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Consider this context-free grammar:
$$G:\\\;\; S\to aSbb|aaSbbb|\lambda$$
Is the string $a^{2020} b^{4020}\in L(G)$?
I try to derive such a string but I can't, how it possible? |
How it possible given string belong to given grammar |
|formal-languages|context-free| |
null |
This is possible, provided there is a solution in nonnegative integers to the following pair of equations.
$$
\left\{
\begin{array}{rcrc@{\qquad}l}
x & + & 2y & = & 2020 \\
2x & + & 3y & = & 4020
\end{array}
\right.
$$ |
{"Voters":[{"Id":167063,"DisplayName":"Kevin Tsou"}],"DeleteType":1} |
{"Voters":[{"Id":167063,"DisplayName":"Kevin Tsou"}]} |
null |
I wrote code for the Regular Simplex (RSM) and the Irregular Simplex (Nelder-Mead Method (N-MM)) in Visual Studio in C++. For given functions (quadratic (QF), Rosenbrock with alpha = 1 (RF1), Rosenbrock with alpha = 12 (RF12)), starting points ({-1, -1}, {2, 2}), accuracies (1e- 2, 1e-5) received the number of iteratio... |
The problem of finding a local minimum of a function |
|optimization|program-optimization| |
In the incomplete [GeeksForGeeks question #8](https://www.geeksforgeeks.org/practice-questions-time-complexity-analysis/), it is only correct when $k > 1, k \in \mathbb{Z}$. The code is equivalent to `for(int i=0;i<n;i=1+i*k)`. We can work out the first few iterations and see:
Iteration #$1$: $i = 1 + 0k = 1$
Ite... |
I am very interested in the following questions, which sprang out from the topological study of loops in surfaces and their intersection numbers.
Consider, over a finite set of boolean variables $X$, a [Boolean system in CNF][1] (conjunctive normal form) whose clauses only contain non-negated literals.
Assume the... |
Monotone boolean satisfiability problem : finding minimal solutions |
null |
null |
In the incomplete [GeeksForGeeks question #8](https://www.geeksforgeeks.org/practice-questions-time-complexity-analysis/), it is only correct when $k > 1, k \in \mathbb{Z}$. The code is equivalent to `for(int i=0;i<n;i=1+i*k)`. We can work out the first few iterations and see:
Iteration #$1$: $i = 1 + 0k = 1$
Ite... |
The complement of the language $L = \{a^nb^n : n \in \mathbb{N}\}$ should actually be linear, if I'm not mistaken.
Take some string $x \notin L$ over the alphabet $\Sigma = \{a, b\}$, since $L \subseteq a^*b^*$ one of three cases must be true:
- $x \notin a^*b^*$, or
- $x = a^nb^m \in a^*b^*$ and $n > m$, or
- $... |
The complement of the language $L = \{a^nb^n : n \in \mathbb{N}\}$ should actually be linear, if I'm not mistaken.
Take some string $x \notin L$ over the alphabet $\Sigma = \{a, b\}$, since $L \subseteq a^*b^*$ one of three cases must be true:
- $x \notin a^*b^*$, or
- $x = a^nb^m \in a^*b^*$ and $n > m$, or
- $... |
The complement of the language $L = \{a^nb^n : n \in \mathbb{N}\}$ should actually be linear, if I'm not mistaken.
Take some string $x \notin L$ over the alphabet $\Sigma = \{a, b\}$, since $L \subseteq a^*b^*$ one of three cases must be true:
- $x \notin a^*b^*$, or
- $x = a^nb^m \in a^*b^*$ and $n > m$, or
- $... |
You can't, because the length of the shortest path between $u$ and $v$ is not known. The purpose of the Bellman-Ford algorithm is to compute such lengths, so you can't assume they are already known. |
I am very interested in the following questions, which sprang out from the topological study of loops in surfaces and their intersection numbers.
Consider, over a finite set of boolean variables $X$, a [Boolean formula in CNF][1] (conjunctive normal form) whose clauses only contain non-negated literals.
