DFA Designing string start and end with same symbol
Duration: 3 min
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The video lecture demonstrates the step-by-step design of a minimal Deterministic Finite Automaton (DFA) that accepts strings over the alphabet {a, b} where every accepted string starts and ends with the same symbol. The instructor begins by writing the problem statement and underlining the critical condition. He constructs the state diagram by defining states based on the first symbol read and the last symbol read, ensuring transitions lead to accepting states only when the start and end symbols match.
Chapters
0:00 – 2:00 00:00-02:00
The instructor writes the problem statement on the whiteboard: 'Design a minimal DFA that accepts all strings over the alphabet Σ = {a, b} such that every accepted string start and end with same symbol.' He underlines the phrase 'start and end with same symbol' to emphasize the constraint. He begins drawing the DFA, starting with an initial state q0. He draws transitions for input 'a' leading to state q1 and input 'b' leading to state q2. He marks both q1 and q2 as accepting states (double circles) because single characters 'a' and 'b' satisfy the condition. He adds self-loops on q1 for input 'a' and on q2 for input 'b' to handle strings like 'aa' or 'bb'.
2:00 – 2:46 02:00-02:46
The instructor completes the transition logic for the DFA. He draws a transition from q1 on input 'b' to a new state q3, and from q2 on input 'a' to a new state q4. He marks q3 and q4 as non-accepting states. He explains that if the machine is in q3 (started with 'a', last seen 'b'), reading 'a' should return it to the accepting state q1. Similarly, from q4 (started with 'b', last seen 'a'), reading 'b' returns to accepting state q2. He adds self-loops on q3 for 'b' and q4 for 'a' to maintain the 'waiting' state. He writes 'qab a' as an example trace to verify the logic, concluding that q3 and q4 are distinct states necessary for the minimal design.
The lesson progresses from identifying the problem constraints to constructing a state machine that tracks the first symbol read. By creating separate paths for strings starting with 'a' and 'b', and using intermediate states to track the last symbol, the instructor ensures that acceptance only occurs when the start and end symbols match. The final diagram illustrates a 5-state DFA (including the start state) that correctly accepts strings like 'a', 'aa', 'aba', 'b', 'bb', 'bab' while rejecting strings like 'ab', 'ba', 'abb', 'baa'.