  We investigate classical bosonic open-string field theory from the
perspective of the Wilson renormalization group of world-sheet theory. The
microscopic action is identified with Witten's covariant cubic action and the
short-distance cut-off scale is introduced by length of open-string strip which
appears in the Schwinger representation of open-string propagator. {\it
Classical open-string field theory} in the title means open-string field theory
governed by a classical part of the low energy action. It is obtained by
integrating out suitable tree interactions of open-strings and is of
non-polynomial type. We study this theory by using the BV formalism. It turns
out to be deeply related with deformation theory of $A_{\infty}$-algebra. We
introduce renormalization group equation of this theory and discuss it from
several aspects. It is also discussed that this theory is interpreted as a
boundary open-string field theory. Closed-string BRST charge and boundary
states of closed-string field theory in the presence of open-string field play
important roles.
