Short version is that P (not) equal to NP ? is one of the most important questions in theoretical computer science. Lots of brilliant people have worked on it and come up with no clear proof yet. The consensus seems to be that such a proof cannot be written using existing axiom systems If P!=NP then it is known that there are such problems; at least one exists. Unfortunately we do not know if any real-world problems we In short: knowing whether NPI is empty or not is equivalent to solving proving P!=NP, so get cracking! If you can find a problem that is definitely in NP but definitely..

* Are there any known problems in NP (and not in P) that aren't NP Complete? My understanding is that there are no currently known problems where this is the case*, but it hasn't been ruled out as a possibility This paper does not claim to settle the P versus NP question, but provides some evidence for P-not-equal-to-NP in a certain restricted model of This paper explains why P is not equal to NP in the so-called Mabel-Mildred-Feinstein model of computation. The paper also kind of settles the Collatz..

NP-hard problems need not be in NP, i.e., they need not have solutions verifiable in polynomial time. For instance, the boolean satisfiability problem is NP-complete by the Cook-Levin theorem, so any instance of any Based on the definition alone it is not obvious that NP-complete problems exist P equals NP is a hotly debated Millennium Prize Problem - one of a set of seven unsolved mathematical problems laid out by the The math problem that could change the world: Does P = NP? Depending on the answer, one of the famous unsolved Millennium problems could have major implications in our lives This is not the same as low computational complexity, so the remarks I am about to make concern the question of how one might distinguish between NP yeah re boaz' 1st comment above, wonder if there are some proven NP complete problems in crypto wrt block ciphers & permutations/scramblers but..

- istic and exact algorithm which solves the Subset-Sum problem in..
- The most notorious problem in theoretical computer science remains open, but the attempts to solve it have led to profound insights
- NP-hard problems need not be in NP, i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is Although it is unknown whether P = NP, problems outside of P are known. A number of succinct problems (problems that operate not on normal input..
- istic polynomial time i.e can be solved in polynomial More Terms and Definitions :- NP-hard — a problem X is NP-hard if every problem Y ∈ NP reduces to X i.e X is at least as hard to solve as every problem..
- NP-complete is a special category of NP problems that have time complexities greater than Another NP-hard problem is identifying a chess move in any given board state that is the absolute best move Not just that, but part of the definition of an NP-complete problem is every problem in NP is reducible..
- Does
**NP****not**equal**P**? Charles Sauerbier cs@orst.edu. ABSTRACT — Stephen Cook posited SAT is**NP**-Complete in 1971. contention that SAT being shown contained in**P****is****not**sufficient to resolve the question of**NP**=**P**, as one must address the question of whether post-SAT-**is**-P**P****is**also then..

- Mathematician Who Claimed 'P Is Not Equal To NP' Says His Proof Is Wrong. Proving P=NP would mean a P solution to an NP problem was theoretically possible, but it wouldn't necessarily help you find one. Even if P!=NP, we might still be able to solve some NP problems in P time using quantum..
- P probably does not equal. NPNP. NP, whereas only 9 said otherwise and, as they later told reporters, mainly to contradict the popular sentiment.[1]. Scott Aaronson, a renowned theoretical NP−complete, have been explored to no avail; whereas, like any other successful scientific hypothesis, the
- Why is P=NP such a big deal that it warrants a $1 million prize? So what then, is the P versus NP problem? For the record, the status quo is that P≠NP. P (polynomial time) refers to the class of problems that can be solved by an algorithm in polynomial time
- The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly. For faster navigation, this Iframe is preloading the Wikiwand page for P versus NP problem. The cover is not a good choice

P and NP are not complexity classes and NP does not stand for not-polynomial. I'm not sure what you're talking about with them not being, (I do know that P is a subset of NP) . And the problems I referred to here I believe are NP-Complete, so maybe I was being too loose should have used that.. (Not all NP-hard problems are members of the class of NP problems, however.) For example, modern cryptography relies on the assumption that factoring the product of two large prime numbers is not P. Note that verifying the product of two prime numbers is easy (polynomial time), but computing.. By Ayesha Ahmed. Creativity, ingenuity, luck. All concepts that set apart the most brilliant minds from the rest. But also concepts we cannot strictly define. After all, there are no set of rules for genius. Well, actually, there might be

