Does O(1) communication complexity imply that a language is regular?

Question

Let's say that we have a function $g(i,j)$, which is an arbitrary boolean-valued function over $i,j \in \{a,b\}^*$, such that $|i| = |j| = m.$ Moreover, we can also say that $g$ has communication complexity $c(m),$ and we let $L = \{ij \mid g(i,j) = 1\}.$

Would it be accurate to say that if $c(m) = O(1),$ then $L$ is regular? I'm not entirely sure that this is the case. I've been trying to think about counterexamples to this statement, but I can't really think of any. I do know that the converse is true, namely that if $L$ is regular, then $c(m) = O(1)$. I've been racking my brain over this in the past few days. Any ideas?

Answer 1

No. Let $L_0$ be a context-free language, say the language of matched parentheses, $L_1 = \Sigma^* \setminus L_0$, and

$$L = \{ij \mid b \in \{0,1\}, i \in L_b, j \in L_b, |i|=|j|\}.$$

Then $L$ is not regular, but it has communication complexity $O(1)$.

Answer 2

Let $L'$ be an arbitrary language, and consider $$ L = \{ \Sigma^{|x|} x : x \in L' \}. $$ Then $L$ has roughly the same complexity as $L'$, but it has communication complexity $1$.