Interaction Grammars are a linguistic formalism that aims to capture both the syntax and the semantics of natural languages with two levels of representation: at the syntactic level, they use tree descriptions and, at the semantic level, they use Directed Acyclic Graphs (DAGs) descriptions for representing under-specified structures. The synchronization between the two levels is realized by a partial function that map syntactic nodes to semantic nodes. Syntactic composition is driven by an operation of cancellation between polarized morpho-syntactic features and in parallel, semantic composition is driven by a similar operation of cancellation between polarized semantic features.