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According to wikipedia and other references there exists complete language $L \in EXP$ such that for every languages $L'$ in $EXP$ there exists a polynomial reduction $f$ that converts instance of $L'$ into $L$ in polynomial time. I believe this definition is fine but I have some observation that are confusing me. I construct a language $L''$ in $EXP$ such that $L''$ can not be convert into $L$ with polynomial time reductions and $L''$ is in $EXP$ with an algorithm M in this way:

let $f_i$ be the iteration of all reductions.

1) $input(i,x)$

2) run $f_i(i,x)$ in time at most $2^n$ if the computation time is greater than $n^{\log n}$ then accept.

3)else accept $(i,x)$ iff $f_i(i,x) \not \in L$.

Obviously $Language(M) \in EXP$ so there exists a polynomial time reduction from $Language(M)$ to $L$ but this makes a contradiction because M prevents the reductions of less than $n^{\log n}$ time to $L$ and it is because of line 3 of algorithm.suppose $f_j$ is polynomial reduction, the membership of $(i,x)$ in $L(M)$ is the opposite of the member ship of $f_j(j,x)$ so $f_j$ is not a reduction from L(M) to L by the line 3 of algorithm.

My question is: where is my mistake?

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Are every problems in EXP karp reducible to any EXP-Complete?

Yes. That's the definition of $\mathrm{EXP}$-completeness.

My question is: where is my mistake?

Your mistake is in believing that $L''\notin\mathrm{EXP}$. You've demonstrated a Turing machine that decides $L$ in time $O(n^{\log n}) = O(2^{(\log n)^2})\subset O(2^n)$. I'm not sure what you mean by "$M$ prevents the reductions" – you're trying to reduce from $L(M)$ to $L$, not from $L$ to $L(M)$.

On the other hand, $L''$ is not $\mathrm{EXP}$-complete, by the time hierarchy theorem. Any language reducible to $L''$ must be in time $O(2^{(\log (n^c))^2})$ for some $c$, but this is just time $O(2^{(c\log n)^2})=O(2^{(\log n)^2})$.

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  • $\begingroup$ Thank you for quick answer. If reduction $f_j(j,x)$ runs in polynomial time for example $n^2$ according to line 3 of algorithm the membership of $(j,x)$ in $L(M)$ will be the opposite of the membership of $f_j(j,x)$ in L so $f_j$ is not a reduction. I think this force the $f_j$ to not be a reduction from L(M) to L and by the limitation of time $n^{\log n}$ the algorithm prevents all polynomial reductions and my exact problem is here. $\endgroup$ Aug 18, 2019 at 9:59

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