https://r2.erweima.ai/mathbot/image/1724431179441n940umqa.jpg

The image depicts a finite automaton (FA) with states \( q_0 \), \( q_1 \), \( q_2 \), and \( q_3 \), and transitions based on input symbols '0' and '1'. The task is to determine which of the given strings are accepted by the FA.

The automaton has the following transitions:

- \( q_0 \xrightarrow{0} q_0 \)
- \( q_0 \xrightarrow{1} q_1 \)
- \( q_1 \xrightarrow{0} q_2 \)
- \( q_2 \xrightarrow{1} q_3 \)
- \( q_3 \xrightarrow{0, 1} q_3 \)

The initial state is \( q_0 \), and the accepting state is \( q_3 \).

Let's evaluate each string to see if it reaches the accepting state \( q_3 \).

### (a) 0111
- Start at \( q_0 \)
- Read '0': Stay in \( q_0 \)
- Read '1': Move to \( q_1 \)
- Read '1': No transition from \( q_1 \) on '1', so the string is **not accepted**.

### (b) 0110
- Start at \( q_0 \)
- Read '0': Stay in \( q_0 \)
- Read '1': Move to \( q_1 \)
- Read '1': No transition from \( q_1 \) on '1', so the string is **not accepted**.

### (c) 10011
- Start at \( q_0 \)
- Read '1': Move to \( q_1 \)
- Read '0': Move to \( q_2 \)
- Read '0': No transition from \( q_2 \) on '0', so the string is **not accepted**.

### (d) 0110110
- Start at \( q_0 \)
- Read '0': Stay in \( q_0 \)
- Read '1': Move to \( q_1 \)
- Read '1': No transition from \( q_1 \) on '1', so the string is **not accepted**.

Thus, **none of the strings are accepted** by the finite automaton.

Would you like further details on any part of this explanation?

### Additional Questions:
1. What are the possible modifications to the automaton that would allow any of the strings to be accepted?
2. How can you determine if a string is accepted by a finite automaton in general?
3. Could this automaton be minimized, and if so, how?
4. How can finite automata be used in regular expression matching?
5. How does this automaton compare to a pushdown automaton or a Turing machine?

**Tip:** When designing finite automata, always consider the language you're trying to accept and carefully define the transitions for each input symbol.

Explore the concept of finite automata and determine which strings are accepted by a given automaton. This detailed explanation covers state transitions, accepting states, and evaluates specific strings using a finite automaton model.

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