All mathematical questions about computer science, including theoretical computer science, formal methods, verification, and artificial intelligence. For questions about Turing computability, please use the (computability) tag instead. For numerical analysis, use the (numerical-methods) tag. For ...

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0
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2answers
221 views

the strings of five decimal digitis

My question is : Consider strings of five decimal digits, such as 00147, or 99999. In each case below, what is the number of such strings satisfying the given property? (a) The string has no repeated ...
0
votes
1answer
66 views

How would I find a minimum weight spanning tree for W?

If I were to let $W$ be the weighted graph formed by taking a complete graph $K_5$ on five vertices 1, 2, 3, 4, 5 with the weight of each edge $\{x,y\}$ given by $(\{x,y\}) = x + y$, how would I find ...
0
votes
1answer
126 views

How to approach this Secret Sharing scheme?

Suppose that I want to break up a secret into shares such that any set of k people can recover the secret, but I’m also worried that some people might be dishonest and may lie about the secrets they ...
0
votes
1answer
244 views

Simultaneous recursion

I have no idea how to even start proving the following theorem: If $f_0, f_1: \mathbb{N}^r \rightarrow \mathbb{N}$ and $g_0, g_1: \mathbb{N}^{r+3} \rightarrow \mathbb{N}$ are primitive recursive, ...
0
votes
3answers
77 views

Induction to prove $2n + 3 < 2^n$

I am having trouble and was wondering if someone could go over the steps slowly to show that: $$2n + 3 < 2^n \ \text{for} \ n \geq 4$$ Any help would be amazing!
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2answers
271 views

Algorithms for solving the discrete logarithm $a^x \equiv b\pmod{n}$ when $\gcd(a,n) \neq 1$

The general discrete logarithm problem is to find $x$ given $a, b$ and $n$ such that $$a^x \equiv b\pmod{n}.$$ Normally one can use the "baby-steps giant-steps" algorithm to solve it fairly quickly. ...
0
votes
1answer
130 views

Determining computational complexity of stochastic processes

I have an program which implements a Markov chain Monte Carlo process on a system of N bits, stopping when the process converges. Let's use T to denote the average number of steps made by the Markov ...
0
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1answer
302 views

Comp Sci Math; Hamming Distance

I've been tasked with this question but I have no idea how to answer it. What is the maximum possible hamming distance between two points from level i in a n-cube?
28
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4answers
999 views

Why do we believe the Church-Turing Thesis?

The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to ...
31
votes
6answers
3k views

Simple “real life” NP-hard problems?

There are many proofs lying around that games like Lemmings or Sudoku or Tetris are NP-hard (generalized version of those games, of course). The proofs, as I recall, are not difficult but not simple ...
17
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2answers
2k views

Density of halting Turing machines

If we enumerate all Turing machines, $T_1$, $T_2$, $T_3,\ldots,T_n,\ldots$, What is $$\lim_{m\to\infty}\frac{\#\{k\mid k\lt m \text{ and }T_k\text{ halts}\}}{m}\quad?$$ Or does this depend on how we ...
17
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4answers
4k views

Number of ways to partition a rectangle into n sub-rectangles

How many ways can a rectangle be partitioned by either vertical or horizontal lines into n sub-rectangles? At first I thought it would be: ...
13
votes
4answers
9k views

Do dynamic programming and greedy algorithms solve the same type of problems?

I wonder if dynamic programming and greedy algorithms solve the same type of problems, either accurately or approximately? Specifically, As far as I know, the type of problems that dynamic ...
12
votes
3answers
802 views

Twenty questions against a liar

Here's one that popped into my mind when I was thinking about binary search. I'm thinking of an integer between 1 and n. You have to guess my number. You win as soon as you guess the correct number. ...
8
votes
4answers
540 views

Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...
6
votes
3answers
794 views

Is learning haskell a bad thing for a beginner mathematician?

Haskell is a programming language which uses some concepts from category theory like functor, monad, etc. My question is: Learning intuitive concepts about category from Haskell will ruin my intuition ...
5
votes
2answers
1k views

Learning Proofs (for Computer Science)

Harvard's math curriculum, for freshmen, is divided into 4 classes beyond the BC Calculus level, Math 21, 23, 25 and 55. Math 21 is your classic plug-and-chug multivariable calculus and linear algebra ...
3
votes
2answers
17k views

What is the 3SAT problem?

I don't get the 3SAT problem. Can someone explain the 3SAT problem as if I were 5 years old, ideally with examples? Thanks!
13
votes
1answer
14k views

Recognizable vs Decidable

What is difference between "recognizable" and "decidable" in context of Turing machines?
12
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5answers
1k views

Why is convexity more important than quasi-convexity in optimization?

In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have ...
10
votes
2answers
432 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) being used in computer science? Any research I have done on this topic leads me to some sort of applied math concept. I know that there ...
5
votes
0answers
57 views

Where can I learn more about the “else” operation / “else monoid”?

(The set of natural numbers $\mathbb{N}$ starts at $0$ for me.) Let $X$ denote a set, and define $X_\bot = X \uplus \{\bot\}.$ Let $\mathbf{else}$ denote the binary operation on $X_\bot$ defined as ...
27
votes
6answers
3k views

Is the set of all valid C++ programs countably infinite?

I have heard that the set of valid programs in a certain programming language is countably infinite. For instance, the set of all valid C++ programs is countably infinite. I don't understand why ...
14
votes
6answers
3k views

The Practical Implication of P vs NP Problem

Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it. Suppose that we can prove all questions that can be ...
11
votes
3answers
289 views

What is necessary to exchange messages between aliens? [closed]

Lets assume that two extreme intelligent species in the universe can exchange morse code messages for the first time. A can send messages to B and B to A, both have unlimited time, but they can not ...
7
votes
1answer
605 views

What is the best way to self-study GAP?