Assume th... |
null |
null |
null |
Is the time complexity of `for(int i=0;i<n;i++){i*=k;}` $O(\log_k n)$?
The problem is number 8 from GeeksForGeeks: <br>
https://www.geeksforgeeks.org/practice-questions-time-complexity-analysis/
I trace the program by hand and got stuck. I could not prove that the answer is $O(\log_k n)$. I made a mistake somewh... |
Is the time complexity of a loop that simultaneously increments and multiplies $O(\log_k n)$ when $k = 1$? |
|time-complexity|asymptotics|proof-techniques| |
null |
null |
**tl;dr**
As Lamport writes at the top of page 561 of [Time, Clocks, and the Ordering of Events in a Distributed System](https://doi.org/10.1145%2F359545.359563):
> To break ties, we use any arbitrary total ordering $\prec$ of the processes.
To elaborate, ties are indeed possible and already anticipated by Lamp... |
Let's address your questions/confusions in order:
1.
> I don't see the authors working on the former equation, only the latter.
That's because $x$ doesn't matter:
- it does not occur in the postcondition ($v \leq 1$) investigated here
- it does not occur in the RHS of the equation for $v$
2.
> Why did the ... |
I was referring to the textbook Artificial Intelligence: A modern approach 3rd by Stuart Russell and Peter Norvig.
what to prove about the general "graph search": (Here I assume "within a factor of b" means multiplier $b$ instead of [addition $\pm b$][1] based on the complexity context. So if time complexity is $O(b... |
How is the best way to go about proving that a greedy algorithm is optimal, inductive vs contradiction, are there parts in the proof that are key to include. I did not feel that the lesson I was given adequately covered how to start a greedy proof, any advise on a general way to go about proving greedy algorithms woul... |
Proving Optimal Greedy Algorithms |
|algorithms|proof-techniques|greedy-algorithms|correctness-proof| |
null |
{"OriginalQuestionIds":[59964],"Voters":[{"Id":755,"DisplayName":"D.W.","BindingReason":{"GoldTagBadge":"algorithms"}}]} |
There's not enough information here to answer the question, but I can prove a pretty good guess based on how tokenizers are typically implemented. Generally the tokenizer will run through the list of patterns, trying each one I sequence. If one of them matches it takes the longest possible match as a token, then star... |
null |
I've implemented a ranking system (based on a score), using a Map<Player,Score> and an improved BST<Score, Set<Player>>.
So now I can have "rank of a player" in O(log(n)), "top xxx players", insert/update/remove in O(log(n))
Now I want to improve by data structure to add a "best rank". So when a player score is up... |
Improving a ranking system with "best rank" |
|binary-trees|search-trees|rank| |
Suppose I have a function $f: \mathbb{R}^n\rightarrow\mathbb{R}$. All I know about the function is, I have a set of pairs of vectors ($\vec{v}_a$, $\vec{v}_b$) for which I know which one is greater (i.e. I know that $f(\vec{v}_a)>f(\vec{v}_b)$). I want to find the minimum of the function (or at least a good approximati... |
Find minimum of a function only knowing the ordering of a set of input points |
|optimization|gradient-descent|topological-ordering| |
null |
I have a problem where I need to connect branches of multiple trees with edges such that every start node is connected to every other start node. Each tree can branch at a second node into 1-3 branches, with the count of branches always being even. I've implemented a solution, but it's not very readable or maintainable... |
Efficient algorithm for connecting branches of multiple trees with edges to ensure full connectivity |
|graphs| |
I'm studying heuristics for a course in my Master's. Our professor has told us that in order to point to a higher grade in the exam, we should check if our algorithm hits a stagnation points and that we have not seen any clear improvements to the optimum we have found. He asked it for some algorithms, especially the Si... |
A stagnation check for heuristics, especially simulated annealing? |
|heuristics|simulated-annealing| |
You can't. The function could be anything. For example, consider
$$f(x) = \begin{cases}
g(x) &\text{if }x \ne \alpha\\
-10^{100} &\text{if } x = \alpha
\end{cases}$$
where $\alpha \in \mathbb{R}^n$ is some obscure point (maybe very far from the origin). Then any reasonable algorithm might never evaluate $f$... |
I've implemented a ranking system (based on a score), using a Map<Player,Score> and an improved BST<Score, Set<Player>>.
