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In a partially ordered set, am I always able to order two random elements out of the set?
Or is it possible that two elements within the set have no order relation to each other?
For example if there are three elements {a, b, c} and $a \leq b$ and $a \leq c$, does either $b \leq c$ or $c \leq b$ have to hold?
I ... |
Are two elements always in a relation within a partially ordered set? |
|programming-languages| |
|programming-languages|typing|functional-programming| |
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|programming-languages|typing|functional-programming| |
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|formal-languages|automata|finite-automata|compilers| |
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|programming-languages|combinatorics|poset| |
In a [partially ordered set][1], there are members which are not comparable. A partial order where all elements are comparable is called a [total order][2].
[1]: http://en.wikipedia.org/wiki/Partially_ordered_set
[2]: http://en.wikipedia.org/wiki/Total_order |
|programming-languages|combinatorics|poset|order-theory| |
|combinatorics|poset|order-theory| |
Functional programming is a programming paradigm which primarily uses functions as means for building abstractions and expressing computations that comprise a computer program. |
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In a [partially ordered set][1], there are members which are not comparable. A partial order where all elements are comparable is called a [total order][2].
We say $a$ and $b$ are comparable when at least one of the following holds:
- $a\leq b$,
- $b \leq a$.
[1]: http://en.wikipedia.org/wiki/Partially_... |
|formal-languages|computability| |
|logic|lambda-calculus| |
|algorithms|dynamic-programming| |
In a [partially ordered set][1], there may be members which are not comparable. A partial order where all elements are comparable is called a [total order][2].
We say $a$ and $b$ are comparable when at least one of the following holds:
- $a\leq b$,
- $b \leq a$.
[1]: http://en.wikipedia.org/wiki/Partial... |
OK, so here is a question from a past test in my Theory of Computation class:
>A useless state in a TM is one that is never entered on any input string. Let $$\mathrm{USELESS}_{\mathrm{TM}} = \{\langle \langle M\rangle, q \rangle \mid q \text{ is a useless state in }M\}.$$
Prove that $\mathrm{USELESS}_{\mathrm{TM... |
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OK, so here is a question from a past test in my Theory of Computation class:
>A useless state in a TM is one that is never entered on any input string. Let $$\mathrm{USELESS}_{\mathrm{TM}} = \{\langle M, q \rangle \mid q \text{ is a useless state in }M\}.$$
Prove that $\mathrm{USELESS}_{\mathrm{TM}}$ is undecida... |
In a [partially ordered set](http://en.wikipedia.org/wiki/Partially_ordered_set) (poset for short), you can have $a \le b$ and $a \le c$ without $b$ and $c$ being comparable (i.e. neither $b \le c$ nor $c \le b$ holds). That's what makes it a partial order and not a [total order](http://en.wikipedia.org/wiki/Total_orde... |
|semantics|poset|order-theory| |
In a partially ordered set, am I always able to order two arbitrary elements out of the set? Or is it possible that two elements within the set have no order relation to each other?
For example if there are three elements $\{a, b, c\}$ and $a \leq b$ and $a \leq c$, does either $b \leq c$ or $c \leq b$ have to hold?... |
|semantics|order-theory| |
|computer-networks|protocols|distributed-systems| |
partially ordered set, a set with a partial order relation |
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A [partially ordered set](http://en.wikipedia.org/wiki/Partially_ordered_set), or poset, is a set with a partial order relation (a relation that is reflexive, transitive and antisymmetric, but not necessarily [total](http://en.wikipedia.org/wiki/Total_order)).
In [denotational](http://en.wikipedia.org/wiki/Denotatio... |
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A [partially ordered set](http://en.wikipedia.org/wiki/Partially_ordered_set), or poset, is a set with a partial order relation (a relation that is reflexive, transitive and antisymmetric, but not necessarily [total](http://en.wikipedia.org/wiki/Total_order)).
Arguably the most common poset is the powerset with incl... |
null |
|order-theory| |
You might want to read the paper:
> Sanjay Mehrotra, On finding a vertex solution using interior point methods, Linear Algebra and its Applications, Volume 152, 1 July 1991, Pages 233-253, ISSN 0024-3795, 10.1016/0024-3795(91)90277-4. [sciencedirect article link](http://sciencedirect.com/science/article/pii/00243795... |
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Simply put, if one were to build a quantum computing device with the power of, say, 20 qubits, could such a computer be used to make any kind of modern hashing algorithm useless?
Would it even be possible to harness the power of quantum computing in a traditional computing application? |
|cryptography|quantum-computing|hashing| |
|discrete-mathematics|order-theory| |
|algorithms|optimization| |
|computability|undecidability|turing-machines| |
|computability|undecidability|formal-methods|turing-machines| |
I'm trying to understand what is wrong with the following proof of the following recurrence
$$
T(n) = 2T(\lfloor\frac{n}{2}\rfloor)+n
$$
$$
T(n) \leq 2(c\lfloor\frac{n}{2}\rfloor)+n \leq cn+n = n(c+1) =O(n)
$$
The documentation says it's wrong because of the inductive hypothesis that
$$
T(n) \leq cn
$$
... |
Error in the use of asymptothic notation |
|complexity-theory|time-complexity|recursion|runtime-analysis| |
Let's say the final goal is to prove `T(n) = O(n)`. You start with the induction hypothesis:
T(i)≤ci for all i < n
And to complete the proof, you have to show that T(n) <= cn as well.
