art with code

2018-07-02

Some more rubbish on P != NP

Feeling crank today, so:

Non-deterministic machines are one-bit oracles. Given a branch in execution, they always follow the correct path. You can use this property to write a universal non-deterministic machine:

readInput();

NonDeterministic done, correctOutput;

while (!done) {
  outputBit(correctOutput);
}

Its execution time is linear to the size of the output. There are still problems that take non-polynomial time compared to the input. For instance, "given an input of n bits, print out n! bits".

The question in P ?= NP then is: "given a problem where the output size is a polynomial of the input size, is it possible to emulate a non-deterministic machine using a deterministic machine with at most a polynomial slowdown". Since the ND machine takes a number of steps linear to the output size, which is polynomial of the input size, it solves the problem in polynomial time to the input. If the deterministic machine can emulate the ND machine at a polynomial slowdown, its running time is a polynomial of the output size, which is a polynomial of a polynomial of the input size.

Maybe you could also ask if it's possible to prove that there exists a problem for which an optimal program running on a deterministic machine requires a number of steps exponential to the size of its output.

Make a bridge from output size to input size? ...exponential to the size of its output and where the output size grows linearly with input size.

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