If a language is regular, then must it also be a DPDA?

Question

$$\{a^nb^m:n \leq m \leq 2n\}$$

The pumping lemma says that this language is not regular, so does that mean that it can't be a DPDA?

what's the rule to determine if a language is DPDA or not?

Answer 1

A language is regular if and only if it can be recognized by a deterministic finite automaton, thus if your language is not regular, there can be no DFA.

But as you said nothing about contextfreeness, there surely can be a PDA or DPDA that recognizes this language.

The regular languages are precisely a subclass of the contextfree ones, these are the ones recognizable by a Nondeterministic Pushdown Automaton.

The contextfree ones have a (real) subclass called "deterministic contextfree", these are precisely the ones recognizable by a deterministic pushdown automaton.

And the regular languages are a real subclass of the deterministic contextfree languages, thus

$$REG \subsetneq DPDA \subsetneq PDA$$ holds.

As you have proven with the pumping lemma, your language is not in $REG$ (which i didn't check btw.), but it can be in $DPDA$. Over all there can be a DPDA for your language, but no DFA.