  This paper augments OBDD with conjunctive decomposition to propose a
generalization called OBDD[$\wedge$]. By imposing reducedness and the finest
$\wedge$-decomposition bounded by integer $i$
($\wedge_{\widehat{i}}$-decomposition) on OBDD[$\wedge$], we identify a family
of canonical languages called ROBDD[$\wedge_{\widehat{i}}$], where
ROBDD[$\wedge_{\widehat{0}}$] is equivalent to ROBDD. We show that the
succinctness of ROBDD[$\wedge_{\widehat{i}}$] is strictly increasing when $i$
increases. We introduce a new time-efficiency criterion called rapidity which
reflects that exponential operations may be preferable if the language can be
exponentially more succinct, and show that: the rapidity of each operation on
ROBDD[$\wedge_{\widehat{i}}$] is increasing when $i$ increases; particularly,
the rapidity of some operations (e.g., conjoining) is strictly increasing.
Finally, our empirical results show that: a) the size of
ROBDD[$\wedge_{\widehat{i}}$] is normally not larger than that of the
equivalent \ROBDDC{\widehat{i+1}}; b) conjoining two
ROBDD[$\wedge_{\widehat{1}}$]s is more efficient than conjoining two
ROBDD[$\wedge_{\widehat{0}}$]s in most cases, where the former is NP-hard but
the latter is in P; and c) the space-efficiency of
ROBDD[$\wedge_{\widehat{\infty}}$] is comparable with that of d-DNNF and that
of another canonical generalization of \ROBDD{} called SDD.
