arXiv subjects

Reiner Czerwinski

Publications and source records attributed to Reiner Czerwinski.

8 recordsLinked to original sources

arXiv2024

The Polynomial Hierarchy does not collapse

The arithmetical hierarchy (AH) is similar to the polynomial hierarchy (PH). Unlike the PH, the AH does not collapse relative to any oracle. A language in the (k + 1)-st level of the AH is computable enumerable (c.e.) relative to the kth level. So, given an oracle in the kth level of the AH, we could use a black-box search to decide whether the input word is in the language. With very large padding arguments, i.e. the paddings grow faster than any relative to the level k of the AH computable function, we would construct a language contained in the k + 1 level of PH, if we use only a finite set of input words. From the oracle in AH, we would construct an analogue oracle at the kth level of PH. For the input words of the finite set, a word is in the language of AH, if and only if it is in the language of PH. And the input word is in the oracle set of AH, if and only if it is in the oracle of PH. As in the language of AH, we must apply a black-box search in the language of PH. So, we would also have exponentially many oracle queries in the language of PH. The PH does not collapse.

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arXiv2024

Separation of PSPACE and EXP

This article shows that PSPACE not equal EXP. A simple but novel proof technique has been used to separate these two classes. Whether an arbitrary Turing machine accepts an input when the running time is limited has been computed in this paper. Then, the limit goes to infinity. Thus, methods of the recursion theory can be applied to problems of computational complexity theory without violating the relativization barrier.

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arXiv2023

NP-hard problems are not in BQP

Grover's algorithm can solve NP-complete problems on quantum computers faster than all the known algorithms on classical computers. However, Grover's algorithm still needs exponential time. Due to the BBBV theorem, Grover's algorithm is optimal for searches in the domain of a function, when the function is used as a black box. We analyze the NP-complete set \[\{ (\langle M \rangle, 1^n, 1^t ) \mid \text{ TM }M\text{ accepts an }x\in\{0,1\}^n\text{ within }t\text{ steps}\}.\] If $t$ is large enough, then M accepts each word in $L(M)$ with length $n$ within $t$ steps. So, one can use methods from computability theory to show that black box searching is the fastest way to find a solution. Therefore, Grover's algorithm is optimal for NP-complete problems.

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arXiv2023

$P\not=NP$ relative to a $P$-complete oracle

The $P$ versus $NP$ problem is still unsolved. But there are several oracles with $P$ unequal $NP$ relative to them. Here we will prove, that $P\not=NP$ relative to a $P$-complete oracle. In this paper, we use padding arguments as the proof method. The padding arguments are not bounded by a computable function. Such as we can use methods from computability theory to separate complexity classes.

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arXiv2023

$L$ is unequal $NL$ under the Strong Exponential Time Hypothesis

Due to Savitch's theorem we know $NL\subseteq DSPACE(\log^2(n))$. To show this upper bound, Savitch constructed an algorithm with $O(\log^2(n))$ space on the working tape. We will show that Savitch's algorithm also described a lower bound under the Strong Exponential Time Hypothesis. Every algorithm for the Connectivity Problem needs $O(\log^2(n))$ space in this case.

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arXiv2022

A Polynomial Time Algorithm for Graph Isomorphism

We claimed that there is a polynomial algorithm to test if two graphs are isomorphic. But the algorithm is wrong. It only tests if the adjacency matrices of two graphs have the same eigenvalues. There is a counterexample of two non-isomorphic graphs with the same eigenvalues.

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arXiv2022

Incomplete sets in P for logspace-reduction

In this article, we investigate the behaviour of TMs with time limit and tape space limit. This problem is in P when the time limit is unary coded. If both limits go to infinity, it is undecidable which limit is exceeded first. Thus logspace-incomplete sets in P can be constructed. This implies L $\not=$ P.

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arXiv2022

Separation of P and NP

There have been many attempts to solve the P versus NP problem. However, with a new proof method, P not equal NP can be proved. A time limit is set for an arbitrary Turing machine and an input word is rejected on a timeout. The time limit goes toward infinity. Due to the halting problem, whether a word is accepted can only be determined at runtime. It can be shown by Rice's theorem, if a finite set of words are to be checked, they all have to be tested by brute force.

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