What Is Model Theory Good for?

June 2019

Model theory, which is one of the most important mathematical tools in the study of language semantics, makes certain things simple: when no assumption is made on the concrete representation of some knowledge (a.k.a. its domain of interpretation), it is possible to manipulate all possible interpretations of a logical statement at once.

In pure logic, the only relevant interpretations are those that make a statement false (or not satisfied by the interpretation). Inference can then be "computed" by iteratively constructing such interpretations.

Model theory is often criticized for not being intuitive enough when it comes to writing algorithms, despite a clean formulation of logical truth in terms of unsatisfiability. However, all the interpretations that are generally ignored by logicians may be appealing to computer scientists, who generally like data structures, beyond inference. Here is a couple of examples of what model theory may be good for.

Numeric Interpretations of Natural Language

The craze around Web-scale language models, like word2vec, GloVE, BERT and friends, redirects the attention of researchers on large numeric vector representations of words and meanings. While the general approach looks elegant (i.e. simple to conceive) and versatile enough to work well on various tasks like machine translation and question answering, it is difficult to construct any sort of explanation to a neural network's behavior when interacting with humans in natural language.

In contrast, procedures that are based on model theory to check the consistency of formal statements naturally provide "counter examples" to what one thought was correct statements, which greatly helps debug a system specification (a set of statement).

One way of taking the best of both is to consider vector representations as actual interpretations of natural language, which suggests that one could then derive another (formal) language that are also satisfied by that vector representation. The benefit of this translation is that machines then have a language to express themselves that has two fundamental properties:

  1. it is not based on numbers but on words or symbols (these words may actually be the same as those included in the language model's training set) and
  2. it includes a notion of causality (and thus of explainability) via logical necessity.

Conflicts of Interpretations in Social Networks

It does not seem to far-fetched to imagine that arguments on social media originate from contradictory interpretations of statements shared online. If we take this analysis literally, one way of analyzing the dynamics of information spreading within social networks is to develop agent models that do not store their beliefs, which are usually understood as Datalog-like propositions, but some internal interpretation that results from reading the statements being shared.

If we introduce the principle that that a domain of interpretation may be extended but never updated. That model-theoretic approach should account for the emergence of ideological communities within social networks characterized by "compatible" interpretations among agents belonging to the same community. The size and nature of a community is then determined by social interactions, i.e. the sharing of statements on social media. It is even possible to check how agents respond to biases if those are modeled as inference rules that do not hold universally: an agent is subject to a bias if its local interpretation satisfies the corresponding rule.

Agent models of this kind to simulate interactions on social media are particularly important these days, as it would be possible to study how fake news spread and what policies reduces their impact without limiting agents' actions.