arXiv subjects

Tayfun Pay

Publications and source records attributed to Tayfun Pay.

5 recordsLinked to original sources

arXiv2026

Even more properties of parity based bit-counting complexity classes

We study several additional properties of parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We first prove that ${\bf MNS}\subseteq{\bf P}^{{\bf B_{|1|\oplus}P}}={\bf P}^{{\bf B_{|0|\oplus}P}}$ and since ${\bf C_=P}={\bf ES}={\bf MNS}$ is already known, we establish that ${\bf C_=P}={\bf ES}={\bf MNS}\subseteq{\bf P}^{{\bf B_{|1|\oplus}P}}={\bf P}^{{\bf B_{|0|\oplus}P}}$. We then prove that ${\bf PP}\subseteq{\bf P}^{{\bf B_{|1|\oplus}P}}$ and ${\bf PP}\subseteq{\bf P}^{{\bf B_{|0|\oplus}P}}$, which consequently yields ${\bf P}^{\bf PP}={\bf P}^{\bf B_{|0|\oplus}P}={\bf P}^{\bf B_{|1|\oplus}P}$. We then demonstrate that the same method can be used to prove ${\bf \# P}\subseteq{\bf FP}^{{\bf B_{|1|\oplus}P}}$ and ${\bf \# P}\subseteq{\bf FP}^{{\bf B_{|0|\oplus}P}}$. We also show that the parity based bit-counting hierarchies contain ${\bf CH}$.

cs.CC↗

arXiv2026

Additional properties of parity based bit-counting complexity classes and hierarchies

We study some properties of the parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We first prove that both of these complexity classes are closed under complement and ${\bf B_{|1|\oplus}P}\subseteq {\bf B_{|0|\oplus}P}$. We then prove that ${\bf US}\subseteq {\bf P}^{{\bf B_{|1|\oplus}P}}$ and ${\bf US}\subseteq {\bf P}^{{\bf B_{|0|\oplus}P}}$. We then study the class defining characteristic functions of the parity based bit-counting complexity classes, where the one associated with ${\bf B_{|1| \oplus}P}$ produces the Prouhet-Thue-Morse sequence. We then prove that a finite contiguous block of these sequences yield the parity of the starting number and then prove that ${\bf \oplus P}\subseteq {\bf P}^{{\bf B_{|0|\oplus}P}}$ and ${\bf \oplus P}\subseteq {\bf P}^{{\bf B_{|1|\oplus}P}}$. We then use the parity based bit-counting complexity classes to define various hierarchies and show that they all contain ${\bf PH}$ and are contained in ${\bf CH}$.

cs.CC↗

arXiv2026

Bit-counting complexity classes

We define bit-counting complexity classes, where the membership depends on the binary profile of the number of accepting paths of non-deterministic polynomial time Turing machines. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the complexity class ${\bf PP}$ is contained in our comparison based bit-counting complexity classes ${\bf B_{|0|=|1|}P}$, ${\bf B_{|0|<|1|}P}$ and ${\bf B_{|0|>|1|}P}$. We further show that all of these complexity classes are Turing equivalent ${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$. We also prove that complexity classes ${\bf NP}$ and ${\bf CoNP}$ are contained in both of our parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$.

cs.CC↗

arXiv2023

A Note On The Natural Range Of Unambiguous-SAT

We discuss the natural range of the Unambiguous-SAT problem with respect to the number of clauses. We prove that for a given Boolean formula in precise conjunctive normal form with n variables, there exist functions f(n) and g(n) such that if the number of clauses is greater than f(n) then the formula does not have a satisfying truth assignment and if the number of clauses is greater than g(n) then the formula either has a unique satisfying truth assignment or no satisfying truth assignment. The interval between functions f(n) and g(n) is the natural range of the Unambiguous-SAT problem. We also provide several counting rules and an algorithm that determine the unsatisfiability of some formulas in polynomial time.

cs.CC↗

arXiv2018

An Overview Of Some Semantic And Syntactic Complexity Classes

We review some semantic and syntactic complexity classes that were introduced to better understand the relationship between complexity classes P and NP. We also define several new complexity classes, some of which are associated with Mersenne numbers, and show their location in the complexity hierarchy.

cs.CC↗