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29
votes
2answers
6k views

What is the difference between quantum TM and nondetermistic TM?

I was going through the discussion on the question How to define quantum Turing machines? and I feel that quantum TM and nondetermistic TM are one and the same. The answers to the other question do ...
29
votes
1answer
16k views

Hash tables versus binary trees

When implementing a dictionary ('I want to look up customer data by their customer IDs'), the typical data structures used are hash tables and binary search trees. I know for instance that the C++ STL ...
29
votes
2answers
939 views

Equivalence of Büchi automata and linear $\mu$-calculus

It's a known fact that every LTL formula can be expressed by a Büchi $\omega$-automaton. But, apparently, Büchi automata are a more powerful, expressive model. I've heard somewhere that Büchi automata ...
29
votes
1answer
2k views

Rice's theorem for non-semantic properties

Rice's theorem tell us that the only semantic properties of Turing Machines (i.e. the properties of the function computed by the machine) that we can decide are the two trivial properties (i.e. always ...
28
votes
5answers
5k views

Boolean search explained

My mother is taking some online course in order to be a librarian of sorts, in this course they cover boolean searches, so they can search databases efficiently, however, she got a question sounding ...
28
votes
7answers
5k views

Are algorithms (and efficiency in general) getting less important?

Since buying computation power is much affordable than in the past, are the knowledge of algorithms and being efficient getting less important? It's clear that you would want to avoid an infinite loop,...
28
votes
2answers
4k views

Church-Turing Thesis and computational power of neural networks

The Church-Turing thesis states that everything that can physically be computed, can be computed on a Turing Machine. The paper "Analog computation via neural networks" (Siegelmannn and Sontag, ...
28
votes
4answers
4k views

What did Turing mean when saying that “machines cannot give rise to surprises” is due to a fallacy?

I encountered below statement by Alan M. Turing here: "The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are ...
28
votes
3answers
2k views

Why is Relativization a barrier?

When I was explaining the Baker-Gill-Solovay proof that there exists an oracle with which we can have, $\mathsf{P} = \mathsf{NP}$, and an oracle with which we can have $\mathsf{P} \neq \mathsf{NP}$ to ...
28
votes
2answers
4k views

Is Lambda Calculus purely syntactic?

I've been reading for a few weeks about the Lambda Calculus, but I have not yet seen anything that is materially distinct from existing mathematical functions, and I want to know whether it is just a ...
28
votes
3answers
3k views

What's really meant by context-free in the term context-free grammar?

I have been studying compilers for a while, and I have been searching what's meant by "context" in grammar and what it means for grammar to be "context-free", but with no result. So can anyone help ...
28
votes
4answers
13k views

Generalised 3SUM (k-SUM) problem?

The 3SUM problem tries to identify 3 integers $a,b,c$ from a set $S$ of size $n$ such that $a + b + c = 0$. It is conjectured that there is not better solution than quadratic, i.e. $\mathcal{o}(n^2)$....
28
votes
2answers
5k views

Why are there more non-computable functions than computable ones?

I'm currently reading a book in algorithms and complexity. At the moment I'm, reading about computable and non-computable functions, and my book states that there are many more functions that are non-...
28
votes
2answers
4k views

Why are the total functions not enumerable?

We learned about the concept of enumerations of functions. In practice, they correspond to programming languages. In a passing remark, the professor mentioned that the class of all total functions (i....
28
votes
4answers
5k views

How to determine likely connections in a social network?

I am curious in determining an approach to tackling a "suggested friends" algorithm. Facebook has a feature in which it will recommended individuals to you which it thinks you may be acquainted with. ...
28
votes
2answers
581 views

Characterization of lambda-terms that have union types

Many textbooks cover intersection types in the lambda-calculus. The typing rules for intersection can be defined as follows (on top of the simply typed lambda-calculus with subtyping): $$ \dfrac{\...
28
votes
3answers
2k views

Do subqueries add expressive power to SQL queries?

