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How can we show that the language $K = \{<M>: M \text{ is a finite automata and L(M) contains a word of form } a^ib^j\}$ is decidable?

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Let $\hat M$ be the DFA for the language $L(a^*b^*)$ (aka, words of the form $a^ib^j$).

Construct the intersection DFA between $M$ and $\hat M$, and call it $M'$. Now, check if $L(M')=\emptyset$. If it does, reject. Otherwise, accept.

Here is useful material about intersection DFA and about checking the empty-ness of the DFA language

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