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I have an easy question about the DPDA. Could you read an input and push a different symbol to the stack.

An example would be A transition from q1 to q2 where read input is v pop is epsilon(empty string) and push is x Both v and x are symbols

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If you mean, you can push a symbol different from what your machine has read from the input, then yes you can. The input and stack symbols are not related and there is no rule that requires them to be the same (see the formal definition here). Most of the time you would see examples where the same symbol is read and pushed just for convenience. You can even have the set of input and stack symbols to be disjoint.

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