Computational complexity, a part of theoretical computer science.

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State-Space Complexity of RISK board game

I want to calculate the (state-space) complexity of the RISK board game. Online I found a thesis that outlines that complexity (page 34). Here is the summary: Let M denote the maximum number of ...
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22 views

Is complexity change when multiply with some number

The power operation with modulo operation $a^b \mod n$ has time complexity $O(\log n)$. what happen when it is multiplied with a number $p·a^b \mod n$, it can be read as $(p \mod n)(a^b \mod n)$. Is ...
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31 views

Understanding an algorithm

I want to understand the above algorithm. My solution says that the algorithm should return $0$ if $n$ is a prime or 1. Otherwise it returns the smallest (positive) non-trivial divisor. Lets ...
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1answer
36 views

Pumping lemma proof of $L = \{a^nb^m \mid 0\leq n<m\}$

Prove the following language is not regular using the pummping lemma $L = \{a^nb^m \mid 0\leq n<m\}$ I tried solving this problem what I don't think I was able to reach an accurate proof. But this ...
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1answer
27 views

Transform a k-CNF formulae to conjunctions of boolean literals

The question comes from Mehryar Mohri's Foundations of Machine Learning. In Example 2.5 the book transform a $k$-CNF formula to conjunctions of boolean literals, but I can't understand the trick in ...
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Theoretical computer science text for mathematician

I am a high school student, I know some basic programming in java,python and visual basic. I love combinatorics and I have encountered various cases in which I have found some problems are really ...
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1answer
35 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
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Determine $P_2 = f(0.7)$ when Neville's method is used to approximate $f(0.5)$

Let $f(x) = \ln(x + 1)$. Neville's method is used to approximate $f(0.5)$, giving the following table. $$x_0 = 0 - P_0 = 0$$ $$x_1 = 0.4 - P_1 = 2/8 - P_{0,1} = 3/5$$ $$x_2 = 0.7 - P_{2=?}- ...
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290 views

Finding maximum of array consisting of an exponential amount of positive integers

Consider an array of an exponential amount of positive integer numbers, let's say $$ x_1, x_2, \ldots, x_{2^k} $$ for some fixed positive integer $k$. The question is the following. What is the ...
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1answer
29 views

Trying to understand the math in a neuroscience article by Karl Friston

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertertable and refer to both physical and ...
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29 views

Analysis of algorithm about complexity

$n$ is $O(\log n)^{\log n}$ ? This is true or false, Give the reasons behind that ? I dont get understand about that $O(\log n)^{\log n}$.
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How to find the asymptotic behavior of these sums?

Let $$X(n) = \displaystyle\sum_{k=1}^{n}\dfrac{1}{k}.$$ $$Y(n) = \displaystyle\sum_{k=1}^{n}k^{1/k}.$$ $$Z(n) = \displaystyle\sum_{k=1}^{n}k^{k}.$$ For the first, I don't have a formal proof but I ...
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35 views

Complexity & Computability & Turing Problem [closed]

Suppose for example Program $\mathfrak{P}$ has number : $2^{46} 5^2 7^{37}-1$ we can ensure that this program has 1 labeled instruction (like goto). I have no idea about this sentence? why? any ...
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1answer
47 views

Computable Set & Function

we know that i read this sentence are true? can anyone say an example for following sentence? there are a non computable set A such that
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1answer
40 views

Logic & Computability Problem

i read this sentence in one exam that be false. anyone could say why? if predicate H(x) become false when a program with code r(x) halt on input l(x), then H be a computable predicate.
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2answers
71 views

Solving the equation $n\log n = 10^9$

This seems very basic (I guess my calculus needs brushing up). Is there a way to find n without a calculator in this one? $10^{9} = n\log(n)$ My Attempt (log is base 2 base on the book convention.) ...
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1answer
635 views

If an unary language exists in NPC then P=NP

I've a question regarding a theorem in Complexity Theory. It is said that if there exists an unary language in NPC then P=NP e.g if {1}* in NPC then the above is correct. It means that there exists ...
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1answer
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How long would it take to test all possible states of a 64x64x64 cube of bits?

Imagining a solid cube of $64 \times 64 \times 64$ bits (each of which can have exactly two states), how long would it take to test all possible states of one of these? Let's also assume we're using ...
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61 views

TAUTOLOGIES NP-Complete Condition

The decision problem TAUTOLOGIES is, Given $\forall x_1 \forall x_2 ... \forall x_n$ $\phi(x_1, x_2, ... x_n)$ a set of universally quantified Boolean variables and a Boolean formula ...
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13 views

(Algorithm complexity) Find properties of ≪ relation

I got a exercises related to algorithm complexity analysis as below ...
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23 views

Splitting a graph into two isomorphic parts

Say a graph $G$ has $2n$ vertices. I'd like to know if I can partition the vertices of $G$ into two parts $X$ and $Y$ such that $G[X]$ is isomorphic to $G[Y]$ ($G[S]$ denotes the subgraph of $G$ ...
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$P=NP \Rightarrow P(S)=NP(S)$?

