Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > BOOLEAN FUNCTION:
Reports tagged with Boolean function:
TR96-022 | 15th March 1996
Martin Sauerhoff, Ingo Wegener, Ralph Werchner

#### Optimal Ordered Binary Decision Diagrams for Tree-like Circuits

Many Boolean functions have short representations by OBDDs (ordered
binary decision diagrams) if appropriate variable orderings are used.
For tree-like circuits, which may contain EXOR-gates, it is proved
that some depth first traversal leads to an optimal variable ordering.
Moreover, an optimal variable ordering and the resulting OBDD
can ... more >>>

TR00-006 | 26th January 2000
E.A. Okol'nishnikiva

#### On operations of geometrical projection and monotone extension

Some operations over Boolean functions are considered. It is shown that
the operation of the geometrical projection and the operation of the
monotone extension can increase the complexity of Boolean functions for
formulas in each finite basis, for switching networks, for branching
programs, and read-$k$-times ... more >>>

TR03-005 | 28th December 2002
Scott Aaronson

#### Quantum Certificate Complexity

Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation ... more >>>

TR06-158 | 8th December 2006
Gyula Gyôr

#### Representing Boolean OR function by quadratic polynomials modulo 6

We give an answer to the question of Barrington, Beigel and Rudich, asked in 1992, concerning the largest n such that the OR function of n variable can be weakly represented by a quadratic polynomial modulo 6. More specially,we show that no 11-variable quadratic polynomial exists that is congruent to ... more >>>

TR08-032 | 18th March 2008
Dmitriy Cherukhin

#### Lower Bounds for Boolean Circuits with Finite Depth and Arbitrary Gates

We consider bounded depth circuits over an arbitrary field $K$. If the field $K$ is finite, then we allow arbitrary gates $K^n\to K$. For instance, in the case of field $GF(2)$ we allow any Boolean gates. If the field $K$ is infinite, then we allow only polinomials.

For every fixed ... more >>>

TR11-130 | 25th September 2011
Sergei Lozhkin, Alexander Shiganov

#### On a Modification of Lupanov's Method with More Uniform Distribution of Fan-out

In this paper we suggest a modification of classical Lupanov's method [Lupanov1958]
that allows building circuits over the basis $\{\&,\vee,\neg\}$ for Boolean functions of $n$ variables with size at most
$$\frac{2^n}{n}\left(1+\frac{3\log n + O(1)}{n}\right),$$
and with more uniform distribution of outgoing arcs by circuit gates.

For almost all ... more >>>

TR12-072 | 5th June 2012
Anindya De, Ilias Diakonikolas, Vitaly Feldman, Rocco Servedio

#### Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces

The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 \cite{Chow:61, Tannenbaum:61} that the (exact values of the) Chow parameters of any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean ... more >>>

TR12-163 | 24th November 2012
Avishay Tal

#### Properties and Applications of Boolean Function Composition

For Boolean functions $f:\{0,1\}^n \to \{0,1\}$ and $g:\{0,1\}^m \to \{0,1\}$, the function composition of $f$ and $g$ denoted by $f\circ g : \{0,1\}^{nm} \to \{0,1\}$ is the value of $f$ on $n$ inputs, each of them is the calculation of $g$ on a distinct set of $m$ Boolean variables. Motivated ... more >>>

TR21-098 | 7th July 2021
Srikanth Srinivasan, S Venkitesh

#### On the Probabilistic Degree of an $n$-variate Boolean Function

Nisan and Szegedy (CC 1994) showed that any Boolean function $f:\{0,1\}^n\to\{0,1\}$ that depends on all its input variables, when represented as a real-valued multivariate polynomial $P(x_1,\ldots,x_n)$, has degree at least $\log n - O(\log \log n)$. This was improved to a tight $(\log n - O(1))$ bound by Chiarelli, Hatami ... more >>>

TR21-111 | 19th July 2021
Aniruddha Biswas, Palash Sarkar

#### Influence of a Set of Variables on a Boolean Function

Revisions: 2

The influence of a set of variables on a Boolean function has three separate definitions in the literature, the first due to Ben-Or and Linial (1989), the second due to Fischer et al. (2002) and Blais (2009) and the third due to Tal (2017). The goal of the present work ... more >>>

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