Consider a sentence as a series of words with an associated set of labels that tell one how information is passed through the sentence - examples include combinatory categorical grammars or Lambek grammars alongside a host of other such structures.
Now consider that each word is given an associated vector - this is supposed to represent the 'meaning of the word' in and of itself - the manner in which it is formed is not relevant. The labeling can be considered a description of how these vectors relate to form sentence meaning.
The problem is how to use the given information to give a sentence meaning which lives in the same space as the word meanings?
An example is given in this paper (https://arxiv.org/pdf/1003.4394.pdf) the issue here is that by turning words with more complex labeling (or typings in the paper) into tensors one can indeed achieve what is being discussed but for real world sentences it is perfectly common to have typings of 5 or 6 dimensions, which result in tensor calculations that are simply too laborious to be practical.
What would be useful would be if one could relate these vectors to elements of a non-commutative non-associative algebra on which one could asses proximity of these elements. This may seem vague so I will give an example of something that gets me halfway. Specifically it gives me non-comutativity from the original vector setup
Before we get to the example of how one can do this I would like to quickly illustrate the importance: consider the two sentences 'man bites dog' versus 'dog bites man'. Clearly they have different meaning so our chosen representation should have the structure to reflect this.
Now to the example structure. If we take each words vector representation and raise it to a density matrices via the outer product, it is then clear that one can multiply these together and get an output 'sentence meaning' which give in the same space as the raised word vectors - it is moreover non-commutative. What it lacks however is the non-associativity - The relational information given by the words labeling is irrelevant - only pure left to right order holds meaning- by introducing non-associativity one could use the labeling to give the manner in which the words combine - think 'Jim likes dogs who like crisps' - with non associativity one could make sure it was clear that one brought together the component 'dogs who like crisps' prior to 'Jim likes' provided the labeling indicated this.
With this outlined I hope it's clear what is required and why - a non-commutative non-associative algebra is a reasonable representation of a sentence - provided it also has a concept of proximity as it would then allow one to consider sentence meanings in relation to words. Any help with suggestions of structures to look into or ideas on the reasonableness of this request would be appreciated, assistance would prevent countless blind alleys as I try to look into this.
