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According to my book the definition of a deterministic Pushdown automaton allows for $\delta(q,\epsilon,Z)$ to be non-empty if $$\forall\sigma\in\Sigma:\,\delta(q,\sigma,Z)\neq\emptyset$$

Can someone please explain/give motivation for this definition ?

Mainly, why is the automaton deterministic if we allow $\epsilon$ movements ? why do we condition it on the (strange) condition that $\forall\sigma\in\Sigma:\,\delta(q,\sigma,Z)\neq\emptyset$ ?

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up vote 3 down vote accepted

That looks like a typo: the usual definition is that if $\delta(q,\epsilon,Z)\ne\varnothing$, then $\delta(q,\sigma,Z)$ must be empty for each $\sigma\in\Sigma$, so that there’s no ambiguity.

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