However, what you're able to deduce is T(n) <= (c+1)n, which is not helpful to complete the proof.
Notice that you are ... |
Let's say the final goal is to prove `T(n) = O(n)`. You start with the induction hypothesis:
T(i) ≤ ci forall i < n
And to complete the proof, you have to show that `T(n) ≤ cn` as well.
However, what you're able to deduce is `T(n) ≤ (c+1)n`, which is not helpful to complete the proof. Therefore, we cannot... |
Error in the use of asymptotic notation |
null |
|time-complexity|recursion|master-theorem| |
Let's say the final goal is to prove $T(n) = O(n)$. You start with the induction hypothesis:
$T(i) \leq ci$ for all $i < n$.
And to complete the proof, you have to show that $T(n) \leq cn$ as well.
However, what you're able to deduce is $T(n) \leq (c+1)n$, which is not helpful to complete the proof. Therefore... |
null |
|cryptography|quantum-computing|hash| |
Let's say the final goal is to prove $T(n) = O(n)$. You start with the induction hypothesis:
$T(i) \leq ci$ for all $i < n$.
And to complete the proof, you have to show that $T(n) \leq cn$ as well.
However, what you're able to deduce is $T(n) \leq (c+1)n$, which is not helpful to complete the proof. Therefore... |
I'm extending the answer already given, perhaps only by explaining my comment in more detail.
As guessing can clearly be difficult and tedious, sometimes better methods exist. One such method is the [Master Theorem](http://en.wikipedia.org/wiki/Master_theorem). Our recurrence is now of the form $T(n) = a T(n/b) + f(... |
Cryptanalysis is the study of cryptographic algorithms (ciphers, hashes, etc.) aiming to find weaknesses (i.e. "break") them. |
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[Cryptanalysis](http://en.wikipedia.org/wiki/Cryptanalysis) is the analysis of cryptographic algorithms with the aim of finding weaknesses, which allow attackers to "break" them. For ciphers, the aim is to get the key or plaintext, for hashes it is to find collisions or preimages, for signatures/MACs it is to create fo... |
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[Cryptography](http://en.wikipedia.org/wiki/Cryptography) studies techniques for secure communication. The purposes of cryptography include [confidentiality](http://en.wikipedia.org/wiki/Confidentiality) (leading to [tag:encryption]), [data integrity](http://en.wikipedia.org/wiki/Data_integrity) (leading to [tag:hash] ... |
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Mathematical function that maps arbitrary-sized data to fixed-size integers, often used to use as keys in hash tables or to help ensure data integrity |
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A method of solving recurrences of a specific form. |
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A [hash function](http://en.wikipedia.org/wiki/Hash_function) maps data structures of any size to fixed-length integers, with the intent that different data map to different integers. Some important kinds of hash functions are
* [tag:hash-tables], where the hash is a small integer used as an array index for fast da... |
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|programming-languages|computer-architecture|cpu-cache| |
3-Partition problem asks whether a set of $3n$ integers can be partitioned into $n$ sets of three integers such that each set sums up to some given integer $B$. Balanced Partition asks whether $2n$ integers can be partitioned into two equal cardinality sets such that both sets have the same sum. Both problems are known... |
Reduction from 3-Partiton problem to Balanced Partition problem |
|np-hard|reductions| |
3-Partition problem asks whether a set of $3n$ integers can be partitioned into $n$ sets of three integers such that each set sums up to some given integer $B$. Balanced Partition asks whether $2n$ integers can be partitioned into two equal cardinality sets such that both sets have the same sum. Both problems are known... |
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You've omitted a few steps. It looks like you're attempting to prove by induction that $T(n) = O(n)$, and your proof goes:
> Suppose $T(k) = O(k)$ for $k<n$. This means $T(k) \le c \, k$ for some $c$. Then $T(n) = 2 T(\lfloor n/2\rfloor) + n \le 2 c \lfloor n/2\rfloor + n\le (c+1) \,n$, so $T(n) = O(n)$.
This pro... |
We talk about **handles** when we talk about **bottom-up parsing**. Bottom-up parsing during a left-to-right scan of the input constructs a rightmost derivation in reverse. Informally, a **handle** is a substring that matches the body of a production, and whose reduction represents one step along the reverse of a right... |
We talk about **handles** when we talk about **bottom-up parsing**. Bottom-up parsing during a left-to-right scan of the input constructs a rightmost derivation in reverse. Informally, a **handle** is a substring that matches the body of a production, and whose reduction represents one step along the reverse of a right... |
|time-complexity|recursion| |
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|algorithms|time-complexity|recursion| |
|algorithms|landau-notation|recurrence-relation| |
|complexity-theory|np-hard|reductions| |
|security|access-control|structured-data| |
|complexity-theory|graphs| |
|algorithms|complexity-theory|graphs| |
|concurrency|distributed-computing|shared-memory|imperative-programming| |
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