Questions tagged [boolean-complexity]
The boolean-complexity tag has no summary.
24 questions
0
votes
1
answer
85
views
3-CNF to 2-CNF reduction
I want to start by saying that I've struggled to find any satisfying answer to this question of mine. I did read this question, but it's slightly different.
My idea is simply that every 3-cnf formula ...
- 101
0
votes
0
answers
34
views
Identifying the DMAC Literals Representing Internal Processes (Message Schedule, T1, T2, etc.) in SHA-256 CNF Conversion Using CGen Tool
When using the CGen tool to convert the SHA-256 algorithm into Conjunctive Normal Form (CNF), the input and output bits are represented by DMAC literals. However, I am specifically interested in ...
0
votes
1
answer
73
views
Relationship between Formula Complexity and Depth Complexity of A Boolean Function
A (de Morgan) formula $\phi$ is a rooted binary tree, whose leaves are identified with
literals of the forms $x_i$ and $\neg x_i$, and whose internal vertices are labeled as AND ($\land$) or OR ($\lor$...
- 651
2
votes
1
answer
68
views
Convert the language into CNF or DNF form
A boolean circuit C has n inputs and m outputs, and is constructed with AND, OR, and NOT gates. Each gate has fan-in 2 except the NOT gate which has fan-in 1. The out-degree can be any number. A ...
- 97
0
votes
0
answers
73
views
Boolean circuit with true value
Let
$$L=\left\{\,\langle\,B_n,\, \,x\,\rangle:\enspace\substack{B_n \text{ is a boolean circuit and }
\\x \in \{0, 1\}^n\text{such that }B_n(x) = 1}\right\}$$
I want to prove that $L$ is $\textbf{P}$-...
- 1
3
votes
0
answers
41
views
Communication complexity of testing balancedness of a Boolean polynomial
The problem I consider is the following: given the $2^n$ coefficients of a Boolean polynomial $f : \{0, 1\}^n \rightarrow \{0, 1\}$, determine if $f$ is balanced namely if the truth table of $f$ ...
1
vote
2
answers
333
views
Boolean circuits with fan-out of each gate is 2
I am following the book of Arora and Barak book.
We consider Boolean circuits as we do, Specifically, inner nodes are either AND, OR (both – fan-in 2), or NOT (fan-in 1) gates. The fan-out of each ...
user172436
2
votes
0
answers
90
views
Approximate the parity function in L1-norm
Consider the parity function $MOD_2(x) = x_1 \oplus \cdots \oplus x_n$ for $x \in \mathbb{F}_2^n$. I am concerned about the degree bounds for a real polynomial $f$ which approximates $MOD_2$ well in ...
- 21
2
votes
1
answer
128
views
Lower Bound on Parity of Boolean Functions
Let's say we have boolean functions $f_1, \cdots, f_n$, each of which operates on pairwise disjoint variables (i.e. the variables for each function are unique to that function). Then, how can we show ...
- 23
1
vote
1
answer
124
views
Algorithm design: Model redundancy in tests
I've run across an interesting problem at work that I'm not quite sure how to grapple.
Broadly, there is a suite of of $n$ tests to ensure the quality of a product. However, the tests are both time-...
- 11
1
vote
1
answer
324
views
How to construct a carry-lookahead adder of the optimal $O(n)$ size
Problem (TL;DR): I'd like to know how to construct a CLA adder that has $O(n)$ size and $O(\log n)$ depth using only fan-in 2 AND gates and XOR gates, as suggested in this answer and this answer.
...
- 31
0
votes
1
answer
235
views
Given a boolean circuit that computes a boolean function, can we always find an equivalent circuit with optimal size?
Let's say that we have a decision problem $P$. Let's also say that $I_n$ is the set of all instances of size $n$ that exist for this problem, and that its cardinality is finite.
There is a sequence of ...
- 145
1
vote
0
answers
36
views
Why is End-Of-The-Line defined in terms of "Arithmetic circuits" instead of "Boolean circuits"
The definition of PPAD (Polynomial parity arguments on directed graphs) revolves around the definition of "End-Of-The-Line"
An exponentially large polynomial-depth arithmetic circuit, $f$, ...
- 327
1
vote
1
answer
133
views
Acyclic boolean circuit (DAG)
If a function f has a while loop or for loop, can I compile this function into an ...
- 13
0
votes
1
answer
62
views
influence of neighourhood points
Im trying to understand the following question. Suppose $h,f:\{-1,1\}^n\rightarrow \{-1,1\}$ satisfy $\sum_x h(x)f(x)\leq 0.5$, then one can rewrite this as $\textsf{Pr}_x [h(x)=f(x)]\leq 3/4$. Can we ...
1
vote
0
answers
73
views
Formula for computing a specific Fourier coefficient of a boolean function
According to O'Donnell's book ``Analysis of Boolean Functions", in order to determine the Fourier coefficient of a boolean function $f$ on a subset $S$, we take an inner product of $\chi_S$ and $...
- 11
0
votes
1
answer
229
views
Is there a notation for boolean algebra complexity?
To represent complexity of an algorithm, Computer Scientist is used to using big-O notation.
How about complexity of boolean algebra?
Boolean algebra is commonly used in digital circuit design with ...
1
vote
1
answer
742
views
Time-complexity of evaluating a CNF formula
Given a Boolean formula over $n$ variables in CNF and a partial assignment to it, all the algorithms I can think of to evaluate the assignment run in time $\Theta(n^2)$. Is it possible to do it in $O(...
- 259
1
vote
1
answer
212
views
How many different boolean functions exist up to permutation of its $n$ variables
i am relatively new here, so if this was asked before, feel free to redirect me. I am searching for an answer in form of a (iterative or recursive) Formula or even better, an algorithm to list them ...
- 13
1
vote
1
answer
129
views
Can Boolean circuits of polylog depth represent all Boolean functions?
Consider a Boolean circuit using (2-input) logical-and, (2-input) logical-or and logical-not as basic components. The depth of the Boolean circuit is the length of the longest path from the input to ...
- 298
3
votes
1
answer
1k
views
Shannon's result that some Boolean functions require exponential circuits
In 1949 Shannon proved, using a non-constructive counting argument, that some boolean functions have exponential circuit complexity, see [1] and many texts on computational complexity. This result has ...
- 8,368
2
votes
1
answer
117
views
Is $f(X)f^d(X) = 0$ for a Boolean function $f$?
I'm currently trying to understand a step in the proof for in the Crama and Hammer book on Boolean Functions. The proof is Proposition 4.12, which claims that the self-dualization of Boolean $f$ is ...
- 429
2
votes
1
answer
65
views
sum of Boolean characters larger degree
I was curious if someone knew the answer/reference for the following. So it is well-known that if $S\in \{0,1\}^n$, then
$$
\frac{1}{2^n}\sum_{x\in \{0,1\}^n} (-1)^{\langle S, x\rangle}=1
$$
if and ...
- 21
1
vote
1
answer
77
views
Boolean function represented as a column vector: easy way to see if it has full degree?
If you have a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ on $n$ variables, you can represent this function as a $2^n$ vector indexed by the input space such that $
f_x = f(x)$. For ...
- 991