How does 'understanding' a problem work, and when do we feel like we understood an explanation or proof? Having an opaque long formal proof often feels insufficient, similarly some arguments feel unsatisfying because they contain many subjectively arbitrary choices.

An explanation is a sequence of claims corroborating a statement. A reasoner understands an explanation (or rather has more understanding of an explanation) when each claim has low surprisal given its mental state after reading all of the previous statements, starting with the problem statement. The aggregate of the surprise over the entire explanation indicates how poorly understood it is. The measure of surprisal is essentially about the reconstructability of the explanation using something like Monte Carlo Tree Search over all possible explanations informed by the mental states of the reasoner.

The surprisal of claims is roughly how long it would take for the reasoner to come up with the claim given its current mental state. Since the space of such claims is exponential in the length of the claim the reasoner has to use some form of attention to guide its search. We can model this attention mechanism by an ontological graph. Such a graph encodes the collection of mental concepts and associative links between them. The mental state of the reasoner is an activation of an ensemble of concepts and the associative links make available other concepts, lowering their surprisal when invoked in the next step of the explanation.

When a step in an explanation is highly surprising some understanding is needed. The reasoner does this by modifying its ontology, creating new concepts or creating associations that make the step more obvious. I call such modifications insights, a good insight gives clarity and makes steps obvious and subjectively trivial.

To examine this situation consider the mutilated chessboard problem[1] and subsequent explanation:

Suppose a standard 8×8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares?

The answer that it is impossible. Opposite corner squares always have the same color so that there are 30 and 32 white or black squares left. A domino placed on the chessboard will always cover one white square and one black square. So there will always be either two white or two black s

...

Posts

Wikitag Contributions

Comments