Questions tagged [notation]

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

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42 views

Same notation/terminology for union of sets and concatenation (Kleene star)?

For the union of sets we use the union operator $\cup$ (or $\bigcup$). And for a concatenation (Kleene star) we also use the union operator. The operations are different, but why the same terminology ...
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1answer
22 views

Is grammar that describes an equation in prefix (Polish) notation always unambiguous?

I recently completed a problem in which I was asked to generate a parse tree for the expression $+ \, 5 \, * \, 4 \, 3$ using the following grammar and rightmost derivation: $$Expr \rightarrow + \, ...
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2answers
45 views

Parentheses after Typing Environment

I've been reading about System F Omega lately, and I keep stumbling across a construct in typing rules that I cannot find an explanation of: Γ(x) = k. For example, ...
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1answer
41 views

Determining asymptotic notation of a complex function

$$ 5n^4\log{n} - \frac{100n^2}{\log_4(n^2)} + 40 $$ I am currently studying algorithm analysis and i need to express this function in terms of big O, theta and omega, so I should find C, and N0 for ...
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1answer
19 views

Transformation Function: Gonzalez and Woods

I have been reading Image Processing from Gonzalez and Woods and in the chapter Image Transformation I have come across this equation $$ T \left(u, v\right) = \sum_{x=0}^{M-1} { \sum_{y=0}^{N-1} ...
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1answer
47 views

Why in the code i ≃ n?

i=2; while(i<n) { write('*'); i=i*i; } Why $n ≃ i$? I mean suppose $n=1000$ and so $i= 2,4,16,32,256,65536$ is in every steps. In the book wrote $2^2 ...
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1answer
66 views

Difference between ⫾ (U+2AFE) and ⫿ (U+2AFF) in the context of Dijkstra's Guarded Command Language?

Continuing https://tex.stackexchange.com/questions/435986/how-to-draw-the-box-of-dijkstras-guarded-command-language, what is the difference in the intended usage of ⫾ (Dijkstra choice, U+2AFE) and ⫿ (...
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2answers
77 views

Typesetting LTL in Unicode?

Unicode has an abundance of open circles, diamonds and boxes. Using these, we can typeset linear-time temporal logic formulas such as $$ (\mathop{⬦}ψ)\ \,\land\ \,\mathop{⬦}(θ\ \mathsf{U}\ \mathop{◻}\...
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1answer
88 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 ...
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2answers
454 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 ...
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1answer
34 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 ...
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1answer
482 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 ...
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3answers
60 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 ...
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1answer
27 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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35 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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72 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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55 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 ...
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1answer
29 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
58 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 ...
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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 ...
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33 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
432 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 ...
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2answers
57 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 $T(n)$ ...
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1answer
113 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
273 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
130 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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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 ...
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1answer
24 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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2answers
75 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
130 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
145 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? ...
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2answers
992 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:...
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1answer
54 views

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

Suppose that $L$ is a language. What does $\overline{L}^*$ signify? Is $\overline{L^*}$ the same as $(\overline{L})^*$?
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3answers
372 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 ...
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1answer
111 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
44 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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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\\...
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0answers
42 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. ...
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1answer
75 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 ...
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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 ...
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1answer
43 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}...
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1answer
64 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?
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2answers
211 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 ...
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1answer
87 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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1answer
162 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.
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1answer
92 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 ...
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1answer
164 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: ...