If $P=NP$, then is it possible to conclude that for every sparse set $S$, $P(S)=NP(S)$? ($P(S)$ means a class of sets for which a Polynomial Deterministic Turing Decider with Oracle Set $S$ exists, ...
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Linear-bounded Turing Machines

What is the class that contains linear-bounded Turing machines? Is it possible to diagonalize on this class? And is there any Universal linear-bounded Turing machine that can simulate linear-bounded ...
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27 views

Hardness of bounded modular square root of 1

If we know any square root of 1 modulo N different from 1 and N-1, then we can find a nontrivial factor of N. So to find such a square root has a certain hardness. In fact, if in general we ask to ...
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math rules when having 2 variables in Big-O

I came across the following in some lecture notes: O(log n) + O(log m) = O(log n + log m ) = O(log (m + n)) that last step to ...
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33 views

Integer programming feasibility is NP-what

What is the complexity class of the general problem of integer programming feasibility? The sources I've looked at are, in my opinion, very confusing. Some say NP-hard, some say NP-complete. Some ...
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1answer
244 views

what is the relationship between the complexity class E(and EXP) and NP?

I want to know any relationship between the complexity class E(and EXP) and NP. I also would like to know whether there is any $DTIME$ formulation or relations of $NTIME(O(n^k))$ where n is the size ...
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what is the computational complexity of solving a quadratic program with linear inequality constraints

I'm aware of several solution methods and have several solvers at my disposal, but I can't for the life of me find analysis on the complexity. In particular, I'm interested in the complexity of ...
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19 views

computational complexity of $\sum_{k=1}^K {n\choose k}$

I would like to know what is the computational complexity of $n\choose k$ and $\sum_{k=1}^K {n\choose k}$ in terms of $\mathcal{O}$.
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1answer
31 views

How on earth will anyone prove $n^3-3n^2+n-1=Θ(n^3)$

I know its homework question.Sorry for that.But i was solving all problems of Skiena and chapter and got stuck to this problem of 2nd chapter 2.10. Its easy to prove $n^3-3n^2+n-1=O(n^3)$ because ...
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Recursive Set and Complement Problem

if we have $$A=\{x:|W_x\ne\phi\}$$ can we say always my tight listed below is true? $A$ is recursive , $A$ is r.e, complement of $A$ is r.e, complement of $A$ is not recursive?
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What's the complexity of expanding a general polynomial?

Suppose I have a polynomial in the form $(a_1 x_1+ a_2 x_2+...+ a_m x_m)^n$, where $x_1,...,x_m$ are the independent variables. I want to expand it to the form of sum of products. What is the ...
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L={(M,W) | M is a Turing Machine that stops on input W } is not R. E.

I've been thinking about how to show this but I'm stuck. on Computability, Complexity, and Languages, Second Edition: Fundamentals of Theoretical Computer Science (Computer Science and Scientific ...
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1answer
105 views

What is the simplest collatz like problem that is undecidable?

I have read that problems resemblings collatz have been shown to be undecidable. Conway proved that apparantly but Im not sure if the proof was constructive. So I wonder : What is the simplest ...
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Double sigma summation is in complexity calculation

Basically i was reading skiena and doing exercise of 2nd chapter.The result of my calculation got differed from the actual solution given on Solution site and there is one thing i don't understand how ...
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1answer
41 views

Polytime implementation of Discrete Log using primitive recursive functions

The primitive recursive functions are defined by Godel as: $z() = 0$ $s(x) = x+1$ $\pi_i(x_1, \dots, x_k) = x_i$ Plus closure under Composition: $h(x_1, \dots, x_m) = f(g_1(x_1, \dots, x_m), ...
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Confused with the definition of NP

In the blog post at: Concrete Nonsense I read: NP is the set of problems that can be solved by polynomial-time non-deterministic algorithms. An equivalent definition of NP is the set of ...
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Polynomial-time reduction and complexcity sets P and NP.

hello I am having difficulties to understand the topic of P,NP and Polynomial-time reduction. I have tried to search it on web and ask some of my friends , but i havent got any good answer . I wish ...
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25 views

Using the pumping lemma to prove that a certain language is not regular

I've been trying to understand the pumping lemma since forever, I just don't know what it does, I have no clue what any of it does. My college professor sucks, he thinks writing a bunch of stuff on a ...
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1answer
24 views

What is the time complexity of an $O((\ln n)^{\ln n})$ algorithm?

How can the time complexity of an $O((\ln n)^{\ln n})$ algorithm be simplified and compared to some other time complexities?
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1answer
58 views

Simple question on complexity

I just started to learn the complexity theory, and I have a simple question: Let's say that there's a language B in NP, such that there's no polynomial reduction from B to a given language A (which ...
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1answer
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Reduction between languages in P

I have a simple question about the class P: Is there exist a polynomial time reduction between every two languages A, B in P?
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266 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
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Complexity of and an algorithm for finding ideals of a ring?

One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R ...
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Generalization of Jacobi symbol for higher powers?

Let $n$ be an odd positive integer of unknown factorization, and let $x$ be relatively prime to $n$. The Jacobi symbol $\left(\frac{x}{n}\right)$ gives me partial information on whether $x$ is a ...
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1answer
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Trying to prove something in complexity

I just started to learn about complexity-theory, and I'm trying to prove this: If P=NP, then every (non-trivial) language in P is NP-complete. Can someone give me a solution please?
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Prove in complexity theory

Given a language A, which is in NP and also not NP-complete, I have to prove that P != NP. [Note: A is not trivial] Any suggestions?
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$O(n^{\log(n)}) $ time algorithms

Is $O(n^{\log(n)}) $ time algorithm considered of exponential time ? Is it applicable ?
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Help understanding Recursive algorithm question

We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$ For $n\geq0$, let $c(n)$ be the total number of additions for calculating $f(n)$ ...
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4k views

Converting to Chomsky Normal Form

I am trying to learn how to convert any context free grammar to Chomsky Normal Form. In the example below, I tried to apply Chomsky Normal Form logic, to result in a grammar, where every symbol either ...