So now I can have "rank of a player" in O(log(n)), "top xxx players", insert/update/remove in O(log(n))
Now I want to improve by data structure to remember the "best rank" of each player. So whe... |
|data-structures|binary-trees|search-trees|rank| |
Suppose $L=\{ww^r:w\in\Sigma^*\}$. Already, we know that we can draw a PDA for $L$ such that accept each $w\in L$ with space complexity at least $|w|$.
My question is how is it possible to draw a PDA that accepts or rejects $w$ with space complexity $\frac{w}{C}$ for some constant $C$? Any help or hint will be appre... |
Accept $L=\{ww^r:w\in\Sigma^*\}$ in less that $|w|$ storage |
|formal-languages|context-free| |
null |
It is (obviously) possible if $|\Sigma| = 1$ or $C = 1$.
It is also possible for $|\Sigma| > 1$ and $C>1$. The idea is that you can use a bigger stack alphabet to encode multiple letters at once.
For example, if $\Sigma = \{a, b\}$, the following PDA recognizes $L$, and the stack is at most $\frac{|w|}2$ for an i... |
null |
In most US universities, undergraduate students must satisfy the requirements of a degree program in order to graduate. The general rules are:
- The university has a minimum number of total credits needed, and a series of subject-based baskets (e.g. sciences, humanities, writing, etc) with their own minimum credit ... |
algorithm to categorize courses into academic degree requirements? |
|algorithms|classification|packing| |
You can't. The function could be anything. For example, consider
$$f(x) = \begin{cases}
g(x) &\text{if }x \ne \alpha\\
-10^{100} &\text{if } x = \alpha
\end{cases}$$
where $\alpha \in \mathbb{R}^n$ is some obscure point (maybe very far from the origin). Then any reasonable algorithm might never evaluate $f$... |
I'm assuming we're testing if a student satisfies the requirements for a specific degree, not whether it satisfies *any* degree.
If so, we can immediately calculate the total number of credits, and whether all required courses have been satisfied with a single loop over the completed courses, using a hashset of requ... |
null |
null |
It is (obviously) possible if $|\Sigma| = 1$ or $C = 1$.
It is also possible for $|\Sigma| > 1$ and $C>1$. The idea is that you can use a bigger stack alphabet to encode multiple letters at once.
For example, if $\Sigma = \{a, b\}$, the following PDA recognizes $L$, and the stack is at most $\frac{|w|}2$ for an i... |
null |
null |
I was referring to the textbook Artificial Intelligence: A modern approach 3rd by Stuart Russell and Peter Norvig.
what to prove about the general "graph search": (Here I assume "within a factor of b" means multiplier $b$ instead of [addition $\pm b$][1] based on the complexity context. So if time complexity is $O(b... |
The answer by [@D.W.](https://cs.stackexchange.com/users/755/d-w) is correct that any search space will need *some* structure in order to make progress. However, I think it would be pessimistic to conclude that the No Free Lunch theorems apply; since they assume uniform sampling from the space of all functions, which i... |
The answer by [@D.W.](https://cs.stackexchange.com/users/755/d-w) is correct that any search space will need *some* structure in order to make progress. However, I think it would be pessimistic to conclude that the No Free Lunch theorems apply; since they assume uniform sampling from the space of all functions, which i... |
While reading an article devoted to the sparse self-attention, I came across a notation that was not very clear:
$$
Attend(X, S) =
\Big(
a(x_i, S_i)
\Big)_{i∈{\{1,...,n}\}}
$$
What means $\Big( \space \space \Big)_{i∈{\{1,...,n}\}}$ ?
It would be great to describe this in more detail.
