> For to define "reasoning"
I'd say reasoning is the process of applying logic to draw inferences from some information/axioms/assumptions. For instance if you're asked "can a fridge fit in a bread-box?" and (implicitly or explicitly) go through:
1. A fridge is much larger than a bread-box
2. Larger objects cannot fit inside smaller objects without flexibility
3. Neither objects are sufficiently flexible
4. Therefore, a fridge cannot fit in a bread-box
Then I'd be happy saying you have used reasoning to reach your answer.
> How can knowledge be encoded in a machine? [...] LLMs say that knowledge is encoded in the relationships between words [...]
I don't think it'd be fully correct to say that knowledge is only encoded by relations between words. The input/output of the model is tokens of text, but internally it'll be converted into high-dimensional semantic vector spaces of concepts.
Different words describing the same concept ("Bread-Box", "breadbin", ...), or even images in the case of multi-modal models, can be associated with the internal representation of a bread-box, from which useful semantic manipulations/inferences can be made about the concept and not just the word used to reference it (like approximating the bread-box's size, a factor potentially learned from images but applied to answer a textual question).