> But the problem is that Church–Rosser Property[0] (proofs ~= programs) and Brouwer–Heyting–Kolmogorov Interpretation[1] (Propositions at types) are NOT binary SAT, and you have concepts like mere propositions[3] that are very different than just BSAT.
The computational limits imposed on program verification are independent of the logical theory used, and depend only on its expressive strength. Many if not most interesting program properties require interleaved properties (forall-exists or forall-exists-forall etc.) which are intractable to verify.
> But CodeSpeak doesn't have formal specifications, so this is irrelevant.
The lack of formalism also doesn't matter to the limitations on correctness. If you wish to know, with certainty, that a program satisfies some property, that knowledge has a cost. But both humans and LLMs write programs at least in part based on inductive rather than deductive reasoning, which cannot be rigorously given a level of confidence. That may not be what we want, but that's what computational complexity says we can have, so we're forced to work in this way.
We should complain about what we don't have, but demanding things we can't have isn't going to help. There's no fundamental reason why AI shouldn't, someday, be able to program as well as humans, but there are fundamental limitations to producing the software we wish we had. Humans can rigorously use deductive methods, with mechanical help, and AI could possibly use it, too. But there's no reason to believe AI could break the size barrier of such problems. People have been able to rigorously verify only very small programs, and maybe AI could do a little better if only because of its tenacity, but if we expect to produce perfect programs by any means, we'll be waiting for a long time.