Does SQL need subqueries? Imagine a sufficiently generalized implementation of the structured query language for relation databases. Since the structure of the canonical SQL ...
28
votes
2answers
4k views

Where to get graphs to test my search algorithms against?

I am implementing a set of path finding algorithms such as Dijkstra's, Depth First, etc. At first I used a couple of self made graphs, but now I'd like to take the challenge a bit further and thus I'...
28
votes
1answer
10k views

How hard is counting the number of simple paths between two nodes in a directed graph?

There is an easy polynomial algorithm to decide whether there is a path between two nodes in a directed graph (just do a routine graph traversal with, say, depth-first-search). However it seems that, ...
28
votes
2answers
652 views

Why do neural networks seem to perform better with restrictions placed on their topology?

Fully connected (at least layer to layer with more than 2 hidden layers) backprop networks are universal learners. Unfortunately, they are often slow to learn and tend to over-fit or have awkward ...
28
votes
1answer
686 views

Subset sum problem with many divisibility conditions

Let $S$ be a set of natural numbers. We consider $S$ under the divisibility partial order, i.e. $s_1 \leq s_2 \iff s_1 \mid s_2$. Let $\qquad \displaystyle \alpha(S) = \max \{|V| \mid V\subseteq S, ...
28
votes
2answers
563 views

How to efficiently determine whether a given ladder is valid?

At my local squash club, there is a ladder which works as follows. At the beginning of the season we construct a table with the name of each member of the club on a separate line. We then write the ...
27
votes
7answers
6k views

What are the simplest examples of programs that we do not know whether they terminate?

The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not. ...
27
votes
6answers
8k views

Are there programs that can 'translate' source code between any two languages?

Are there programs that can 'translate' source code between any two languages (assuming the translator has access to the requisite libraries)? If there are, how do they work (techniques used, ...
27
votes
2answers
6k views

How do I construct reductions between problems to prove a problem is NP-complete?

I am taking a complexity course and I am having trouble with coming up with reductions between NPC problems. How can I find reductions between problems? Is there a general trick that I can use? How ...
27
votes
3answers
6k views

What happens when an internet connection is faster than the storage write speed?

If one attempted to download a file at a speed of 800 Mb/s (100 MB/s) onto a hard drive with a write speed of 500 Mb/s (62.5 MB/s), what would happen? Would the system cap the download speed?
27
votes
2answers
4k views

What is a batch compiler?

I have the following quotation from my compiler's course (in the context of graph coloring): Because it is slow, graph coloring tends to be used in batch compilers, while linear scan tends to be ...
27
votes
2answers
3k views

Why is C's void type not analogous to the empty/bottom type?

Wikipedia as well as other sources that I have found list C's void type as a unit type as opposed to an empty type. I find this confusing as it seems to me that <...
27
votes
3answers
58k views

Why is selection sort faster than bubble sort?

It is written on Wikipedia that "... selection sort almost always outperforms bubble sort and gnome sort." Can anybody please explain to me why is selection sort considered faster than bubble sort ...
27
votes
4answers
4k views

Clear, intuitive derivation of the fixed-point combinator (Y combinator)?

The fixed-point combinator FIX (aka the Y combinator) in the (untyped) lambda calculus ($\lambda$) is defined as: FIX $\triangleq \lambda f.(\lambda x. f~(\lambda y. x~x~y))~(\lambda x. f~(\lambda y. ...
27
votes
2answers
10k views

Counting binary trees

(I'm a student with some mathematical background and I'd like to know how to count the number of a specific kind of binary trees.) Looking at Wikipedia page for Binary Trees, I've noticed this ...
27
votes
6answers
8k views

Generating Combinations from a set of pairs without repetition of elements

I have a set of pairs. Each pair is of the form (x,y) such that x,y belong to integers from the range [0,n). So, if the n is 4, then I have the following pairs: <...
27
votes
2answers
25k views

How to prove greedy algorithm is correct

I have a greedy algorithm that I suspect might be correct, but I'm not sure. How do I check whether it is correct? What are the techniques to use for proving a greedy algorithm correct? Are there ...
27
votes
4answers
69k views

What exactly is the difference between supervised and unsupervised learning?

