  We study generic properties of string theory effective actions obtained by
classically integrating out massive excitations from string field theories
based on cyclic homotopy algebras of $A_\infty$ or $L_\infty$ type. We
construct observables in the UV theory and we discuss their fate after
integration-out. Furthermore, we discuss how to compose two subsequent
integrations of degrees of freedom (horizontal composition) and how to
integrate out degrees of freedom after deforming the UV theory with a new
consistent interaction (vertical decomposition). We then apply our general
results to the open bosonic string using Witten's open string field theory.
There we show how the horizontal composition can be used to systematically
integrate out the Nakanishi-Lautrup field from the set of massless excitations,
ending with a non-abelian $A_\infty$-gauge theory for just the open string
gluon. Moreover we show how the vertical decomposition can be used to construct
effective open-closed couplings by deforming Witten OSFT with a tadpole given
by the Ellwood invariant. Also, we discuss how the effective theory controls
the possibility of removing the tadpole in the microscopic theory, giving a new
framework for studying D-branes deformations induced by changes in the closed
string background.