Background: This year I'll do another Group Theory course ( Open University M336 ). In the past I have used Mathematica's AbstractAlgebra package but (although visually appealing ) this is no longer ...
6
votes
1answer
418 views

Importance of Constructible functions

A function $f$ is called fully time-constructible if there exists a Turing machine $M$ which, given a string $1^n$ consisting of $n$ ones, stops after exactly $f(n)$ steps. Analogously, we can call a ...
5
votes
3answers
669 views

Are computers going to be able to discover and prove important mathematics theorems? [closed]

Are computers going to be able to discover and prove important mathematics theorems within, say, 20 years?
3
votes
0answers
77 views

Computability and continuous real functions

I have found somewhere the following statement: "Every computable real function has to be continuous," but I'm not able to prove it and the "proofs" that I found in some blog posts don't seem ...
5
votes
1answer
328 views

Matrix Chain Multiplication?

The following are questions about using dynamic programming for matrix chain multiplication. Pseudocode can be found in the Wikipedia article on matrix chain multiplication. 1) Why is the time ...
5
votes
1answer
7k views

Degeneracy in Linear Programming

Consider the standard form polyhedron, and assume that the rows of the matrix A are linearly independent. $$ \left \{ x | Ax = b, x \geq 0 \right \} $$ (a) Suppose that two different bases lead to ...
5
votes
5answers
707 views

Can a polynomial size CFG over large alphabet describe a language, where each terminal appears even number of times?

Can a CFG over large alphabet describe a language, where each terminal appears even number of times? If yes, would the Chomsky Normal Form be polynomial in |Σ| ? EDIT: What about a language where ...
4
votes
2answers
119 views

What is the algorithm hiding beneath the complexity in this paper?

So, I am a computer scientist (at least, I'm working to become one..) and I asked a question on here concerning some mathematics behind the Mandelbrot set. A reply I recieved pointed me to this paper. ...
4
votes
3answers
323 views

Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows: build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...
2
votes
1answer
105 views

Combinatorics/Task Dependency

Here is a competitive programming question: You have a number of chores to do. You can only do one chore at a time and some of them depend on others. Suppose you have four tasks to complete. For ...
2
votes
2answers
548 views

Are axioms and rules of inference interchangeable?

There is an equivalence between cellular automata and formal systems, you can code one into the other and vice versa. But in the the cellular automata (CA) the rules of inference are fixed and are ...
1
vote
3answers
97 views

Combination Problem Understanding

How many ways can a Doctor go to the Hospital on $5$ days of January (which has $31$ days) such that no two visits are on consecutive days? I think the solution is: $\displaystyle\binom{27}{5}$ But ...
0
votes
0answers
46 views

Can linear execution time be achieved [duplicate]

The SELECT algorithm determines the $i$th smallest of an input array of $n>1$ distinct elements by executing the following steps. Divide the $n$ elements of the input array into $\lfloor ...
13
votes
3answers
380 views

Etymology of “topological sorting”

This may be a dumb question, but what's "topological" about topological sorting in graph theory? I thought topology was related to geometry and deformations.
9
votes
5answers
478 views

Is there any mathematical operation on Integers that yields the same result as doing bitwise “AND”?

I'll provide a little bit of a background so you guys can better understand my question: Let's say I have two positive, non-zero Binary Numbers.(Which can, obviously, be mapped to integers) I will ...
8
votes
1answer
494 views

Throwing balls into $b$ buckets: when does some bucket overflow size $s$?

Suppose you throw balls one-by-one into $b$ buckets, uniformly at random. At what time does the size of some (any) bucket exceed size $s$? That is, consider the following random process. At each of ...
7
votes
0answers
186 views

How to maximize the number of operations in process

In my research project I have encountered the following problem, concerning a tuple of words in the formal language $L=\{0,1\}^*$, with $\epsilon$ denoting the empty word. If we are given an ordered ...
7
votes
2answers
916 views

Finding the 2,147,483,647th prime number

In computer science an array is indexed by an integer (int). Unlike in mathematics, the computer science integer (int) has a ...
4
votes
1answer
133 views

Why do we divide or multiply by 2 when converting binary?

Trying to understand the fundamentals of binary rather than just following steps, I wanted to know why do we multiply by 2 to convert a decimal (0.5, 0.25) to a binary and why do we divide by 2 when ...
4
votes
3answers
3k views

Finite automaton that recognizes the empty language $\emptyset$

Since the language $L = \emptyset$ is regular, there must be a finite automaton that recognizes it. However, I'm not exactly sure how one would be constructed. I feel like the answer is trivial. ...
4
votes
2answers
532 views

How to calculate π [duplicate]

Possible Duplicate: Simple numerical methods for calculating the digits of Pi How do people/computers calculate π? Im sure long ago, someone just took a measurement of the circumference of ...
4
votes
1answer
309 views

Reasoning the calculation of the Hilbert's distance

I'm not a mathematician, I'm a computer science student, and I'm attending to a course called Advanced Functional Programming. There's this homework where I need to implement the Hilbert R-tree data ...
4
votes
5answers
15k views

Merge Sort time complexity analysis

How can I prove that $T(n) = 2T(n/2) + n$ is $O(n \log n)$ ?
4
votes
4answers
9k views

Reduction from Hamiltonian cycle to Hamiltonian path

I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone ...
3
votes
1answer
108 views

Prove that Y is not context free.

2.42 Let $$Y = \{w\mid \text{$w=t_1\#t_2\#\cdots\#t_k$ for $k \ge 0$, each $t_i \in 1^*$, and $t_i \ne t_j$ whenever $i \ne j$}\}.$$ Here $\Sigma = \{1,\#\}$. Prove that $Y$ is not context free. ...