Practice Question - 2
Duration: 4 min
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This educational video features an instructor from Knowledge Gate solving a problem in automata theory. The task is to identify the language accepted by a Pushdown Automaton (PDA) given its transition mapping. The instructor systematically converts the mathematical notation of the transition function into a visual state diagram. He then analyzes the paths within this diagram to determine the structural properties of the accepted strings, such as their ending character and the parity of specific symbols. By testing specific examples against the diagram, he eliminates incorrect options and identifies the correct language definition.
Chapters
0:00 – 2:00 00:00-02:00
The instructor begins by presenting the problem on a slide titled 'Q consider the following mapping and find the correct language?'. The slide lists seven transition rules for a PDA, such as $\delta(q_0, a, z_0) = (q_1, z_0)$ and $\delta(q_1, \epsilon, z_0) = (q_f, z_0)$. He starts drawing the state diagram on the right side of the screen. He draws circles for states $q_0, q_1, q_2$ and adds arrows representing the transitions. He draws an arrow from $q_0$ to $q_1$ labeled 'a', and from $q_0$ to $q_2$ labeled 'b'. He continues by adding a self-loop on $q_1$ for 'a' and an arrow from $q_1$ to $q_2$ for 'b'. He also draws a self-loop on $q_2$ for 'a' and an arrow from $q_2$ back to $q_0$ for 'b'. Finally, he draws an epsilon transition from $q_1$ to a final state $q_f$. The multiple-choice options are visible at the bottom: (a) ending with a, (b) ending with a, contain even number of a, (c) ending with a, contain odd number of a, (d) ending with a, contain even number of b.
2:00 – 4:19 02:00-04:19
The instructor analyzes the diagram to find the correct option. He notes that the only way to reach the final state $q_f$ is via an epsilon transition from $q_1$. This implies the string must end with a transition that lands in $q_1$. Since transitions to $q_1$ occur on input 'a' (from $q_0$ or $q_1$), the string must end with 'a'. He then tests the string 'bba'. Tracing the path: $q_0 \xrightarrow{b} q_2 \xrightarrow{b} q_0 \xrightarrow{a} q_1 \xrightarrow{\epsilon} q_f$. This string is accepted. It has one 'a' (odd) and two 'b's (even). This eliminates option (b) which requires an even number of 'a's. He tests 'aa': $q_0 \xrightarrow{a} q_1 \xrightarrow{a} q_1 \xrightarrow{\epsilon} q_f$. This is accepted and has two 'a's (even). This eliminates option (c) which requires an odd number of 'a's. He observes that to return to $q_0$ from $q_2$, a 'b' is needed. Since $q_2$ is entered by 'b' from $q_0$, 'b's must come in pairs to return to the start state. Thus, the number of 'b's must be even. This confirms option (d) 'ending with a, contain even number of b' is the correct answer.
The lecture demonstrates a methodical approach to solving PDA problems by converting transition mappings into state diagrams and testing sample strings to deduce the language properties. The key takeaway is understanding how state transitions constrain the input string structure, specifically how 'b' transitions must occur in pairs to return to the start state, while 'a' transitions allow entry into the accepting state.