  One of the fundamental requirements for models of semantic processing in
dialogue is incrementality: a model must reflect how people interpret and
generate language at least on a word-by-word basis, and handle phenomena such
as fragments, incomplete and jointly-produced utterances. We show that the
incremental word-by-word parsing process of Dynamic Syntax (DS) can be assigned
a compositional distributional semantics, with the composition operator of DS
corresponding to the general operation of tensor contraction from multilinear
algebra. We provide abstract semantic decorations for the nodes of DS trees, in
terms of vectors, tensors, and sums thereof; using the latter to model the
underspecified elements crucial to assigning partial representations during
incremental processing. As a working example, we give an instantiation of this
theory using plausibility tensors of compositional distributional semantics,
and show how our framework can incrementally assign a semantic plausibility
measure as it parses phrases and sentences.
