For a string xx ∈ {aa, bb}∗ with |xx| = nn, how many states are required for an FA accepting xx and no other strings? For each of these states, describe the strings that cause the... For a string xx ∈ {aa, bb}∗ with |xx| = nn, how many states are required for an FA accepting xx and no other strings? For each of these states, describe the strings that cause the FA to be in that state. ii. For a string xx ∈ {aa, bb}∗ with |xx| = nn, how many states are required for an FA accepting the language of all strings in {aa, bb}∗ that begin with xx? For each of these states, describe the strings that cause the FA to be in that state.
Understand the Problem
The question is asking about finite automata (FA) and how many states are needed for two specific scenarios involving strings composed of 'aa' and 'bb'. The first scenario involves an FA that accepts exactly one specific string 'xx' of length 'nn', while the second scenario involves an FA that accepts any string that starts with 'xx'. For both scenarios, we need to determine the states needed and the conditions under which the FA transitions into those states.
Answer
n+1 states for both scenarios, where n is the length of string xx.
For part (i), the finite automaton accepting only string xx requires n+1 states, where n is the length of the string xx, and each state represents reading an additional character of xx. For part (ii), the FA accepting all strings starting with xx also requires n+1 states; each state represents partially matching the prefix xx.
Answer for screen readers
For part (i), the finite automaton accepting only string xx requires n+1 states, where n is the length of the string xx, and each state represents reading an additional character of xx. For part (ii), the FA accepting all strings starting with xx also requires n+1 states; each state represents partially matching the prefix xx.
More Information
In both scenarios, the DFA requires additional states to transition through each part of the pattern being matched. Each transition effectively enforces the deterministic nature of transitioning for each symbol in the string or pattern prefix.
Tips
A common mistake is forgetting to count the start state when calculating the total number of states.
Sources
- Number of states of a finite automaton recognizing all words - math.stackexchange.com
- Construct a DFA that Start With aa or bb - GeeksforGeeks - geeksforgeeks.org
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