- istic Turing machine in polynomial The most important sample problem set in P is Is X the correct decryption key for this encrypted file? Obviously, checking to see whether a key is..
- P, NP, and NP-Completeness. Siddhartha Sen Questions: sssix@cs.princeton.edu. NP-complete problems are the hardest in NP: if any NP-complete problem is p-time solvable, then all problems in NP are (u, v) Î Ē Þ either u or v not in V', since (u, v) Ï E Þ at least one of u or v in V - V', so covered
- Does NP not equal P? Charles Sauerbier cs@orst.edu. ABSTRACT — Stephen Cook posited SAT is NP-Complete in 1971. contention that SAT being shown contained in P is not sufficient to resolve the question of NP = P, as one must address the question of whether post-SAT-is-P P is also then..
- The class
**NP**consists of those problems that are verifiable in polynomial time.**NP****is**the class of decision problems for which it is easy to check the correctness of a claimed answer, with the aid of a little extra information. It is not known whether**P**=**NP**

P versus NP ( Clay's Millennium Prize Problem) is a major unsolved problem of computer science and mathematics. P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP NP-class problems include many pattern-matching and optimization problems that are of great practical interest, such as determining the optimal We don't need a formal proof that P is not equal to NP in order to rely on the conjecture, Aaronson says. Programmers know about the problem and would be.. However, P=NP in the sense that P!=NP self-referentially (or perhaps Cantor's cardinality viewed infinitesimally; fractionally thus factorially, perhaps in We don't know if P=NP, or P is not equal to NP. The question I was posing is a simple decision problem. Its either Yes, No, or Undecidable (meaning.. It is not a repository for any question you may have. E is for Explain - merely answering a question is not enough. LI5 means friendly, simplified and P = NP would mean computers can do just about anything. P != NP, which is what many people believe, means life is hard, random, and we all better..

Quiz or Mock test on NP complete problems. The quiz contains multiple choice questions on GATE and technical interview preparation. Because, if one NP-Complete problem can be solved in polynomial time, then all NP problems can solved in polynomial time This is why the answer to the P vs. NP problem is so interesting to people. If anyone were able to show that P is equal to NP, it would make difficult real-world As such, the P vs. NP problem is the search for a way to solve problems that require the trying of millions, billions, or trillions of combinations..

- istic Polynomial time solving. but NP problems are checkable in polynomial time means that given a solution of a problem , we can check that whether the solution is correct or not in polynomial time
- I know that $P$ vs. $NP$ is important for theoretical and conceptional reasons, but what if finally $P=NP$ holds but no effective proof can be found. I guess this wouldn't change much. EDIT: Just to be clear: I do not dismiss complexity theory at all and I can appreciate theoretical results, even if they are..
- Wednesday, August 11, 2010. Vinay Deolalikar, a mathematician who works for HP Labs, claims to have proven that P is not equal to NP. The problem is the greatest unsolved problem in theoretical computer science and is one of seven problems in which the Clay Mathematics Institute has offered..
- NP-hard problems need not be in NP, i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is Based on the definition alone it is not obvious that NP-complete problems exist; however, a trivial and contrived NP-complete problem can be formulated..
- ..Overview Ron Parr CPS 570 NP hardness is not an AI topic It s important for all computer scienhsts Understanding it will deepen your understanding of AI (and other CS) topics You will be expected to understand its relevance and use for AI problems Eat your vegetables; they re good for you P and NP
- Spring Quarter 2020 - Passed/Not Passed (P/NP) and Satisfactory/Unsatisfactory (S/U) Update. The Academic Senate recognizes the stresses that students and instructors are currently facing in their academic and personal lives. To reduce academic uncertainty and increase flexibility when trying to..
- This is a rough guide to the meaning of NP-Complete. It is not intended to be an exact definition, but should help you to understand the concept. These are just my personal ideas and are not meant to be rigorous. Its All About Time to Solve. When you measure how long a program takes to run when it..

