Questions tagged [boolean-algebra]

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-2 votes
2 answers
76 views

Building a XOR gate out of NOR gates

Can someone explain for me how to build a XOR gate using 5 NOR gates? The thing I'm looking for is a proof similar to this: $A^B\ =\ (!A)B\ +\ A(!B)$ $=\ !!((!A)B)\ +\ !!(A(!B))$ $=\ !(!!A\ +\ !B)\ +\ ...
3 votes
1 answer
403 views

What is special about a canonical representation of Boolean functions?

My textbook (Saurabh's Introduction to VLSI Design Flow) mentions while discussing formal verification that a representation of a Boolean function is said to be canonical if the following holds: If a ...
0 votes
1 answer
74 views

Prove that it is impossible to construct the toffoli gate using only CNOT gates

Show that it is not possible to construct the toffoli gate using only CNOT gates, given we are allowed to choose any number of ancilla bits. My Attempt The action of a toffoli gate can be defined as, ...
3 votes
1 answer
31 views

Is boolean formula equivalence problem for 2-CNFs $\mathsf{coNP}$-hard?

The problem: Given two boolean formulas in 2-CNF, decide if they are equivalent. I know that the problem is $\mathsf{coNP}$-hard when at least one formula is in 3-CNF. However, the same proof of $\...
4 votes
2 answers
2k views

Do Karnaugh maps yield the simplest solution possible?

I'm learning to use a Karnaugh map, but I'm not sure if I obtained the simplest expression possible. Have a look at this example: Truth table (inputs are A B C; output is F): ...
5 votes
0 answers
160 views

Can minimal CNF contain clause longer than initial CNF?

Let $\Phi$ be a k-CNF and $\Phi_{min}$ be a minimal CNF (one that contains smallest amount of literal occurences) that is equal to $\Phi$. Can $\Phi_{min}$ contain a clause of size $m > k$? What I ...
1 vote
0 answers
58 views

Prove or disprove that the Quine-McCluskey method generates the circuit with the minimum inputs and minimum gates?

Recently, when I self-learnt Discrete Mathematics and Its Applications 8th by Kenneth Rosen, it says in 12.4 Minimization of Circuits which uses the Karnaugh Map or the Quine-McCluskey method: ...
1 vote
1 answer
86 views

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 ...
0 votes
3 answers
59 views

Is there an algorithm to implement N-input-gates using smaller gates?

To borrow part of a description from a similar but distinct question: there exist 2^(2^N) different functions which accept N binary inputs and return a 1 bit output. For the purposes of my question I ...
0 votes
1 answer
60 views

what is the smallest 3-CNF possible that enforces the boolean expression: a = b + c?

What is the smallest 3-CNF system of equations possible that enforces the boolean expression: a = b + c for boolean variables a, b, c? 'Smallest' can be defined as: (1) Number of 3 CNF clauses. (2) ...
1 vote
0 answers
22 views

CNF Horn-renamability to 3-CNF Horn-renamability reduction?

A CNF formula is Horn-renamable if you can invert variables in such a way that each clause has at most one positive literal. There is an algorithm based on a reduction to 2-SAT given in Renaming a Set ...
2 votes
1 answer
51 views

Number of n-variable symmetric boolean functions that are linear

How many symmetric boolean functions exist that are linear? Let $f$ be a $n$-variable boolean function. $f$ is said to be symmetric if it is unchanged by any permutation of its variables, i.e. for 2-...
0 votes
0 answers
30 views

Expressing Boolean Functions In Terms of Another Function

Given two boolean functions f1 and f2, are there any tools available that could be used to automate the process to represent f1 in terms of f2? I understand the process of this doing this by hand ...
6 votes
1 answer
709 views

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 ...
0 votes
2 answers
71 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 ...
7 votes
1 answer
723 views

Why do Karnaugh maps work?

The question is quite straightforward: Why do Karnaugh maps work? What was the reasoning that led Maurice Karnaugh to come up with these maps? At first glance, it doesn't seem a natural approach, ...
2 votes
0 answers
24 views

Equivalent-symmetric (E-symmetric) variables check for Boolean functions

In [1], it states that checking $f(..., x_i, ..., \bar{x}_j, ...) = f(..., \bar{x}_j, ..., x_i, ...)$ (variables $x_i$ and $x_j$ are E-symmetric) is equivalent to checking $f_{x_ix_j} = f_{\bar{x}_i\...
1 vote
1 answer
142 views

Why is conjunctive normal form (CNF) "better" for SAT than disjunctive normal form (DNF)?

