Classical BV algebra is the underlying algebraic structure of quantum fields theory in its classical limit, and is both DGA and DGLA with a suitable compatibility condition. On its cohomology there are structures of mimimal A_\infty-algebra and minimal L_\infty-algebra induced from the DGA part and DGLA part, respectively. I will explain both mathematical and physical meaning of those homotopy algebras and discuss another homotopy algebraic structures induced from 'the compatibility part'. Then I will explain how those structures to become the limiting case of quantum flat structure.