  We introduce the Hochschild extensions of dg algebras, which are
$A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric
Hochschild extensions, more precisely, every exact Hochschild extension of a
finite dimensional complete typical dg algebra is a symmetric
$A_\infty$-algebra. Moreover, we prove that the Koszul dual of trivial
extension is Calabi-Yau completion and the Koszul dual of exact Hochschild
extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of
the trivial extension of a finite dimensional complete dg algebra is the
Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact
Hochschild extension of a finite dimensional complete typical dg algebra is the
deformed Calabi-Yau completion of its Koszul dual.
