Questions tagged [notation]

Use this tag whenever you have a question about what some particular symbol/notation means.

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1answer
66 views

Exact meaning of $2^{\mathcal{O}(f(n))}$

In Sipser's Introduction to the Theory of Computation he uses the notation $2^{\mathcal{O}(f(n))}$ to denote some asymptotic running time. For example he says that the running time of a single-tape ...
3
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2answers
256 views

Weight functions in graph algorithms

In text books, for instance in the 3rd edition of Introduction to Algorithms, Cormen, on page 625, the weights of the edge set $E$ is defined with a weight function $w:E\rightarrow \mathbb{R}$. Why ...
3
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1answer
192 views

Properties of Reverse Polish Notation expressions that are algebraically invariant

The RPN expressions a b + c * and f d e + * are algebraically equivalent, though the names of the variables are different ...
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1answer
29 views

Asymptotic notation and random variables

I have two random variables $X$ and $Y$ and I want to bound the value of one in terms of the other (for now, I don't care about the actual distribution of their values). Suppose that the two ...
1
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1answer
61 views

Meaning of L* if L is a language

I can't find anywhere the meaning of $L^*$, given that $L$ is a language. I know $^* $ means repetition, for example $0^*$ = $\{ \epsilon, 0, 00, 000, \dots \}$. Or if $A$ is an alphabet $A^*$ are all ...
3
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3answers
56 views

Why is subarray $A[p..k-1]$ empty when $k=p$?

I'm working through a proof of correctness for merge sort. I'm given a loop invariant for a for loop, which makes reference to a subarray $A[p..k-1]$. During the initialization step of the ...
2
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1answer
22 views

When and what must be present on the left-hand side of the turnstile in metalogics?

Let me show the problem on an example... An actual task from one of the former exams: Consider a simple functional language: $$e::= x|n|e_1e_2|\lambda x.e$$ With typing rules: $$\...
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0answers
31 views

Substituting a term for a variable in a context

At this link you can read Nicola Gambino's slides on one way to approach the formal syntax of Martin-Löf dependent type theory. (They are concise and very readable.) On slide 10, he gives a standard ...
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0answers
17 views

What's the difference between Acfg and ALLcfg

In computational theory, and talking about CFGs, Turing Machines, and so forth I haven't a satisfactory explanation or definition for what ATM means versus ALLTM or the same or similar uses with ...
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0answers
49 views

What is this weird gate?

This came from a picture of something that I'm supposed to make, and I can't find it in the program I'm supposed to use (LogicWorks). It looks like it 'not's only one of its inputs, but that doesn't ...
1
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1answer
28 views

What is the equivalent of the integers symbol Z for n bit only integers?

We refer to the set of all integers as $\mathbb{Z}$. Now suppose we have a set of integers that can be held within a computer variable of $n$ bits width. Clearly they can only be of $2^{n}$ range, ...
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1answer
41 views

Can the notation for polynomial reduction, A ≤p B be reversed in computability theory?

I don't know this is a proper question on this forum but I was reading about computability theory and I saw the reduction concept and its notation like this: $A \le_pB$. I just wanted to know is this ...
5
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2answers
2k views

Why do ¬, ∀ and ∃ have the same precedence?

I thought the order of precedence of operators and quantifiers was arbitrary, but I don't really understand why those three have the same "strength" in relation to other operators (e.g., ¬ will have ...
1
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1answer
153 views

What is the Haskell-style type signature called (i.e., who is it named after)?

A type signature in Haskell is written in the following format: functionName :: arg1Type -> arg2Type -> returnType There's a (hyphenated, after a person or ...
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0answers
31 views

Did Menezes et al. switch the letters $\mu$ and $\lambda$ in Floyd's cycle detection note 3.8?

The letters $\mu$ and $\lambda$ are usually used to represent, respectively, the length of the tail of the graph and length of the cycle in the graph. But Menezes on note 3.8, page 91 (or PDF-page 6),...
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1answer
196 views

How I can find all equivalence classes by Myhill-Nerode?

first of all I'm sorry for my bad English and second I'm sorry for my mistakes of understanding the following topic, I still going to school and learning this for interest. The topic is Myhill-Nerode ...
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1answer
54 views

Constructing a new graph. G'. What does it mean v ∈ S_{i+1}?

