Mental arithmetic as a test for Sentient AI

Sentient AI differs from Generative AI in supporting continual learning and reasoning, mimicking human cognition. Sentient AI agents are thus able to acquire new skills. One way to explore this is to model how children learn basic arithmetic, something that is trivial for computers, but challenging for young children.

Here is an explanation of how to add whole numbers:

Step I: We arrange the given numbers in columns, ones under ones, tens under tens, hundreds under hundreds and so on.


Step II: We add the digits in each column taking the carry over, if any, to the next column to the left, and adding it along with the digit in that column. We continue this process till we add the digits in all the columns.

Taken from: https://www.math-only-math.com/addition-of-whole-numbers.html

This explanation brushes over the details, e.g. how to add pairs of digits, where we can use our memory for a given pair, or if that fails, work it out step by step, by adding one progressively.

Children are taught to write the numbers down, but as they gain confidence, are able to perform the algorithm mentally.  How can we replicate that with artificial neural networks?

Christian Lebiere’s Ph.D thesis (1999) “The dynamics of cognition: An ACT-R model of cognitive arithmetic” [1], provides a symbolic model of mental addition. ACT-R is based upon the idea of “chunks” as sets of name/value pairs, along with the means to apply a sequence of transformations on chunks.

[1] https://www.researchgate.net/publication/220173218_The_dynamics_of_cognition_An_ACT-R_model_of_cognitive_arithmetic

Can ACT-R provide useful insights into how to design an artificial neural network with similar capabilities?  The idea here is to model addition in terms of understanding and generating sequences of tokens, sequential transformations applied to latent semantics, and an implementation of human-like memory.

To demonstrate cognition, we require the agent to vocalise the sequence of steps the agent makes when working on the task (i.e. the chain of thought). This points to the idea of using autoregressive training as a basic for learning the cognitive steps. The teacher provides a worked example, and the student attempts to duplicate it. Repetition improves performance.

Memory is key to all of this, including working memory and skill memory. For artificial neural networks this corresponds to the current values and model parameters. Continual learning implies that the model parameters are being continually updated as cognition proceeds. In principle, this can be arranged using a mix of local learning rules and gradient descent.  Explicit memory involves cue based retrieval at the same level of the network.

I am now looking for smaller steps to progress towards a full proof of concept for learning mental arithmetic. This could include memorising single digit addition, and understanding the columns for the digits in whole numbers.

This overall approach seeks to mimic human cognition using modest levels of resources in stark contrast to approaches based upon exploiting large language models. This will allow us to build a whole new range of AI systems that can be taught like humans, learn on the job, explain themselves, reflect on their performance and so forth.

Your comments are welcomed!

Dave Raggett <dsr@w3.org>