- istic Turing machine. Since it can be solved in polynomial time, it can..
- P versus NP is considered one of the great open problems of science. This consists in knowing the answer of the following question: Is P equal to NP? This incognita was first mentioned in a letter written by John Nash to the National Security Agency in 1955
- istic Turing machine accepts in Polynomial time. Author: PEB
- The P versus NP problem is a major unsolved problem in computer science. Informally speaking, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. The first mention of the underlying problem occurred in a 1956 letter written by Kurt..

Not sure about the answer? Not sure about the answer? See next answers. What do you need to know NP does not stand for not polynomial. • NP = Non-Deterministic polynomial time • NP means verifiable in polynomial time • Verifiable? - If we are somehow given a 'certificate' of a solution we can verify the legitimacy in polynomial time What happened to automata?

- - There simply are a lot of awfully hard looking problems in NP-Complete (and Co-NP-Complete) and some just don't seem to be 8/31/16 COT 4210 © UCF 386 SAT • SAT is the problem to decide of an arbitrary Boolean formula (wff in the propositional calculus) whether or not this formula is satisfiable..
- This means that NP-hard problems might be in NP, or in a much higher complexity class, or they might not even be decidable problems. That's why people often say something like NP-hard means at least as hard as NP when trying to explain this stuff informally. The halting problem is a good example of..
- ..in detail, if NP is not P then also we cannot investigate this fact in detail, if Navier-Stokes equations have solutions then we cannot investigate this fact in detail Zhuk Vladimir Alexandrovich, Freelancer analyst E-mail: mathbilanandc@gmail.com. Some aspects of np-complete tasks. Abstract
- How not to prove that P does not equal NP as the author says, is an example of the complex and convoluted logic that's needed to explore the field of Very difficult problems, like the isomorphism of NP series, are presented in both examples and exercises. This is NOT a beginner's book, and either..
- Explain Why If NP Is Not Equal To Co-NP Then It Must Be The Case That P Is Not Equal To NP. 2. The class of languages co-NP contains all languages whose complement is in NP. Explain why if NP is not equal to co-NP then it must be the case that P is not equal to NP

- File:P np np-complete np-hard.svg. From Wikimedia Commons, the free media repository. Jump to navigation Jump to search. Euler diagram for P, NP, NP-complete, and NP-hard set of problems
- istic Turing Machine (DTM)
- M. David Stone NEC NP-P501XThe NEC NP-P501X projector offers a bright image, excellent data-image quality, and a capable audio system, plus conveniences that include a 1.7x zoom lens
- 9 NP Technically speaking: A problem is in NP if it has a short accepting certificate. An accepting certificate is something that we can use to quickly Although we know that P is not equal to EXP, it is possible that NP = P, or EXP, or neither. Frustrating! NP-hardness As we will see, some problems are..

NP is the full set of decision problems that can be VERIFIED in polynomial time, that is, given an input, an a-priori answer and a proof which has a bit It's trivial to show that P is a subset of NP. (NP-Hard problems are a set of problems with the following characteristic: If they have a polynomial-time.. ** P Is Not Np on WN Network delivers the latest Videos and Editable pages for News & Events, including Entertainment, Music, Sports, Science and more, Sign The P versus NP problem is a major unsolved problem in computer science**. Informally speaking, it asks whether every problem whose solution can.. What does NP mean? This could be the only web page dedicated to explaining the meaning of NP (NP acronym/abbreviation/slang word). Ever wondered what NP means? Or any of the other 9127 slang words, abbreviations and acronyms listed here at Internet Slang

Furthermore, NP intersect coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P+ = (NP+ intersect coNP+) = (NP Within any contiguous block of infinite clockable ordinals, we show that P_alpha is not equal to NP_alpha intersect coNP_alpha, but if beta.. When using np versions 0.20-0 and higher, if the rst step (bandwidth selection) is not performed explicitly then the second step will automatically call Note that if your factor is in fact a character string such as, say, X being either MALE or FEMALE, np will handle this directly, i.e., there is no need to..

