Godel proved there are true statements in arithmetic that can't be proven true in any sufficiently strong Formal Axiomatic System (FAS). The authors of this paper use similar arguments to prove there are TM's that belong to a certain complexity class that cannot be proven to belong to that class. They refer to these type of algorithms as "hidden machines".


What happens to the quesion of P?NP if there exists a TM that can solve any satisfiable 3SAT instance in polynomial steps, yet, it is impossible to prove this TM does so?


In some sense, nothing much. We still have P = NP. It's possible for there to be statements that are true even though we can't prove them true.

In another sense, something interesting happens: in your scenario, we can explicitly write down a new algorithm that is guaranteed to solve 3SAT in polynomial time, using Levin universal search. See https://cs.stackexchange.com/a/92095/755. So, if there is a non-constructive proof that 3SAT is in P, then you can find a constructive proof that 3SAT is in P.

See also Is P decidable?, Assuming P = NP, how would one solve the graph coloring problem in polynomial time?, Explicit algorithms and algorithms involving unknowns.

  • $\begingroup$ The articla also shows there are hidden non-constructive algorithms. So, we couldn't prove P=/=NP. We might still be able to prove P=NP is undecidable. $\endgroup$ – Russell Easterly Sep 4 '20 at 20:58

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