1. $A = \{ <M> | M $ is a $DFA$ that accepts some string with more 1s than 0s $ \}.$
Decidable.
Logic:
Create a PDA $P$ for β$L(P)$ = More 1βs than 0βsβ language.
For given DFA $M,$ Create a PDA $N$ for $L(P) \cap L(M)$ (We can do this as we know that CFL intersection Regular = CFL)
Check if $L(N)$ is Empty or Not. (Emptiness of CFLs is Decidable)
2. $A = \{ <M> | M $ is a $DFA$ that accepts EVERY(any) string with more 1s than 0s $ \}.$
Decidable.
Logic:
Create a DPDA $D$ for β$L(D)$ = More 1βs than 0βsβ language.
For given DFA $M,$ Create a DPDA $N$ for $L(D) \cap L(M)$ (We can do this as we know that DCFL intersection Regular = DCFL)
Check the Equivalence of $D, N$. (Equivalence of DCFLs is Decidable)
NOTE:
From GATE exam point of view:
In $99.99% $ cases, If the question is about deciding some property of Regular Languages, then it is Decidable.
In $99.99% $ cases, If the question is like βGiven a FA $M$, Does $L(M)$ have some property $P$β, then it is Decidable.
There are some Undecidable problems concerning Regular Languages, BUT those are of Research Level, NOT of basic automata theory level..
So, For GATE exam, you can consider that All properties of Regular Languages are Decidable.
https://cs.stackexchange.com/questions/81542/undecidable-problem-for-regular-languages
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