Are DFAs with a unary alphabet strictly less powerful than DFAs with a binary alphabet?

Question

Are DFAs with a unary alphabet strictly less powerful than DFAs with a binary alphabet? Is this even a meaningful question?

For example, if $\Sigma = \{\texttt{0}, \texttt{1}\}$, we can encode any larger alphabet using $\Sigma$, but if $\Sigma = \{\texttt{0}\}$, this can define a DFA (that say, recognizes $L = \{ \texttt{0}^k \mid k > 0\}$)... but such a DFA would never be able to recognize more "complex" regular expressions. For example, there is no way to encode $\texttt{0011}$ using a unary alphabet that a DFA would recognize (we could use, say, Godel numbering, but that would require a more powerful machine that could "count").

If DFAs with a unary alphabet less powerful than DFAs with a binary alphabet, is there a name for this language/grammar? I recognize this is kind of an odd question, since the DFA that recognizes $L = \{ \texttt{0}^k \mid k > 0\}$ recognizes all unary languages... but technically there still are a countably infinite number of DFAs in this class ($L = \{ \texttt{0}^1 \}$, $L=\{\texttt{0}^2\}$, etc.)

Note I am of course assuming that for $\Sigma = \{ \texttt{0} \}$, that it does not contain the empty symbol $\varepsilon$.

Answer 1

Take the set $A=\{2^k \mid k \ge 1\}$. The language $$ L=\{ \text{bin}(a) \mid a \in A\} = \{10^k\mid k \ge 0 \}$$ is clearly a regular language, whereas $$L=\{ 1^a \mid a \in A\} $$ is not regular. Proving the latter is a standard textbook application of the regular pumping lemma.

Answer 2

It appears as if they are indeed less powerful.

Consider the language $L = \{1 b (0|1)^* b \mid b \in \{0,1\}\}$ for a binary alphabet.

The language can be recognized by a five-state DFA.

It shouldn't be too hard to show that $L' = \{0^i \mid i \text{ is the unary encoding of some string in } L\}$ cannot be recognized by a DFA, as the states reached for some $0^i \neq 0^j$ cannot be merged for $i \neq j$. The basic idea is that a DFA for $L'$ has to keep track of how many digits the word encoded by $i$ and $j$ has already to not forget the second digit. But then merging states for $i$ and $j$ would lose this information.

This is not a formal proof, but it should be a suitable starting point for a formal proof.