Systems for natural language understanding are now quite good and becoming commonplace. Unfortunately, some of the most powerful are also quite opaque: there is no satisfactory theory for why they work. However, ongoing work on semantics is trying to rectify this.

The authors here make progress on this problem by leveraging various powerful tools, namely compact closed categories, pre-group algebras, the category of finite-dimensional vector spaces with completely positive maps, entropy, and Kullback-Leibler divergence. This machinery is then used to build a syntax-to-semantics system that has good properties, namely a reasonable notion of entailment between sentences and a reasonable notion of composition. In other words, the meaning of a sentence comes from the meaning of its parts.

The paper further demonstrates, both in theory and through toy examples, that the density matrix approach (another name for completely positive maps) is superior to using just vectors. Some experiments with “real” data are also presented, but these are marred by too much human intervention in choosing what to report to be convincing.

Although this paper tries to be self-contained, it is really for experts in the domain. Even the examples, which are great for helping one’s intuition, are not always complete, but can be completed easily enough if you already know the material. Furthermore, there are many aspects of linguistics (such as quotations, anaphora, and so on) that cannot be captured by this kind of semantics. Nevertheless, this is still a solid step forward. The prose is clear, the material is well organized, the relevant literature is cited, and a valiant effort has been made to bridge theory and practice.