When hand-manipulating algebra DNF (sum of products) is easier than CNF (product of sums). Possibly because factoring is more difficult than expanding. So why is it the opposite for computational ...
0 votes
0 answers
17 views

Complexity Class of the Problem: Existence of Unsatisfying Interpretations in Boolean Formulas

What is the complexity class of the problem if there exist two different interpretations that do not satisfy a given Boolean formula? I believe the problem of existence of an interpretation that does ...
0 votes
0 answers
34 views

SOP Boolean expression of state variable and output of a Moore FSM

Disclaimer: I'm studying for an exam, but I don't have to hand in anything. I have the state transition diagram of the FSM and its states encoding: I want to find the sum-of-products (SOP) Boolean ...
1 vote
1 answer
44 views

Why do simple Logical Gates have an Irrational amount of Bits?

Suppose $2$ bits are used to encode a message, A and B. If you know $A$ is $1$, you have one bit of information. If you know $A\land B$ is $1$, you have two bits of information. If you know $A\land B$...
-3 votes
2 answers
58 views

!(ab)*(!a+b)(!b+b)=!a Boolean Algebra

How can I prove that !(ab)(!a+b)(!b+b)=!a* in boolean algebra? This is an exercise from a past paper of my teacher but I can't really find the way to the answer. I would appreciate some help.
0 votes
1 answer
72 views

How to solve boolean SAT with equality constraints

Say I have boolean formula in form of a CNF(x1,x2,...) with $x_i$ being boolean variables. Testing the satisfiability of the CNF is the SAT problem, i.e. determine ...
5 votes
5 answers
982 views

Prove HAKMEM Item 23: connection between arithmetic operations and bitwise operations on integers

Prove that for $A, B \in \mathbb{Z}$, $A + B = (A \operatorname{\&} B) + (A \mid B) = (A \oplus B) + 2(A \operatorname{\&} B)$ where $\&$ is bitwise AND, $|$ is bitwise OR and $\oplus$ is ...
0 votes
0 answers
75 views

Is this a valid method to merge clauses in CNF formulas?

Assume $$C_1 = (x_1 \lor x_2 \lor \lnot x_3)\hspace{0.2cm} C_2 = (x_4 \lor x_5 \lor x_3) \hspace{0.2cm} C_3 = (x_3 \lor x_5 \lor x_6)$$ Let $$ \phi_1 = C_1 \land C_2 \land C_3$$ and $$ \phi_2 = (x_1 \...
20 votes
3 answers
6k views

Is it possible to write an AND gate using XOR gates?

How could I express an AND gate using only XOR gates ?
0 votes
0 answers
27 views

Is this language PSPACE complete

Prove $PSPACE$-completeness of the language $READALLEXACT$ = $\{$$(M, x, 1^ s , t)$ | $A$ deterministic Turing machine $M$ on input $x$ reads all bits of the input in exactly $t$ steps and using no ...
0 votes
1 answer
22 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 ...
0 votes
2 answers
92 views

is duality principle in boolean algebra is true for every expression

Let say A = 1 and B = 1 and then A+B = 1 now by using duality(replacing or gate by and gate and 1 by 0) we can say that, A.B = 0 but this is not 0, because 1.1 = 1, so please anyone clear my ...
0 votes
1 answer
43 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:
0 votes
2 answers
178 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.
1 vote
2 answers
240 views

For a logic gate to be universal, must it necessarily be able to perform duplication?

It is said that a gate that can simulate AND and NOT is universal and able to recreate any classical circuit. I was looking at some of the circuits simulated by NAND, and for some of them, we need to ...
1 vote
0 answers
88 views

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 ...
0 votes
2 answers
125 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 ...
0 votes
1 answer
83 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 ...
2 votes
1 answer
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)....
1 vote
1 answer
158 views

Sorting by boolean algebra (hardware) instead of algorithm (software)

Consider there's an 5 elements list that foreach element are 2-bits. Forexample [01,00,10,00,11], if the list is sorted, we hope the output like this [00,00,01,10,11] Maybe that case seems complicated,...
1 vote
2 answers
275 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 ...
0 votes
0 answers
107 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
1 answer
76 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
1 answer
55 views

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
0 answers
34 views

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
2 answers
245 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
1 answer
69 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 ...
1 vote
1 answer
22 views

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
2 answers
7k views

How do i simplify this SOP expression?

Hi i have derived the following SoP (Sum of Products) expression , by analyzing the truth table of a 3 bit , binary to gray code converter. I ask for verification, because i feel as though this answer ...
4 votes
2 answers
77 views

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-...
2 votes
1 answer
67 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
1 answer
56 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.
0 votes
1 answer
43 views

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.

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