulo 7 . We take $\operatorname{prod}(\varepsilon)=1$, where $\varepsilon$ is the null string.
For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$.
Define $L=\left\{x \in \Sigma^{\star} \mid \operatorname{prod}(x)=2\right\}$.
The number of states in a minimum state DFA for $L$ is _________ (Answer in integer)
Consider the following two languages over the alphabet $\{a, b\}$ :
$$\begin{aligned} & L_1=\left\{\alpha \beta \alpha \mid \alpha \in\{a, b\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \\ & L_2=\left\{\alpha \beta \alpha \mid \alpha \in\{a\}^{+} \text {AND } \beta \in\{a, b\}^{+}\right\} \end{aligned}$$
Which ONE of the following statements is CORRECT?
Consider the following deterministic finite automaton (DFA) defined over the alphabet, $\Sigma=\{a, b\}$. Identify which of the following language(s) is/are accepted by the given DFA.
Consider a finite state machine (FSM) with one input $X$ and one output $f$, represented by the given state transition table. The minimum number of states required to realize this FSM is ________ . (Answer in integer)
| Present state | Next state | Output $f$ | ||
|---|---|---|---|---|
| $X = 0$ | $X = 1$ | $X = 0$ | $X = 1$ | |
| A | F | B | 0 | 0 |
| B | D | C | 0 | 0 |
| C | F | E | 0 | 0 |
| D | G | A | 1 | 0 |
| E | D | C | 0 | 0 |
| F | F | B | 1 | 1 |
| G | G | H | 0 | 1 |
| H | G | A | 1 | 0 |
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