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

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 $$\varepsilon$$.