Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$ -fold composition of the base function $h$ . The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of the form $h^d$ , with $h$ being any base function and $d= Ω(\log\log n)$ . (II) The inner function is a recursive function of the form $h^d$ , with $h$ being any constant arity base function (other than AND and OR) and $d= Ω(\log\log n)$ , where $n$ is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$-fold composition of the base function $h$. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of the form $h^d$, with $h$ being any base function and $d= Ω(\log\log n)$. (II) The inner function is a recursive function of the form $h^d$, with $h$ being any constant arity base function (other than AND and OR) and $d= Ω(\log\log n)$, where $n$ is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.

arXiv:2407.08385Leibniz international proceedings in informaticsJan 1, 20240 citations

Overview

O. Costa de Beauregard

Boston studies in the philosophy of scienceJan 1, 19870 citations

A family quasi-distribution function representation in phase space

Yuan Tong-quan, 北京理工大学理学院物理系,北京 100081

A family quasi-distribution function representation is defined in this article. This family quasi-distribution function representation is constructed from the family wave function of the Schrdinger equation in phase space in which the definitions of the operators are α=αp-i?q and α=(1-α)q+i?p. Two interesting relationships are found. The first one is that the family wave function of the Schrdinger equation in phase space is a “Window” Fourier transform of the function φ(λ)exp［i(1-α)qp/?］. The second one is that different choices of the window functions result in different distribution functions. When the window function g(λ) is a Gaussian function the distribution function is the Husimi-like distribution function. When the window function g(λ) is a plural function representing an ellipse, the quasi-distribution function is the Ellipse distribution function; and finally when the plural function g(λ) is supplemented with the additional condition α=0, it will result in the standard ordering, anti-standard ordering distribution function and Wigner function. In this case g(λ) is a function depieting a rectangular window with width λ and height 1/12π?.A family quasi-distribution function representation is defined in this article. This family quasi-distribution function representation is constructed from the family wave function of the Schrdinger equation in phase space in which the definitions of the operators are α=αp-i?q and α=(1-α)q+i?p. Two interesting relationships are found. The first one is that the family wave function of the Schrdinger equation in phase space is a “Window” Fourier transform of the function φ(λ)exp［i(1-α)qp/?］. The second one is that different choices of the window functions result in different distribution functions. When the window function g(λ) is a Gaussian function the distribution function is the Husimi-like distribution function. When the window function g(λ) is a plural function representing an ellipse, the quasi-distribution function is the Ellipse distribution function; and finally when the plural function g(λ) is supplemented with the additional condition α=0, it will result in the standard ordering, anti-standard ordering distribution function and Wigner function. In this case g(λ) is a function depieting a rectangular window with width λ and height 1/12π?.

Acta Physica SinicaJan 1, 20061 citation

Overview

Yupeng Wang, Wen‐Li Yang, Junpeng Cao +1 more

Springer eBooksJan 1, 20150 citations

Methods

Pierre Bouillot

Springer thesesDec 13, 20120 citations

Determination of a Wave Function Functional

Xiao‐Yin Pan, Viraht Sahni, Lou Massa

We propose expanding the space of variations in traditional variational calculations for the energy by considering the wave function $\ensuremath{\psi}$ to be a functional of a set of functions $\ensuremath{\chi}\ensuremath{\mathbin:}\ensuremath{\psi}=\ensuremath{\psi}[\ensuremath{\chi}]$, rather than a function. A constrained search in a subspace over all functions $\ensuremath{\chi}$ such that the functional $\ensuremath{\psi}[\ensuremath{\chi}]$ satisfies a sum rule or leads to a physical observable is then performed. An upper bound to the energy is subsequently obtained by variational minimization. The rigorous construction of such a constrained-search--variational wave function functional is demonstrated.We propose expanding the space of variations in traditional variational calculations for the energy by considering the wave function $\ensuremath{\psi}$ to be a functional of a set of functions $\ensuremath{\chi}\ensuremath{\mathbin:}\ensuremath{\psi}=\ensuremath{\psi}[\ensuremath{\chi}]$, rather than a function. A constrained search in a subspace over all functions $\ensuremath{\chi}$ such that the functional $\ensuremath{\psi}[\ensuremath{\chi}]$ satisfies a sum rule or leads to a physical observable is then performed. An upper bound to the energy is subsequently obtained by variational minimization. The rigorous construction of such a constrained-search--variational wave function functional is demonstrated.

arXiv:physics/0402066Physical Review LettersSep 21, 200420 citations

h^d

denote the standard

d

-fold composition of the base function

h

. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function

f:\{0,1\}^n\to \{0,1\}

is a recursive function of the form

h^d

, with

h

being any base function and

d= Ω(\log\log n)

. (II) The inner function is a recursive function of the form

h^d

, with

h

being any constant arity base function (other than AND and OR) and

d= Ω(\log\log n)

, where

n

is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.