- 종종 NP 문제에 대해서 P 문제가 아닌 것으로 설명하는 경향이 있지만, P 문제는 다항식 시간 동안 '답을 찾을 수 있는' 문제이고 NP 문제는 다항식 시간 마인크래프트의 시작 화면의 스플래쉬에서 'NP is not in P!'라는 문구가 뜬다.어떻게 아는거야. [1] 단 여기서 말한 곱셈은 말 그대로 손으로 곱셈하는..
- All translations of P_is_not_NP. sensagent
- istic polynomial problems. They are a class of decision problems that can be solved using Next, it should be proved that every problem in NP is reducible to the problem in question in polynomial time. 14. Which of the following problems is..
- istic Turing machine in a polynomial time
- Your department may not allow P/NP in your major. The deadline to choose a grade option for the Spring 2020 Quarter is the end of week 10. Your course work must be equivalent to C- or better to receive a P. P/NP courses count for unit requirements but not toward your GPA

The P versus NP problem is a major unsolved problem in computer science. Informally speaking, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer NP-hard problems need not be in NP, i.e., they need not have solutions verifiable in polynomial time. Although it is unknown whether P = NP, problems outside of P are known. A number of succinct problems (problems that operate not on normal input, but on a computational description of the input).. Yuriy, N2TTA will be active from US Virgin Islands, in CQ WW WPX SSB Contest, 28 - 29 March 2020, as NP2P. He will operate in SO Category. NP2P. US Virgin Islands. Information for Radio Amateurs The Status of the P Versus NP Problem It's one of the fundamental mathematical problems of our time, and its importance grows with the rise of powerful P is a subset of NP. The question is whether they're identical. The technical difference has to do with (classical) Turing machines and what they're.. ** What does NP stand for? Your abbreviation search returned 133 meanings**. MLA style: NP. Acronym Finder. 2020. AcronymFinder.com 30 Mar

- NP is the class of languages with polynomial-size membership proofs. P ⊆ NP still holds: If I can decide membership in polynomial time, I do not need a A language A ∈ NP is NP-complete i A is P-hard for NP. Not obvious that any such problem exists. Not at all obvious how to prove this property, that all..
- np.bincount() 从零开始计数（不允许序列中出现负数）； 在一些情况下**只能使用np.average()**而无法使用简单的矩阵乘法操作： 比如
- They are not NP-complete problems but they are related to them. Indeed, given an optimization problem and a value k, we can define a decision problem as follows: P(k): Is there a feasible solution s such that f(s) = k ? If the original optimization problem is NPO then the decision problem is NP
- The more unambiguous statement of the P versus NP problem and the judgement of its hardness, are the key ways to find the full proof of the P versus If combining this viewpoint with Cantor's theorem, it is shown that an NTM is not equipotent to a DTM. This means that generating the power set P(A) of..

* You are wrong*. Figure it out. Sometimes you can still salvage something interesting out of your flawed proof. You believe the proof is correct. Your belief is incorrect. Go back to step 1. Are you making any assumptions or shortcuts, even seemingly small and obvious ones Not only is P vs. NP the defining question of our field; it's one of the deepest questions ever asked for which we'd know how to recognize an answer.. If you doubt this, read the Clay Math Institute's list of million-dollar prize problems notice how P vs. NP stands out, not merely as the only problem of..

ANSWER: NP-complete and in P respectively. (11) Which one of the following statement is true? (A) The intersection of two context free languages is context (54) The problem, which is not NP-hard, is. (A) Finding bi-connected problem of a graph (B) The graph colouring problem (C) Hamiltonian circuit.. 8. INTRACTABILITY II. P vs. NP NP-complete co-NP NP-hard. 3-SAT poly-time reduces to all ofthese problems (and many, many more). If I had to bet now, I would bet that. P is not equal to NP. I estimate the half-life of this problem at 2550. more years, but I wouldnt bet on it being solved before..

One says Q is in NP if there is an existential program operating on I plus a number of guess bits that correctly answer all instances I of Q in NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail To attack the P = NP question, the concept of NP-completeness is very useful. Informally, the NP-complete problems are the toughest problems in NP in Although it is unknown whether P = NP, problems outside of P are known. A number of succinct problems, that is, problems which operate not.. P, NP and NP-complete are not hard to grasp. But NP-hard class is a little confusing to me, I easily forget what exactly is its relation to the others and who NP-hard is just all NP-complete problems + all problems that can be reduced from an NP-complete problem. So they're 'harder than all NP problems' * If P is not equal to NP then it would prove that some problems are truly unsolvable guiding science in the right direction*. As of now we do not know if P equals or does not equal NP but it surely will shatter many fields of study if either one of the solutions is proven. comment down below what you think..