John lives in a city whose streets has the same length. His apartment is located at a specified node H. John need to do his errands where he visits each of k different shop in order. However, each ...
0
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2answers
47 views

Terminology for worst-case N-complexity on $O(1)$ insert after amortisation

Normally, when discussing amortisation and worst-case complexity, amortisation negates the worst-case scenarios, and the BigO describes the average for the operation (the way it's used in interviews ...
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10answers
11k views

O(·) is not a function, so how can a function be equal to it?

I totally understand what big $O$ notation means. My issue is when we say $T(n)=O(f(n))$ , where $T(n)$ is running time of an algorithm on input of size $n$. I understand semantics of it. But $T(n)$ ...
1
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1answer
39 views

What is the meaning of max() in intro. to algorithms?

I'm reading chapter 3(growth functions) of CLRS and in giving an example of proving theta for a standard quadratic function the book gives the following value for $n_0 = 2 \cdot max(|b|/a, \sqrt{|c|/a}...
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2answers
268 views

Clarify the steps: what happened in this mathematical modelling of TSP?

Source: http://examples.gurobi.com/traveling-salesman-problem I don't get this part: (look at the source) $$\sum_{i,j\in\{1,2,3\},i\neq j} x_{ij}=3>2=|\{1,2,3\}|-1$$ I get that $x_{ij}$ is ...
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1answer
111 views

How the language $\{a^nb^mc^nd^m | n \geq1, \ m\geq1\}$ is used to check whether formal and actual parameters are equal?

How does the language $L=\{a^nb^mc^nd^m \mid n \geq1, m\geq1\}$ abstract the problem of checking that the number of formal parameters in the declaration of a procedure agrees with the number of actual ...
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0answers
12 views

How to denote a graph class which allows only $k$ instances of a certain induced subgraph?

Suppose that a graph class $\mathcal{C}$ is defined as follows: A graph $G$ belongs to $\mathcal{C}$ if, and only if $G$ is chordal, but has at most $k$ $5$-cycles. I am aware that the ...
4
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1answer
21 views

In ISGCI, unit interval graphs are denoted as ($C_{n+4}$,$S_3$,claw,net)-free. Is this an accurate notation?

When I search unit interval graphs in ISGCI, it says that the unit interval graphs (UIG) are equivalent to ($C_{n+4}$,$S_3$,claw,net)-free graphs. I am confused about the definition of an $S_3$ graph....
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1answer
80 views

Does big-Oh notation in optimization follow the same convention as in CS?

I first learned big-Oh (little-Oh, big-Theta.....) complexity for growth of functions using CLRS in a computer science class. Now I am doing a project on optimization. In our optimization class, we ...
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2answers
63 views

Is $n^3$ an asymtotically tight bound of $(n^{2.99}).(\log_2n)$? How? [duplicate]

Is $n^3$ an asymtotically tight bound of $(n^{2.99}).(\log_2n)$? If so then how?
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1answer
69 views

$\tilde \Omega$ for division by logarithmic factor

Is $\Omega \left(\frac{n}{\log{n}} \right)\subset \tilde\Omega(n)$?
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2answers
1k views

Double exponentials vs single exponentials

Here are four tenets I cannot reconcile: Double exponential time algorithms run in $O(2^{2^{n^k}})$ time with $k \in \mathbb{N}$ constant Exponential time algorithms run in $O(2^{n^k})$ with $k \in \...
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1answer
93 views

Convert for loop into math sigma notation

Have some code (written in C), need the equivalent notation for mathematics. Is it possible using if-statements and sigma notation for this, and if so, how? ...
4
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2answers
983 views

Understanding the Reasoning Behind These Typing Rules

So the expression $\Gamma \ \vdash \ e:\sigma$ states that under assumptions $\Gamma$, the expression $e$ has type $\sigma$. Then we have the following rules: \begin{array}{cl}\displaystyle {\frac {x:...
0
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1answer
53 views

What does $\overline{L}^*$ signify?

Suppose that $L$ is a language. What does $\overline{L}^*$ signify? Is $\overline{L^*}$ the same as $(\overline{L})^*$?
4
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3answers
201 views

Meaning of the notation Typ := TVar | (Typ → Typ)

In the paper "introduction to type theory" by Herman Geuvers, it states the following definition for the type theory of propositional logic (I've added "convention 2" just for reference): Given ...
5
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1answer
109 views

Is there a formal difference between $f:X \to X$ and $f\in X \to X$?

We can denote by $X\to X$ the set of all functions from $X$ to $X$. Therefore, we can use the following statement to say that $f$ is a function from $X$ to $X$: $$f\in X\to X$$ But we usually state ...
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2answers
42 views

shortest notation for “set of all maps from $A$ to $B$”?

