Is there a relationship between Computer Algebra and NLP?

Question

My intuition is that there is some overlap between understanding language and symbolic mathematics (e.g. algebra). The rules of algebra are somewhat like grammar, and the step-by-step arguments get you something like a narrative. If one buys this premise, it might be worth training an AI to do algebra (solve for x, derive this equation, etc).

Moreover, when variables represent "real" numbers (as seen in physics, for example) algebraic equations describe the real world in an abstracted, "linear," way somewhat similar to natural language.

Finally, there are exercises in algebra, like simplifying, deriving useful equations, etcetera which edge into the realm of the subjective, yet it is still much more structured and consistent than language. It seems like this could be a stepping stone towards the ambiguities of natural language.

Can anyone speak to whether this has either (1) been explored or (2) is a totally bogus idea?

Answer 1

There isn't, really. Natural language is way more complex and irregular than algebra, which is far more formalised and unambiguous.

So far, in NLP, most success/progress has been made in little toy domains, which exclude most of the complexities of real life, including many ambiguities.

When you say the rules of algebra are somewhat like grammar, then that is because it is essentially a formal language, for which we can specify a grammar. There is currently no complete grammar for any human language (and I doubt there ever will be), let alone a formal one that can be processed by computer.

This was one of the reasons why the first AI boom, where a lot of over-hyped promises where made about being able to translate Russian into English automatically, failed abysmally: natural languages are more than just formal grammars of lexical items.

Stochastic approaches have gone some way towards pragmatic solutions, but when it comes to understanding language they are basically a fudge. And don't get me started on deep learning approaches to NLP.

So the only relationship is that we use the term 'grammar' for the descriptive formalisms in both cases; a formal grammar of algebra would be very different from a grammar for a human language.

This doesn't mean, however, that approaches developed in the field of NLP cannot be applied to algebra: even those which failed in NLP because they were overly limiting. To find out more about this, look for Chomsky Hierarchy -- that describes the different expressive powers of formal languages.

But I would argue that human language is outside of that, because it is not a formal language.

Answer 2

One of the ways to ask if this two problmes are related is to ask, could we solve math/algebra equations with NLP approaches, and the answer is yes, it's an absolutely valid idea and it was approached by many researchers.

For example in the "Deep learning for symbolic mathemathics" paper by facebook researchers, the NLP-based approach was used to solve calculus-level math.

Or in this paper authors propose a method to extract semantics from math problems in words and solve them.

In fact, even very simple approaches like a small LSTM network could work for simple strictly stated problems

Answer 3

A lot of natural language processing software are (in 2021) using statistical approaches.

Read The Deep Learning Revolution (by T.Sejnowski), Artificial Beings, the Conscience of a conscient machine (by J.Pitrat), Introduction to Deep Learning (by E.Charniak).

However, mixed approaches (like in RefPerSys) can also be used.

The so-called frame-based approaches can, and have been used, in NLP. From an abstract point of view, they are close to algebra. And probably are still one of the best approaches for generating natural language sentences (in written form).

Pitrat has for dozen of years taught and advocated a symbolic AI approach to NLP. You could read his French book Métaconnaissances, futur de l'IA (metaknowledge, future of AI).

Read also books like Knowledge Representation and Reasoning (Braqueman & Leveque)

You might also read (if you can read French) Nicolas Bourbaki's Théorie des Ensembles or (in English) Interactive Theorem Proving and Program Development (by Bertot & Castéran).

Symbolic AI approaches are definitely "algebra-like". In practice, they don't need powerful GPGPUs, or even don't do a lot of floating point operations.

I assume you want to process human language text available as textual files (e.g. HTML, XML, UTF8 encoded plain text, etc...). If you want to process sounds and speech (for example, an audio signal coming from a microphone), you do need much more signal processing and floating-point operations. If you want to process images (e.g. handwritten text), it is also different. In France, I would recommend Professor Mohamed Daoudi.

Answer 4

Yes there is a relationship, but it's not an exact one like I think is envisioned in the question.

Fuzzy numbers and fuzzy logic translates natural language expressions into quantitative values and mathematical logic. See PRUF—a meaning representation language for natural languages by Zadeh.

Dr. Zadeh, the 'inventor' of fuzzy numbers and fuzzy logic, provided a mathematical framework for taking simple language expressions and expressing them in mathematical form. The underlying basis is that natural language is inherently imprecise. To quote this paper, "First, a basic assumption underlying PRUF is that the imprecision that is intrinsic in natural languages is, for the most part, possibilistic rather than probabilistic in nature."

While fuzzy numbers and fuzzy logic are ubiquitous in many applications, for example fuzzy controllers in engineering, I personally have not seen a groundswell of using PRUF in NLP. Not that it isn't possible. I have been pursuing research off-and-on in this area, and I think that there is a possibility of bridging mathematical logic and NLP, particularly when using concept-based NLP (e.g., something along the lines of Cognitive Linguistics).