In this talk, we present, after an analysis of the BI models, a theory of semantic proof-search for BI, that is based on the use of algebras of labels in order to solve the resource-distribution and sharing problems. Therefore, we deduce methods that generate proofs or countermodels in BI. As consequences, we have two strong results for propositional BI: decidability and finite model property, and we also propose, a new resource semantics, based on partially defined monoids, which generalizes the semantics of BI's pointer logic and for which BI is complete. We analyse the consequences of these proof-theoretic and semantics results about BI in the context of separation and spatial logics (specifications and proofs of programs or distributed concurrent systems).