If the intent is not easy to understand, it's information sparse. Because it takes a lot of CPU (or brainpower) to interpret.
You can run gzip on an English sentence to make it more textually dense, but clearly it is not more information dense in this context.
The prompt was also “easier to understand”, purely in the sense that the response is more or less guarantee to be what I wanted it to say, which was the point behind the demonstration. I went into more detail on it in another comment around here.
Thus from first principles it's most likely that content which is more understandable to humans is also likely to be more understandable to LLMs. Of course they are still capable of interpreting very obscure structures too, but usually at the cost of cognitive performance.
I'm open to being wrong about this, and I'm sure it's being researched.
(Specifically for text representations)
To your point, at some level of intelligence an LLM will be able to infer the intent of your prompt consistently without thinking enabled, in which case interpretability to a human matters less. But for complex tasks you aren't likely to get optimal performance with prompts that are difficult for humans to understand. And yes, you'd see that with thinking enabled as it churns over thousands of tokens trying to "mentally expand" a compressed prompt.
Interesting discussion though!
1. Information density is subjective, lexical complexity is how you measure it. The OP is talking “weight”, I’m talking “mass and gravity.” One of them will get you the other in most situations, so for the causal physicist it doesn’t matter, but if you’re getting into tweaking the universe then your mental model and approach matters significantly. My comment right now could be seen by some as being information dense, since I’m staying roughly on topic and tossing many concepts out, but “lexical complexity” might be the most lexical complexity in the whole thing and taken word-for-word I’m sure less than 1% of it is domain specific. “The program must use parallel processing on the CPU.” That seems decently information dense, but “the” is found in nearly every block of text ever written, “program” - are we talking television? Theater?, “must” is no better than “the”, and so on. Compare it to “#include <immintrin.h>“
2. Most people don’t realize how far that goes with LLMs. The vocabulary it has is dictated by the words in the conversation. If I ask you “what time is it?” you don’t respond “shoelace” because you’d sound crazy, although you could say it if you wanted, but the model absolutely won’t say it because that word literally does not exist yet. The end result feels the same, but the difference matters and it’s why it’s suggested not to use negating instructions. For example: “Do not mention elephants.” Well that mathematically wasn’t possible until you said it. Not having the word in the list of possibilities is a lot better than hoping it adheres to the “do not mention” part. My example prompt took that same idea from the opposite direction. The model must respond, it will be grammatically complete and coherent, and as much as possible the only words it has are the ones tightly associated with making my point for me. It didn’t ramble about baking a chocolate cake because it can’t, and making that the case is the goal with prompting, not specifically density. Word density > language density; feels similar, very different.
Perhaps this comment itself is the irony you were seeking. I spent several meandering paragraphs and included analogies to drive home the point that you should focus on the words that matter most.
Claude is terrible at this! Probably for the same reason that its writing style in prose is so annoying and full of claudisms.