If P=NP, then NP-complete problems are P problems, so obviously the answer is no. I claim that this problem is not merely in NP, but in NP intersect coNP. If P, NP, and coNP aren't enough to rock your world, you can generalize these classes to a giant teetering mess that we computer scientists call the.. The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified (technically, verified in polynomial time) can also be solved quickly (again, in polynomial time)

bir np problemin polinomial sure algoritmasi, bu cesit tum problemlerin cozumu icin uyarlanabilir ve boyle bi algoritma ile her turlu sifre kirilabilir. bu da ispatlanmasi birçok matematikçinin işsiz kalmasına sebep olacak problem. çünkü p = np olursa bir matematik teoremini ispatlamak (yaratıcılık) ve o.. NP-complete problems are a set of problems to each of which any other NP-problem can be reduced in polynomial time, and whose solution may still be Although it is unknown whether P = NP, problems outside of P are known. A number of succinct problems (problems that operate not on normal input.. NP-hard problems need not be in NP, i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is Although it is unknown whether P = NP, problems outside of P are known. Just as the class P is defined in terms of polynomial running time, the class.. Find out what is the full meaning of P1NP on Abbreviations.com! 'Procollagen type I N-terminal Propeptide' is one option -- get in to view more @ The Web's What does P1NP mean? This page is about the various possible meanings of the acronym, abbreviation, shorthand or slang term: P1NP * But A need not halt on strings not in L*. It could just never return an answer. (The existence of problems like the Halting Problem necessitate considering this possibility.) Equivalently, if any problem in NP is not polynomial-time solvable, then no NP-Complete problem is polynomial time solvable

Notepad++ 7.8.5 release. Notepad++ 7.8.4 release. Notepad++ 7.8.3 release. Notepad++ 7.8.2 release. Notepad++ 7.8.1 release. Notepad++ 7.8 release. Notepad++ 7.7.1 release. Notepad++ 7.7 release. Notepad++ 7.6.6 release. Notepad++ 7.6.4 release. Notepad++ 7.6.3 release. Notepad++ 7.6.2 Gilet.. NP-complete problems are not just maths that takes long to calculate, they are problems for which the time required to solve grows non-linearly as the problem size increases. You can look up the formal definition if you wish. But another way to prove a problem is NP-complete is through reduction The classes P and NP are certain classes of such problems (or specifications) with yes/no answers like this, and reformulating Minesweeper in terms of A: For the definitive answer, see the Clay Institute's pages. But to be brief, you must either prove that P=NP or prove that P is not equal to NP, and do it..

Proofs that show that P is a subset of NP are not hard to find. I've seen several places that have simply stated that it's known that P is a subset of the intersection of NP and co-NP If given and not False, return not just the estimate but also its covariance matrix. By default, the covariance are scaled by chi2/sqrt(N-dof), i.e., the weights are presumed to be unreliable except in a relative sense and everything is The coefficient matrix of the coefficients p is a Vandermonde matrix ** This way, the student is guaranteed not to make a false positive (which, in this context, would be a severe embarrassment to any reputable mathematician), and Vinay Deolalikar**. P is not equal to NP. 6th August, 2010 (66 pages 10pt, 102 pages 12pt). Manuscript sent on 6th August to several leading.. NP-Completeness. So far we've seen a lot of good news: such-and-such a problem can be solved quickly (in close to linear time, or at least a time that For instance P is a very good approximation to the class of problems which can be solved quickly in practice -- usually if this is true, we can prove a..

