GATE CS 2018 Set 1 — Question 62
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Theory of Computation → Finite Automata & Regular Languages → Closure Properties (Regular)
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Question
Given a language , define as follows:The order of a language is defined as the smallest such that .
Consider the language (over alphabet 0) accepted by the following automaton.
The order of is _____.
Consider the language (over alphabet 0) accepted by the following automaton.
Correct answer
2 to 2
Solution
The language accepted by the automaton consists of strings that lead to the accepting state (the middle state). Assuming the start state (leftmost) is also accepting (which is required for the order to be finite, as otherwise and would have disjoint sets of string lengths), the language is:Let's compute the powers of :
1.
2..
Possible sums of lengths ():- ()
- ()
- ()
- ()
- ()
- ()
- ()
- And so on. We can form any integer length .
3.. Since , . Since , . Thus .
Comparing the powers:- (e.g., but ).
- .
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