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Can you say more? Is there a thermodynamic interpretation of, say, the Halting problem? (Or P vs. NP?)

This is in relation to the ‘say more’ part, not specifically about the halting problem or P/NP.

Since computation is physical, there are theoretical bounds on the energy consumption of computation. (Landauer’s principle [0]).

Physical limits computation are at least one way to think about the relationship between thermodynamics and software.

The answer to your question is not straightforward, but it is not a definite "no". Here is an interesting (albeit amateur) discussion on the subject on Stack Exchange: https://cs.stackexchange.com/questions/26049/is-there-a-conn...