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Say there is a push-down automata, in this example I'll use a deadfish-like set:

+: increase x by 1

0: set x to 0

ln: set x to ln(x) <-- real valued result

With x being an infinite precision real-valued variable, does this allow said machine to have more power than if it was operating on integers? Or am I misunderstanding something?

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Since none of your instructions can influence the behavior of the pushdown automaton (they can only influence the value of $x$), this does not change which languages the pushdown automaton can accept.

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