GATE CS 2014 Set 2 — Question 46
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Theory of Computation → Finite Automata & Regular Languages → DFA
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Question
Let . Let . Which one of the following is TRUE?
Correct answer
(A) L₁ is regular but not L₂
Solution
1.For : In any binary string, the number of occurrences of the pattern '110' and '011' can differ by at most 1. This is because '110' represents a transition from a block of 1s (length ) to a 0, and '011' represents a transition from a 0 to a block of 1s (length ). Since we can only alternate between these transitions, the counts are always balanced within a difference of 1. This property can be tracked with a finite state machine, making regular.
2.For : requires comparing the total count of '000' substrings with '111' substrings. Since these counts can be arbitrarily large and are not structurally constrained to be nearly equal (unlike ), a machine would need infinite memory (a stack or more) to keep track of the counts. Thus, is not regular.
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