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Suppose that there exists a function $:\{0,1\}^ → \{0,1\}^$ such that, is computable in polynomial time; and the following task cannot be computed in polynomial time (that is, there are $ ∈ \{0,1\}^$, for which, it cannot be computed efficiently): Given $ = ()$, find any $\hat{}∈ \{0,1\}^$ such that $(\hat{}) = .$

Define the language, $$ = \{(, ) | ∃\hat{} ∈ \{0,1\}^ \text{ such that } (\hat{}) = \text{ and } \hat{} < \}.$$ The comparison $<$ is with respect to the integers of which $\hat{}, $ are the respective binary representations (in general, $$ is viewed as a map from integers to integers).

Is the language $$ is in $NP\setminus P?$

My attempt:

  1. Guess a candidate $\hat{x}∈\{0,1\}^n$

  2. Verify that $f(\hat{x})=y.$ Since is computable in polynomial time, we can compute $f(\hat{x})$ and check if it equals in polynomial time.

  3. Verify that $\hat{x}<z$ .Comparing two -bit integers can be done in polynomial time as well.

Thus, a nondeterministic Turing machine can guess $\hat{x}$ and verify both conditions in polynomial time, which implies that $L∈NP.$

If my $NP$ logic is right, how can I show $L$ isn't in $P$? Or any other way to show $$ is in $NP\setminus P?$

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    It seems to me that, if $L$ were in P, then you could find $\hat x$ (as in your first paragraph) in polynomial time by a binary search. The assumption about $f$ in your first paragraph and similar hypotheses have been extensively studied under the heading of "one-way functions". – Andreas Blass Sep 24 '24 at 23:49
  • I assume you mean to write ${0,1}^*,$ not ${0,1}^n$? Per Andreas's comment, see here. – spaceisdarkgreen Sep 25 '24 at 00:11
  • Essentially the same proof that I sketched in my comment here is in the answer by Cyker to the question linked in spaceisdarkgreen's comment. – Andreas Blass Sep 25 '24 at 15:46
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    Yes, your nondeterministic algorithm shows that $L$ is in P. – Andreas Blass Sep 25 '24 at 16:05
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    @AndreasBlass the linked answer language L = { (x', y) s.t. x' <= x and f(x) == y }. And L is different from it. How they will be related? –  Sep 25 '24 at 16:07
  • The more complicated $L$ in the linked answer incorporates the binary search in my answer. – Andreas Blass Sep 25 '24 at 16:11

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