Year: 2024
Type: preprint
Abstract: The arithmetical hierarchy (AH) is similar to the polynomial hierarchy (PH). Unlike the PH, the AH does not collapse relative to any oracle. A $Σ_{k+1}$-complete language is computably enumerable (c.e.) relative to the k-th level of the AH. So, given an oracle in the k-th level of the AH, we require exhaustive search to recognize whether the input string is in the language. We can define a language in AH with a quantified formula and a computable relation. The encoding lengths of the bounded variables are unbounded. However, if we observe only finitely many input strings, then there exist integers such that the encoding lengths of the bounded variables can be limited to these integers without changing the result of the formula for each input string. By adding padding arguments of length equal to those of the bounded variables and padding of length equal to the runtime required to compute the relation, we construct an equivalent language in PH. An OTM operating on the language with padding arguments in PH with a padded oracle has the same program flow as it does on the original language with the unpadded oracle in the AH. The number of oracle queries does not change. We still need brute-force search therefore, the required number of oracle queries is exponential. The PH does not collapse.
Source: arXiv (Cornell University)
Author: Czerwinski, Reiner
Institution: Technische Universität Berlin
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4399197776