  We consider the problem of searching for proofs in sequential presentations
of logics with multiplicative (or intensional) connectives. Specifically, we
start with the multiplicative fragment of linear logic and extend, on the one
hand, to linear logic with its additives and, on the other, to the additives of
the logic of bunched implications, BI. We give an algebraic method for
calculating the distribution of the side-formulae in multiplicative rules which
allows the occurrence or non-occurrence of a formula on a branch of a proof to
be determined once sufficient information is available. Each formula in the
conclusion of such a rule is assigned a Boolean expression. As a search
proceeds, a set of Boolean constraint equations is generated. We show that a
solution to such a set of equations determines a proof corresponding to the
given search. We explain a range of strategies, from the lazy to the eager, for
solving sets of constraint equations. We indicate how to apply our methods
systematically to large family of relevant systems.
