Mathematics

Questions tagged [boolean]

For questions related to Boolean function (whose arguments and result assume values from a two-element set).

In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {$\text{true}, \text{false}$}, {$0,1$} or {$-1,1$}). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory.

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how many semantically different boolean functions are there for n boolean variables?

In short, this is an assignment question for a course I am taking - the exact wording is this: "Given n Boolean variables, how many 'semantically' different Boolean functions can you construct?" Now, I had a crack at this myself - and got pretty…
Zack Newsham
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Are all monotone Boolean functions weighted threshold functions?

$\DeclareMathOperator{\th}{th}$ A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is monotone if for all $a_1,\dots,a_n,a'_1,\dots,a'_n$ , if $a_i \leq a'_i$ for all $1 \leq i \leq n$ then $f(a_1,\dots,a_n) \leq f(a'_1,\dots,a'_n)$ . Some obvious…
mathperson314
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Symmetries of truth functions

The symmetry group of a given $n$ -ary truth function $f:\{0,1\}^n\to\{0,1\}$ is the permutation group consisting of all the $f$ -preserving permutations of the $n$ input variable names: $\{\sigma\in S_n:f^\sigma=f\}$ , where…
Karl
  • 13,505
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Understanding which functions on $\{0,1\}^n$ are non-boolean

I recently came across this paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear…
Paul
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$\{0,1\}$-solutions for integer equations via lattice base reduction?

I would like to find $\{0,1\}$-solutions of a system of equations of the form $$\left\{\begin{array}{c}\sum_{i\in I_1}x_i=1\\\sum_{i\in I_2}x_i=1\\\vdots\\\sum_{i\in I_k}x_i=1\end{array}\right.$$ where $x_1,\ldots,x_n\in\{0,1\}$ are the variables…
user1111929
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Minimal number of binary operators to generate an arbitrary truth table

Considering: $$G(X,A,B)=(X∧A)∨(¬X∧B)$$ For all boolean unary operator $F(X)$ , there exists a unique pair of $A,B$ so that $G(X,A,B)=F(X)$ for all $X$ . That is, by assigning correct $A,B$ , function $G$ can generate any unary boolean operator.…
Xiaoyou Zhai
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Is there a Turing complete cellular automaton buidable from XOR and 1?

I recently learned in my last question that rule 110, while being Turing-complete, can be built from a set of falsity-preserving functions (such as $\{\lor, \oplus\}$ ) which isn't functionally complete . I am now wondering, is there a Turing…
Lysandre Terrisse
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Boolean function recognition by circuit family

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 gate is 2. The size of a circuit is the number of inner nodes (gates) in it. We know that any Boolean…
user1326505
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Applying De Morgans on Product of Sums

I want to write the Product of Sums from of the boolean function $$F(A, B, C) = (A*B) + (A'*A) + (B'*C)$$ I can cancel out $(A'*A)$ and then I'm wondering how I can apply De Morgans law on $$(AB)\cup (B'C)$$ Is it even valid? I just need to get it…
user832075
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Proof-Theoretic Advantages to Using Only NANDs in Infinitary Logics

This question comes out of a question on Philosophy Stack Exchange, and a particular difference of opinion in regards to the initial question stated in 'Is there any major benefit to using NAND in infinitary logic?' . The question revolves around an…
J D
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Finite Commutative ring with 100 elements where $x^2=x$?

Does there exist a finite Commutative ring with 100 elements where $x^2=x$ for every $x\in R$ ? I know finite Boolean rings has the property this property but they have cardinality $2^n$ , for some $n$ . Also Boolean rings are rings with identity.…
Learner
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Convert list of Boolean functions into logic circuit

Assume that I have a list of variables $x_1,...,x_n$ and a list of Boolean functions $f_1(x_1,...,x_n),...,f_m(x_1,...,x_n)$ What I would like to do is create a circuit gate graph $G=(V,E)$ which satisfies: if $x\in V$ then there must be one of the…
Shore
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Name for logical algebra in which $A \lor B = A + B - AB$?

An answer from 5 years ago provides simple rules for converting logical propositions to ordinary elementary algebraic expressions: $$\lnot{A} = 1 - A$$ $$A \land B = AB$$ $$A \lor B = A + B - AB$$ Where $A$ and $B$ can only take on values 1 for…
Mkanders
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construct the circuit which checks whether the $j$-th bit of the sum of given binary numbers $k$ and $m$ is 1

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 circuit is not allowed to have any cycles. For example…
Monte_carlo
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Given the output frequencies create a truth table with minimal boolean function

I have a black box with eight inputs and three outputs and I would like to write minimized boolean functions for them. I know how to simplify the boolean expressions given a truth table, but I don't have a truth table to start with. I only know how…
Andrea Marenco
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