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I believe that I have a misunderstanding in the P=NP problem while I was thinking of how can it be proved in a non-constructive manner.

We know that we can build an infinitely large text file with solutions for each of the solutions for every possible combination for that problem (in a similar manner of building a deterministic tree construction of proportional formulas in propositional logic) where then we know the retrieval of every solution for any instance can be just retrieved via polynomial searching.

My question is: Why isn't this a non-constructive proof that P=NP since it shows that they exists an infinite listing of text that allows getting solution to NPC problems?

I know this is most likely a fundamental error in my understating of the problem but I can't seem to get it so I'd appreciate the help.

TokieSan
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    How do you do polynomial searching in an infinite set of data? – Nathaniel Mar 29 '24 at 16:53
  • In theory and in practice algorithms must have finite size. An algorithm with an infinite table encoded in it as you describe would not have finite size. – Neal Young Mar 29 '24 at 17:08
  • @Nathaniel, that's not hard if the table is encoded correctly and you have countably infinite space. E.g. with a lookup table, or a trie, you could look up the answer for a formula in time proportional to the number of bits it takes to encode the formula. – Neal Young Mar 29 '24 at 17:11
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    @NealYoung That would require a lookup operation in $\mathcal{O}(1)$ time, which does not seem possible with infinite data. But again, as you said, all data must be finite for algorithms, so maybe this discussion is irrelevant. – Nathaniel Mar 29 '24 at 17:14
  • @Nathaniel, not $O(1)$ time, rather, time proportional to the number of bits needed to encode the formula. Whether this is possible will depend on the machine model, which OP has not specified. I believe there are theoretical RAM-like models with infinite memory where the time to address the $i$th bit of memory is $\Theta(\log i)$. On the other hand, I'm not sure what criteria one would use to decide whether any given model with infinite memory (e.g. a Turing machine) is "realistic". – Neal Young Mar 29 '24 at 17:48
  • ps. OP you may be interested in Turing machines with oracles, which are one formalism for TM's with a sort of "infinite lookup table". – Neal Young Mar 29 '24 at 19:13

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