# Questions tagged [boolean-algebra]

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### Representing binary functions with a finite gate set without exponential blow-up?

It is pretty well taught that any binary function can be represented using CNF. But conversion to CNF can take an exponential number of gates. The truth table is exponentially sized relative to the ...
58 views

### Polynomial representations of Boolean functions

The AND boolean function $AND(x)$ can be represented using the polynomial $P(x) = x_1x_2\cdots x_n$. I have a few questions: Is there a similar polynomial for the PARITY boolean function? Is there a ...
1 vote
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1 vote
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### Boolean Integer Linear Optimization/Programming

Trying to solve an ILP optimization problem with a number of potential boolean variables and then express constraints on these variables based on those boolean results. Let's say I am doing 5 coin ...
35 views

### 7 segment decoder Combinational Logic Circuits using Logic Gates

How can a 7 segment decoder operation be implemented using boolean NOR gate ONLY? Question: Here's my truth table and k-map: Normal circuit diagram:
119 views

### Two's complement using logic gates

How can a 4-bit two's complement operation be implemented using boolean NOR gate? I search lots of 4-bit two's complement videos and articals, but most of them are using XOR gate.
24 views

### Reed-Muller Form

I'm having trouble with the Reed-Muller Form, I'm using this notation to make it easier: + = or, * = and, ^ = xor. This is the given expression (a + !b) * (b + !c). a + !b = a + (b ^ 1) = a ^ b ^ 1 ^ ...
20 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
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### Method for simplifying complex logic tables

Not sure if this is the right StackExchange site, but back in college (20 years ago) I took a Digital Systems Design class where we learned how to reverse engineer a boolean function to meet the ...
86 views

### Boolean Logic when one component switches from 0 to 1

I recently was constructing boolean logic for all sorts of examples from Morris Mano's "Digital Logic and Circuit Design". I noticed that it is possible to construct a boolean logic wrt the ...
54 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 ...
1k views

### Why can't 3-SAT be solved efficiently if you convert all clauses (x ∨ y ∨ z) into (u ∨ z) by introducing a variable?

Let $a_i$, $b_i$, etc., be a literal, i.e., a variable or the negation of a variable. 3-SAT concerns formulas in CNF form: $(a_1 \vee a_2 \vee a_3) \wedge \dots \wedge (b_1 \vee b_2 \vee b_3)$ (3-CNF)....
95 views

### Time complexity of negating a CNF formula

Suppose i have a CNF formula. If i negate the CNF formula, then i obtain DNF formula. However, i can't find anywhere on internet that mention the time complexity. What is the time complexity of ...
1 vote
185 views

### Combinational logic check if bits is prime

I wonder if there's Digital Logic Circuit (using combinatorial logic gates) that check if number is prime or not. For example given input fixed 8-bit that will produce 1-bit output. 00000101 will ...
1 vote
58 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 ...
1 vote
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### Why do we use {+1, -1} in place of {0, 1} for the Fourier analysis of boolean functions?

I want to know what will change if we keep on using {0,1} for our Fourier analysis of boolean functions? What are the things, which can not be performed with {0,1} and can be done with {+1, -1}?
1 vote
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### Fourier Dimension of Boolean functions

I was recently reading about Fourier dimension of Boolean functions. What I understand is that if we take the Fourier expansion of $f: \{\pm1\}^n \to \{\pm1\}$ and consider the monomials with non zero ...
1 vote
230 views

### Clarification regarding linear boolean functions!

I am a little confused when it comes to linear boolean functions. According to this post: What is a simple way of explaining what a linear boolean function means in boolean algebra and relating it to ...
1 vote
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### How to come up with combination a short-circuit evaluation table?

(a || b) || (c && d)) Given the above, how do I derive the table below: a b c d output T - - - TRUE F T - - TRUE F F T T TRUE F F T F FALSE F F F - FALSE I'm told that this is short ...
1 vote
58 views

### Influence of a variable in composition of Boolean functions

Suppose $f$ and $g$ are Boolean functions without a constant term, and where every variable has the same influence. How to show every variable will have the same influence in $f \circ g$? To me it ...
63 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 ...
1 vote
54 views

### Is [F(a, b, c) = a' + b] functionally or logically complete?

I'm having a problem determining whether [F(a, b, c) = a' + b] is functionally(logically) complete or not. I would really appreciate it if you could help me. P.S: I can't have 1 or 0 as input.
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### Can element occur in a CNF formula?

For example, is $(X \vee 1)$ a valid formula in conjunctive normal form (CNF)? If yes, then I would have to consider such formulas when trying to prove a statement about all CNF formulas.
1 vote
30 views

### Is there an algorithm for generating non comparable boolean vectors?

First some context: A Boolean Network of $n$ components is a function $f$ from the set $\{0,1\}^n$ (set of vectors of $n$ components whose values are 0 or 1) to itself. The dynamical behavior of a ...
25 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 ...
33 views

### Is there an efficient way to generate a pseudo-boolean function from a linear constraint?

I would like to define a pseudo-boolean function $f$ such that $f(x) = 0$ for all logically valid combinations of $x\in{0,1}$ and $f(x) > 0$ for all logically invalid combinations of $x\in{0,1}$. ...
36 views

### graph representation of a Boolean function

I'm trying to classify a certain family of Boolean functions, and need to represent the function as a graph. Is there any well-known graph representation for a Boolean function that captures the ...
69 views

### Stalmarck's method: x ≡ x → z, does z have to be true?

I have been researching Ståmarck's method 1. In the paper cited here, some rules are given. Rules are made of triplets (x, y, z) such that: y $\to$ z $\equiv$ x where x, y and z are booleans which ...
1 vote
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### Is there a known relationship between Kolmogorov Complexity of a binary string and the logic optimization of the corresponding Boolean function?

I haven't thought about how to go about proving it or finding a counterexample (I probably don't have the right background), but it seems intuitive to me that, given some representation of a Boolean ...
31 views

### Can anyone solve this? Is the answer 4 or 7. I'm confused

I'm trying to solve this but I'm confused with different answers. I'm getting 4 but the answer written is 7. Please guide me.
1 vote
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### Four Queens Problem to a Conjunctive Normal Form

Given a chessboard with 4 rows and 4 columns (4x4) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Assign a Boolean variable to each cell of the board as below (1, 2, 3, etc. are variable names) If a ...
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### Compact representation for quantified boolean formula

I got black-box (too big to analyze) boolean formula f(...) with 3 sets of input arguments: $x_1... x_i, y_1... y_j, z_1... z_k$. And I want to find such values for x-arguments that for every y-...
327 views

### What are the limits of Boolean Algebra?

Any decision problem algorithm can be represented as a boolean expression. The rules of boolean algebra (De Morgan's law, distributivity, etc.) can be used to manipulate and simplify that expression, ...
1 vote