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Revision #1 to TR21-049 | 19th April 2021 20:42
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#### Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations

**Abstract:**
We call any consistent and sufficiently powerful formal theory that enables to algorithmically verify whether a text is a proof \textbf{algorithmically verifiable mathematics} (av-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of av-mathematics. Our main results are as follows. \\

"$\P \subsetneq \NP$ or for any deterministic, polynomial time compression algorithm $A$ there exists a nondeterministic, polynomial time compression machine $M$ that reduces infinitely many binary strings logarithmically stronger than $A$." \\

"$\P \subsetneq \NP$ or f-time resource bounded Kolmogorov complexity of any binary string $x$ can be computed in deterministic polynomial time for each polynomial, time constructible function $f$."\\

For computing models with "efficient" interpreters we prove the following theorem:\\

"For each polynomial, time constructible function $f$, $\TIMEf \subsetneq \NTIMEf$ or one can essentially stronger compress words nondeterministically in time $\Oh{f(n)}$ than deterministically in time $f(n)$."

**Changes to previous version:**
strengthening some of the results

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TR21-049 | 1st April 2021 14:32
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#### Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations

**Abstract:**
We call any consistent and sufficiently powerful formal theory that enables to algorithmically in polynomial time verify whether a text is a proof \textbf{efficiently verifiable mathematics} (ev-mathematics). We study the question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of ev-mathematics. Our main results are as follows. \\

"$\P \subsetneq \NP$ or for any deterministic, polynomial time compression algorithm $A$ there exists a nondeterministic, polynomial time compression machine $M$ that reduces infinitely many binary strings logarithmically stronger than $A$." \\

"$\P \subsetneq \NP$ or f-time resource bounded Kolmogorov complexity of any binary string $x$ can be computed in deterministic polynomial time for each polynomial time constructible function $f$."