I usually write the following: $$\{m| m:A\to B\}$$ Is there a more succint notation?
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0answers
91 views

Where did our present regex notation originate?

First question, and my apologies if it is off-topic! This question suggests to me I may not be totally off-base. Where did our present regex notation originate? I am particularly wondering how <...
1
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1answer
32 views

Syntax of recurrence formula not clear

I have this task for Analysis of Algorithms module in university $$ \begin{align*} W, V &= \{v_1, v_2, v_n\}, S = \{s_1, s_2, s_n\}\\ G(W) &= \max~_{i:w_i\leq w} ~\{G(W-s_i) + v_i\}, G(0)=0\\...
0
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0answers
40 views

Banker's algorithm

I have a question regarding the banker's algorithm step 2. When it says the Need[i] <= work. Need is an n * m matrix and work is a vector of length m. I want to know what Need[i] <= Work means. ...
2
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1answer
132 views

What does $|w|$ mean in the statement “ $|w|$ where $w \in L \subset \Sigma^*$”?

In my assignment I find these symbols: $|w|$ where $w \in L \subset \Sigma^*$ I believe it means the lexikografic value of a word, but I am not 100% certain. Let me phrase my question this way: ...
0
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1answer
45 views

How to represent a map data structure mathematically

Wondering how to represent a map object such as the following mathematically: var foo = { a: 10, b: 'bar', c: true } You could say that it was a function ...
2
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1answer
27 views

Notation for field access in Algorithm

While writing array index in an algorithm $a[i]\gets v$ is conventionally used. Is the notation $a_{i}\gets v$ also used ? If the algorithm uses struct or ...
1
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1answer
41 views

Notation for expressing function call count during a program run

Which notation can be used for expressing how many times that function or subroutine called during a run of a program? Example program pseudocode: $ a \gets 0 \\ \textbf{While } a \neq 5 \textbf{ do}...
1
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1answer
60 views

what is the 'x :=' part mean in a hoare triple?

In Hoare logic, there's a thing called a Hoare triple, e.g.$$ \begin{array}{ccccc} \{x = 2\} & & x := x+1 & & \{x = 3\} \end{array}. $$ What does '$x :=$' mean?
3
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2answers
166 views

What does the $O^*$ notation mean?

I'm recently reading some papers on the maximum independent set problem, all the algorithms' time complexity is donated by $O^*()$ notation, like $O^*(1.0836^n)$. One paper says "the $O^*$notation ...
16
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3answers
10k views

What is the purpose of using NIL for representing null nodes?

In my Algorithms and Data Structures course, professors, slides and the book (Introduction to Algorithms, 3rd edition) have been using the word NIL to denote for ...
5
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1answer
161 views

When it is said an algorithm runs in exponential time, is it meant it has complexity $O(2^n)$, or $2^{O(n)}$?

Also, are they equivalent or are they different? Examples of algorithms/Turing Machines that run in complexity of one but not the other would be much appreciated.
5
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3answers
265 views

Help understand the notation

I'm reading a computer science paper, this one: https://arxiv.org/abs/1608.03960. It uses certain notation in one of the illustrations (Figure 9, page 8 in the PDF version) that I don't understand how ...
2
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1answer
75 views

Is this a standard representation of a 'space' character?

I recently came across a handwritten note with directions to push a string of text into a database. There was a strange character in the string I've never seen before, a lower case 'b' with a forward ...
-1
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1answer
39 views

Why $\left\lceil lgn \right\rceil <lgn+1\le 2lgn\quad for\quad all\quad n\ge 2$

I have some confusion about 3.2-4 in CLRS. Here is the question : Is the function $\left\lceil \log { n } \right\rceil !$ polynomially bounded? Is the function $\left\lceil \log { \log { n } } \...
0
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1answer
27 views

A commonly-used symbol for the set of all heads of directed edges?

Let $G = (V, E)$ be a digraph, and consider the set $\left\{ v \in V \; \middle| \; \left(\{v\} \times V \right) \cap E \neq \varnothing \right\}$, the set of vertices which form edges with other ...
5
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1answer
167 views

What does $n^{O(1)}$ mean?

I read an example that said explain what "$f(n)$ is $n^{O(1)}$" means. I can't interpret the $n^{O(1)}$ syntax. I know what Big $O$ notation is, its just that this example looks odd to me.