# Tag Info

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I think the Wikipedia articles $\mathsf{P}$, $\mathsf{NP}$, and $\mathsf{P}$ vs. $\mathsf{NP}$ are quite good. Still here is what I would say: Part I, Part II [I will use remarks inside brackets to discuss some technical details which you can skip if you want.] Part I Decision Problems There are various kinds of computational problems. However in an ...

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Part II Continued from Part I. The previous one exceeded the maximum number of letters allowed in an answer (30000) so I am breaking it in two. $\mathsf{NP}$-completeness: Universal $\mathsf{NP}$ Problems OK, so far we have discussed the class of efficiently solvable problems ($\mathsf{P}$) and the class of efficiently verifiable problems ($\mathsf{NP}$). As ...

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Translating Code to Mathematics Given a (more or less) formal operational semantics you can translate an algorithm's (pseudo-)code quite literally into a mathematical expression that gives you the result, provided you can manipulate the expression into a useful form. This works well for additive cost measures such as number of comparisons, swaps, statements,...

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Before I answer your general question, let me first take a step back, give some history background, and answer a preliminary question: Do non-computable functions even exist? [notational note: we can relate any function $f$ with a language $L_f=\{ (x,y) \mid y=f(x) \}$ and then discuss the decidability of $L_f$ rather than the computability of $f$] ...

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For simplicity, I'll begin by only considering "decision" problems, which have a yes/no answer. Function problems work roughly the same way, except instead of yes/no, there is a specific output word associated with each input word. Language: a language is simply a set of strings. If you have an alphabet, such as $\Sigma$, then $\Sigma^*$ is the set of all ...

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Landau notation denotes asymptotic bounds on functions. See here for an explanation of the differences among $O$, $\Omega$ and $\Theta$. Worst-, best-, average or you-name-it-case time describe distinct runtime functions: one for the sequence of highest runtime of any given $n$, one for that of lowest, and so on.. Per se, the two have nothing to do with ...

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First your integer numbers are converted into binary numbers. For example, the integer 2 is converted to 0010. The CPU uses a digital comparator: A digital comparator or magnitude comparator is a hardware electronic device that takes two numbers as input in binary form and determines whether one number is greater than or less than or equal to the ...

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Execution Counts of Statements There is another method, championed by Donald E. Knuth in his The Art of Computer Programming series. In contrast to translating the whole algorithm into one formula, it works independently from the code's semantics on the "putting things together" side and allows to go to a lower level only when necessary, starting from an "...

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More than useful mentioned answers, I recommend you highly to watch "Beyond Computation: The P vs NP Problem" by Michael Sipser. I think this video should be archived as one of the leading teaching video in computer science.! Enjoy!

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Consider the following algorithm (or procedure, or piece of code, or whatever): Contrive(n) 1. if n = 0 then do something Theta(n^3) 2. else if n is even then 3. flip a coin 4. if heads, do something Theta(n) 5. else if tails, do something Theta(n^2) 6. else if n is odd then 7. flip a coin 8. if heads, do something Theta(n^4) 9. else if ...

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