# DCFL substitution closure

Deterministic context-free languages are not in general closed under substitution. I have been looking at the DK-test (it is described in the 3rd edition of Sipser's Intro. to Theory of Computation, 2012) which can be used to check if a given CFG is deterministic, and it seems to me this can be used to show that DCFLs are closed under substitution when the substituted languages are disjoint DCFLs. Is this a known result? Or is anything known about the substitution closure of DCFLs with other restrictions on the substituted languages?

Since $ab$ is a DCFL, your result would imply that DCFLs are closed under concatenation, which they aren't. So your result can't hold.
• @sjmc Only if the word before substitution contains no substring $aa$ for any $a\in \Sigma$. – FrankW Mar 10 '14 at 14:22