Define NFA. Draw NFA for the language of all strings that
conttain substring “1010”
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A Non-deterministic Finite Automaton, or NFA, is a computational model that extends the concept of finite automata. Unlike deterministic finite automata, an NFA can have multiple possible transitions from a single state on the same input symbol. It can also include epsilon transitions, which allow state changes without consuming input. An NFA accepts a string if there exists at least one computational path from the start state to an accept state.
Now let's design an NFA for the specific problem of recognizing all strings that contain the substring "1010". Our strategy is to create states that track our progress in matching this substring. We need five states: q0 for the start where we haven't matched anything, q1 for matching "1", q2 for matching "10", q3 for matching "101", and q4 as our accept state when we've found the complete substring "1010".
Here is the complete NFA diagram for recognizing strings containing the substring "1010". From the start state q0, we transition to q1 on input '1' and stay in q0 on input '0'. From q1, we move to q2 on '0' or stay in q1 on '1'. From q2, we advance to q3 on '1' or return to q0 on '0'. From q3, we reach the accept state q4 on '0', completing "1010", or go back to q1 on '1'. Once in q4, we stay there on any input since the substring has been found.
Let's trace through an example to see how our NFA works. Consider the input string "11010". We start at q0. On the first '1', we move to q1. On the second '1', we stay in q1. On '0', we move to q2. On '1', we advance to q3. Finally, on '0', we reach the accept state q4, having found the substring "1010". The string is accepted.
In summary, we have successfully designed an NFA that recognizes all strings containing the substring "1010". The key properties of NFAs include non-deterministic transitions and the ability to accept a string if any computational path leads to an accept state. Our NFA correctly handles overlapping patterns and accepts strings like "1010", "11010", and "010101010" while rejecting strings that don't contain the required substring. This demonstrates the power and flexibility of non-deterministic finite automata in pattern recognition.