scholarly journals A Measure in Which Boolean Negation is Exponentially Powerful

1982 ◽  
Vol 11 (154) ◽  
Author(s):  
Sven Skyum
Keyword(s):  
Complexity Theory ◽  
Boolean Functions ◽  
Combinatorial Complexity ◽  
Long Time

The power of negation in combinatorial complexity theory has for a long time been an intriguing question. In this paper we demonstrate that in the more restricted setting of projections among families of Boolean functions, negation can be exponentially powerful.

scholarly journals CATEGORICAL COMPLEXITY

2020 ◽  
Vol 8 ◽  
Author(s):  
SAUGATA BASU ◽  
UMUT ISIK
Keyword(s):  
Complexity Theory ◽  
Boolean Functions ◽  
Circuit Complexity ◽  
Polynomial Rings ◽  
Common Interest ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Complexity Classes ◽  
Graded Algebras ◽  
Projective Varieties

We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes.

Affine equivalence of monomial rotation symmetric Boolean functions: A Pólya’s theorem approach

2016 ◽  
Vol 10 (3-4) ◽  
Author(s):  
Thomas W. Cusick ◽  
K. V. Lakshmy ◽  
M. Sethumadhavan
Keyword(s):  
Boolean Functions ◽  
Affine Transformation ◽  
Equivalence Classes ◽  
Affine Equivalence ◽  
Symmetric Boolean Functions ◽  
Rotation Symmetric ◽  
Long Time ◽  
Input Variables ◽  
Large Groups ◽  
Rotation Symmetric Boolean Functions

AbstractTwo Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

scholarly journals On a hierarchy of Boolean functions hard to compute in constant depth

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Anna Bernasconi
Keyword(s):  
Harmonic Analysis ◽  
Complexity Theory ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Combinatorial Property ◽  
Constant Depth ◽  
Models Of Computation ◽  
Mathematical Properties ◽  
International Audience ◽  
Mathematical Techniques

International audience Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.

A Theory of Learning for the Creation and Management of Knowledge in Learning Communities and Organizations

2013 ◽  
pp. 28-44
Author(s):  
Ton Jörg
Keyword(s):  
Social Sciences ◽  
Learning Communities ◽  
Complexity Theory ◽  
Theory Of Change ◽  
New Paradigm ◽  
The Past ◽  
Communities Of Learners ◽  
Educational Crisis ◽  
Fundamental Challenge ◽  
Long Time

Social sciences have been in crisis for a long time, partly by being the captive of the Newtonian paradigm, and partly through the effects of this paradigm on practice. This crisis was recognized in the past by the Russian psychologist and philosopher Lev Vygotsky, and continues to this day. The educational crisis is just one instance. It is hard to imagine how to escape this crisis, and a real shift of paradigm is needed. In this article, such a shift toward the paradigm of complexity is advocated. The shift implies a reframing of complexity and a new kind of thinking in complexity. The new paradigm implies the development of a causally generative complexity theory of change and development. Ultimately, the fundamental challenge is to harness the complexity of complex, generative learning in the communities of learners in learning organizations.

A Theory of Learning for the Creation and Management of Knowledge in Learning Communities and Organizations

2010 ◽  
Vol 1 (1) ◽  
pp. 27-42 ◽  
Author(s):  
Ton Jörg
Keyword(s):  
Social Sciences ◽  
Learning Communities ◽  
Complexity Theory ◽  
Theory Of Change ◽  
New Paradigm ◽  
The Past ◽  
Communities Of Learners ◽  
Educational Crisis ◽  
Fundamental Challenge ◽  
Long Time

Social sciences have been in crisis for a long time, partly by being the captive of the Newtonian paradigm, and partly through the effects of this paradigm on practice. This crisis was recognized in the past by the Russian psychologist and philosopher Lev Vygotsky, and continues to this day. The educational crisis is just one instance. It is hard to imagine how to escape this crisis, and a real shift of paradigm is needed. In this article, such a shift toward the paradigm of complexity is advocated. The shift implies a reframing of complexity and a new kind of thinking in complexity. The new paradigm implies the development of a causally generative complexity theory of change and development. Ultimately, the fundamental challenge is to harness the complexity of complex, generative learning in the communities of learners in learning organizations.

A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

2008 ◽  
Vol 8 (10) ◽  
pp. 943-950
Author(s):  
R. de Wolf
Keyword(s):  
Complexity Theory ◽  
Boolean Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Quantum Algorithms ◽  
Minimal Degree ◽  
Close Connection ◽  
Worst Case

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree $deg_{\eps}(f)$ among all polynomials (over $\mathbb{R}$) that approximate a symmetric function $f:\01^n\rightarrow\01$ up to worst-case error $\eps$: $ deg_{\eps}(f)=\widetilde{\Theta}\left(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}\right).$ In this note we show how a tighter version (without the log-factors hidden in the $\widetilde{\Theta}$-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.

200kv superconducting cryo electron microscope

1987 ◽  
Vol 45 ◽  
pp. 642-643
Author(s):  
M. Iwatsuki ◽  
Y. Kokubo ◽  
Y. Harada ◽  
J. Lehman
Keyword(s):  
Electron Microscope ◽  
Structure Analysis ◽  
Organic Materials ◽  
Strong Interactions ◽  
Specimen Preparation ◽  
Great Effort ◽  
Electron Microscopic ◽  
Crystal Fields ◽  
Special Skill ◽  
Long Time

In recent years, the electron microscope has been significantly improved in resolution and we can obtain routinely atomic-level high resolution images without any special skill. With this improvement, the structure analysis of organic materials has become one of the interesting targets in the biological and polymer crystal fields.Up to now, X-ray structure analysis has been mainly used for such materials. With this method, however, great effort and a long time are required for specimen preparation because of the need for larger crystals. This method can analyze average crystal structure but is insufficient for interpreting it on the atomic or molecular level. The electron microscopic method for organic materials has not only the advantage of specimen preparation but also the capability of providing various information from extremely small specimen regions, using strong interactions between electrons and the substance. On the other hand, however, this strong interaction has a big disadvantage in high radiation damage.

A Stationary Solution of High-Energy Electron Reflection (HEER) for An Arbitrary Surface

1990 ◽  
Vol 48 (2) ◽  
pp. 374-375
Author(s):  
YIQUN MA
Keyword(s):  
Stationary Solution ◽  
High Energy ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
High Energy Electron ◽  
Bulk Materials ◽  
Periodic Surface ◽  
Electron Transmission ◽  
Electron Reflection ◽  
Long Time

For a long time, the development of dynamical theory for HEER has been stagnated for several reasons. Although the Bloch wave method is powerful for the understanding of physical insights of electron diffraction, particularly electron transmission diffraction, it is not readily available for the simulation of various surface imperfection in electron reflection diffraction since it is basically a method for bulk materials and perfect surface. When the multislice method due to Cowley & Moodie is used for electron reflection, the “edge effects” stand firmly in the way of reaching a stationary solution for HEER. The multislice method due to Maksym & Beeby is valid only for an 2-D periodic surface.Now, a method for solving stationary solution of HEER for an arbitrary surface is available, which is called the Edge Patching method in Multislice-Only mode (the EPMO method). The analytical basis for this method can be attributed to two important characters of HEER: 1) 2-D dependence of the wave fields and 2) the Picard iteractionlike character of multislice calculation due to Cowley and Moodie in the Bragg case.

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