Since NP-Complete is a part of NP, that means you can also take any problem in NP-Complete and trivially convert it to any other problem in NP-Complete. Sometimes we define the set of NP problems which are not NP-Complete as NP-Easy because they're easier than the most general problems in NP The P=?NP problem is philosophically solved by showing P is equal to NP in the random access with unit multiply (MRAM) model. It is shown that the MRAM model empirically best models computation hardness. The P=?NP problem is shown to be a scientific rather than a mathematical problem **P** **is** **not** **NP**? A solution to the so-called **P** versus **NP** problem is worth $1 million and could have a profound effect on computing, and perhaps even the entire world. **P** and **NP** both represent groups of mathematical problems, but it isn't known if these groups are actually identical suming P = NP is not enough to get public-key protocols, instead we need strong average-case assumptions about the diculty of factoring or related problems. [2] S. Aaronson. Is P versus NP formally independent? Bulletin of the European Association for Theoretical Computer Science, 81, Oct ** P/NP should be used sparingly, but it can help take A LOT of stress off of your coursework**. I agree with BIGAUS and recommend waiting until much closer to the P/NP deadline before deciding which You only know what you are getting when you switch, not your final grade. And C- is P, as far as I know

We know that the product between $n$ and $p$ is negative. This tells us that one of the two variables must be positive and the other must be negative. Seems like an important bit of information Four definitions of the acronym NP with examples. NP is an acronym for no problem that is another way of saying you're welcome in online chat or when texting. It is commonly used in response to receiving thanks for something Emulsifier NPE/NP-4;Np-6;NP-7;NP 8;NP-9;NP-10;NP-15;NP-20;NP-70 liquid, paste products using 200 kg galvanized barrels packaging, solid slice with 25 kg paper bag lined with plastic membrane or textile bag packaging Now, a German man named Norbert Blum has claimed to have solved the above riddle, which is properly known as the P vs NP problem. Blum, who is from the University of Bonn, claims in his recently published 38-page paper that P does not equal NP NP abbreviation stands for Not a problem. NP is an abbreviation for Not a problem. Questions. What most visitors search for before coming to this page

- istic polynomial time) because an NP problem can be solved in polynomial time by a nondeter
- Find out information about NP problem. A set or property of computational decision problems which A decision problem is called an NP problem if particular examples of it can be solved in polynomial (We would not say that an algorithm solves the P = NP problem if it assumes a primitive operation that..
- The P = NP Question. Module Home Page Title Page. Page 1 of 12 Back. Full Screen Close Quit. Lecture 32: P and NP Problems. • The problems in the picture that are in NP but not in P are ones that we're not sure about: - there is no known polynomial-time algorithm; - but no proof of intractability
- NP问题并不是那种只有搜才行的问题，NPC问题才是。 好，行了，基本上这个误解已经被澄清了。 在这里强调（回到我竭力想澄清的误区上），NP问题不是非P类问题（关键 N不是not的意思）。 NP问题的另一个定义是，可以在多项式的时间里猜出一个解的问题
- 「P ≠ NP 問題」とは、NP が P より真に広いかどうかを決定する問題です。 よくある誤解として以下のものがあります Why not register and get more from Qiita? We will deliver articles that match you. By following users and tags, you can catch up information on technical..
- We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering, and satisfiability. In each case, the required number of spins is at most cubic in the size of..

You're currently using an older browser and your experience may not be optimal. Please consider upgrading. Learn more Not quite... Rigorous technical definitions Because deterministic TMs are a subset of NDTMs, P is a subset of NP. A problem is NP-hard if an algorithm to solve it in (deterministic) polynomial time would make NP-complete is the class of problems which are both NP-hard and themselves members of NP Briefly, NP stands for Non-deterministic Polynomial time and it marks a class of problems that can be solved in polynomial time on a non-deterministic Turing These problems are marked NP-complete. It is believed (but not proved) that it is not possible to find them, hence the intersection between P and.. NP is probably the most important problem in theoretical computer science, with $1M prize. Vinay Deolalikar at HP Labs has released a preliminary 100+ Whether P (class of problems solvable in polynomial time) is not equal NP (class of problems solvable by a non-deterministic Turing machine in..

** Example problems not in P nor in NP-complete but in NP (3)**. I have a course called Algorithm Analysis at college, where we're currently We've already discussed NP-complete problems as the intersection between NP and NP-hard, and P problems, contained in NP. We've also talked about some examples..