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How can I design a Turing machine that takes string $a^n b a^m$ and check if $n>m$ then convert the string to $a^{n-m}$, otherwise reject it?

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Start with placing tape over $b$, then go to the rightmost $a$, put blank, go back to $b$ and delete the rightmost $a$.
Every time you scratch out $a$s on both sides, $n - m$ does not change until one side is blank.
Check if the left side is not empty, delete $b$ (and if it is required, go back to the beginning of the tape) and accept, otherwise reject.

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  • $\begingroup$ Hi @Evil thanks a lot for your response . it's very helpful for me. $\endgroup$ – yaodao vang Oct 7 '18 at 15:47

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