Link to the a... |
Strange notation in the article about sparse self-attention |
|machine-learning|natural-language-processing| |
|machine-learning|linear-algebra|natural-language-processing| |
Suppose $L$ is regular and we define $L'=\{x:\exists y\in L \wedge \text{ y be a subsequence of x}\}$. Could we conclude that $L'$ is regular or not?
I think it's not regular because if $L=a^*b^*c^*$ then $L'$ could be $a^nb^nc^n$ so it can't be regular. |
Is the $L'$ regular or not? |
|formal-languages|regular-languages| |
I think it's actually regular.
Suppose $L$ is recognised by the DFA $A = (Q, \Sigma, \delta, q_0, F)$, define the NFA $A' = (Q, \Sigma, \delta', q_0, F)$ s.t.
$$\delta'(q, a) = \{\delta(q, a)\} \cup \{q\}$$
for all $q \in Q, a \in \Sigma$.
Intuitively, you can think of $A'$ as being the same as $A$ with the add... |
The notation
$$(x_i)_{i \in \{1,\dots,n\}}$$
is short-hand for
$$(x_1,x_2,\dots,x_n).$$
Just two different ways to write the same thing.
It is analogous to the relationship between $\sum_{i \in \{1,\dots,n\}} x_i$ and $x_1 + x_2 + \dots + x_n$. Two different ways to write the same thing. |
{"OriginalQuestionIds":[77672],"Voters":[{"Id":755,"DisplayName":"D.W."}]} |
I think it's actually regular.
Suppose $L$ is recognised by the DFA $A = (Q, \Sigma, \delta, q_0, F)$, define the NFA $A' = (Q, \Sigma, \delta', q_0, F)$ s.t.
$$\delta'(q, a) = \{\delta(q, a)\} \cup \{q\}$$
for all $q \in Q, a \in \Sigma$.
Intuitively, you can think of $A'$ as being the same as $A$ with the add... |
Suppose $L$ is regular and we define $L'=\{x:\exists y\in L \wedge \text{ y be a proper subsequence of x}\}$. Could we conclude that $L'$ is regular or not?
I think it's not regular because if $L=a^*b^*c^*$ then $L'$ could be $a^nb^nc^n$ so it can't be regular. |
```
public void m1(int n) {
sum = 0;
for (int i = n/2; i <= n; i++) {
for (int j = 1; j <= n; j = 2 * j) {
for (int k = 1; k <= n; k = k * 2) {
sum++;
}
}
}
}
```
So, I know that the innermost loop and second loop are log(n) and the outer... |
Big-O time complexity for this code snippet |
|complexity-theory|time-complexity|asymptotics|java| |
null |
You are right, the two innermost loops perform $\Theta(\log n)$ iterations each, so we have a total of $\Theta(\log^2 n)$ iterations, which are repeated $\Theta(n)$ times in the outer loop, which implies that we have a total of $\Theta(n\log^2 n)$ iterations.
You just have to multiply the number of iterations of ea... |
I am implementing Earley parser and using algorithm from [Elizabeth Scott's paper "SPPF-Style Parsing From Earley Recognisers"][1] (section 4).
Author says
> A family of children of u will consist
of one or two nodes.
However algorithm as described in paper fails at following grammar:
<S> ::= <A> | <B> ... |
Can Shared Packed Parse Forest have more than 2 families? |
|context-free|formal-grammars|parsers| |
You are right, the two innermost loops perform $\Theta(\log n)$ iterations each, so we have a total of $\Theta(\log^2 n)$ iterations, which are repeated $\Theta(n)$ times in the outer loop, which implies that we have a total of $\Theta(n\log^2 n)$ iterations.
You just have to multiply the number of iterations of ea... |
i'm trying to make a game in which the player faces an infinite (finitely specified) series of enemies and has to specify a strategy that provably defeats all of them (ie defeats enemy n in finite time for all n). i'm intending to do it by having the player describe their strategy in a decidable language, which is then... |
can a model of computation with infinitely many states be nontrivially decidable? |
|decidability|infinity| |
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