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I have this confusion if there are two states of a machine $p$ and $q$. Let $x$ be an input string such that length of $x = k$, $g$ be the output function and let $g(p,x)$ and $g(q,x)$ be the output when the input $x$ is applied at state $p$ and $q$.

How can I prove that

$p \overset{k}\equiv{} q \iff \forall x, |x|=k, g(p,x)=g(q,x)$

i.e state p is k equivalent to state q iff and only if the given condition above

My definition of k equivalence is $p \overset{k}\equiv{} q \iff \forall x,|x|\le k, g(p,x)=g(q,x)$

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    What is your definition of $\overset{k}\equiv{}$? Without more information it looks like what you've quoted _is_ the definition, and therefore is not subject to proof.2012-09-26
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    @user34790 Your definition doesn't make sense2012-09-26
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    @Henning. I have given the definition2012-09-26

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