Q:

complexity of the half language

For any language $L$ over $\Sigma^*$, define 
$$L_{1/2} = \{x \in \Sigma^* : xy\in L, y\in\Sigma^{|x|} \}.$$
In words, $L_{1/2}$ consists of all $x$ for which there is a $y$ of equal length such that $xy\in L$.
An exercise in Sipser's book asks to show that $L_{1/2}$ is regular whenever $L$ is.  I have seen two distinct solutions, and both involve an exponential blow-up of states. 
Question: can anyone construct a family of languages $\{L_n\}$ such that the canonical automaton for $(L_n)_{1/2}$ is significantly (say, exponentially) larger than that for $L$?  My best efforts so far only increase the state size by $+1$! 

A:

See Mike Domaratzki's paper, State complexity of proportional removals
http://dl.acm.org/citation.cfm?id=782471
http://www.cs.umanitoba.ca/~mdomarat/pubs/sc_jalc.ps

