Practice Questions DFA
Duration: 2 min
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The video lecture focuses on determining the minimum number of states required for Finite Automata to recognize specific languages based on counting properties. The instructor presents a problem involving a binary alphabet $\Sigma = \{0, 1\}$ and asks for the number of states needed to accept words with an odd number of 1s. He then transitions to a second problem concerning the length of strings divisible by a specific number, illustrating the general rule for constructing minimal automata for modular arithmetic conditions.
Chapters
0:00 – 1:42 00:00-01:42
The instructor starts by displaying a multiple-choice question on the screen: 'Let $\Sigma = \{0, 1\}$, then an automaton A accepting only those words from $\Sigma$ having an odd number of 1's requires _______ states including the start state'. The options provided are 2, 3, 4, and 5. He analyzes the condition mathematically, writing $|w|_1 = 1 \pmod 2$ on the whiteboard to represent the requirement that the count of 1s must be odd. He explains that modulo 2 arithmetic implies two distinct equivalence classes: even and odd. Consequently, he deduces that the automaton requires exactly 2 states—one to track the even count (start state) and one to track the odd count (accepting state). He underlines option (a) 2. Next, he introduces a second question: 'The smallest finite automaton which accepts the language $\{x \mid ext{length of } x ext{ is divisible by } 500\}$ has _______ states'. The options are 200, 300, 400, and 500. He writes the condition $|w| = 0 \pmod{500}$ to formalize the divisibility rule. He explains the general principle that for a language requiring the length of the string to be divisible by $n$, the minimal automaton requires $n$ states to count the length modulo $n$. He emphasizes that the number of states equals the modulus. Therefore, for divisibility by 500, he concludes that 500 states are necessary. He underlines option (d) 500 as the correct answer.
The lesson demonstrates a fundamental property of Finite Automata: the number of states required corresponds to the number of distinct remainders needed to track a specific property. For counting parity (odd/even), 2 states suffice. For counting length modulo $n$, $n$ states are required. This establishes a direct relationship between the modulus of the condition and the state complexity of the automaton.