I am trying to understand clustering methods. What I I think I understood: In supervised learning, the categories/labels data is assigned to are known before computation. So, the labels, classes or ...
27
votes
4answers
2k views

How to find a superstar in linear time?

Consider directed graphs. We call a node $v$ superstar if and only if no other node can be reached from it, but all other nodes have an edge to $v$. Formally: $\qquad \displaystyle $v$ \text{ ...
27
votes
1answer
28k views

Which machine learning algorithms can be used for time series forecasts?

Currently I am playing around with time series forecasts (specifically for Forex). I have seen some scientific papers about echo state networks which are applied to Forex forecast. Are there other ...
27
votes
2answers
3k views

What does 'true concurrency' mean?

I often hear phrases like 'true concurrency semantics' and 'true concurrency equivalences' without any references. What does those terms mean and why are they important? What are some examples of ...
27
votes
3answers
629 views

Measuring the difficulty of SAT instances

Given an instance of SAT, I would like to be able to estimate how difficult it will be to solve the instance. One way is to run existing solvers, but that kind of defeats the purpose of estimating ...
27
votes
1answer
8k views

Which combinations of pre-, post- and in-order sequentialisation are unique?

We know post-order, post L(x) => [x] post N(x,l,r) => (post l) ++ (post r) ++ [x] and pre-order ...
27
votes
2answers
765 views

Are there any specific problems known to be undecidable for reasons other than diagonalization, self-reference, or reducibility?

Every undecidable problem that I know of falls into one of the following categories: Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the ...
27
votes
1answer
1k views

Is there a 'string stack' data structure that supports these string operations?

I'm looking for a data structure that stores a set of strings over a character set $\Sigma$, capable of performing the following operations. We denote $\mathcal{D}(S)$ as the data structure storing ...
27
votes
1answer
879 views

Asymptotics of the number of words in a regular language of given length

For a regular language $L$, let $c_n(L)$ be the number of words in $L$ of length $n$. Using Jordan canonical form (applied to the unannotated transition matrix of some DFA for $L$), one can show that ...
27
votes
0answers
765 views

Graph problem known to be $NP$-complete only under Cook reduction

The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a ...
26
votes
12answers
8k views

Why is overfitting bad?

I've studied this lots, and they say overfitting the actions in machine learning is bad, yet our neurons do become very strong and find the best actions/senses that we go by or avoid, plus can be de-...
26
votes
6answers
11k views

Efficient compression of simple binary data

I have a file containing ordered binary numbers from $0$ to $2^n - 1$: 0000000000 0000000001 0000000010 0000000011 0000000100 ... 1111111111 7z did not compress ...
26
votes
7answers
7k views

Are all turing complete languages interchangeable

Note, while I know how to program, I'm quite a beginner at CS theory. According to this answer Turing completeness is an abstract concept of computability. If a language is Turing complete, then ...
26
votes
6answers
5k views

Is there a physical analogy to the Turing Machine?

Recently in my CS class I've been introduced to the Turing Machine. After the class, I spent over 2 hours trying to figure out what is the relationship between a tape and a machine. I was ...
26
votes
4answers
19k views

Is Morse Code binary, ternary or quinary?

I am reading the book: "Code: The Hidden Language of Computer Hardware and Software" and in Chapter 2 author says: Morse code is said to be a binary (literally meaning two by two) code because ...
26
votes
4answers
27k views

The time complexity of finding the diameter of a graph

What is the time complexity of finding the diameter of a graph $G=(V,E)$? ${O}(|V|^2)$ ${O}(|V|^2+|V| \cdot |E|)$ ${O}(|V|^2\cdot |E|)$ ${O}(|V|\cdot |E|^2)$ The diameter of a ...
26
votes
5answers
34k views

Difference between a turing machine and a finite state machine?

I am doing a presentation about Turing machines and I wanted to give some background on FSM's before introducing Turing Machines. Problem is, I really don't know what is VERY different from one ...

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