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In 1928, Tesla received his last patent, U.S. Patent 1,655,114, for a biplane capable of taking off vertically (VTOL aircraft) and then be "gradually tilted through manipulation of the elevator devices" in flight until it was flying like a conventional plane. Tesla thought the plane would sell for less than $1,000.:251 Although the aircraft was probably impractical, it may be the earliest known design for what became the tiltrotor/tilt-wing concept as well as the earliest proposal for the use of turbine engines in rotor aircraft.[improper synthesis?]
Starting in 1934, the Westinghouse Electric & Manufacturing Company began paying Tesla $125 per month as well as paying his rent at the Hotel New Yorker, expenses the Company would pay for the rest of Tesla's life. Accounts on how this came about vary. Several sources say Westinghouse was worried about potential bad publicity surrounding the impoverished conditions their former star inventor was living under. It has been described as being couched in the form of a "consulting fee" to get around Tesla's aversion to accept charity, or by one biographer (Marc Seifer), as a type of unspecified settlement.
In 1935, in an annual birthday celebration interview, Tesla announced a method of transmitting mechanical energy with minimal loss over any terrestrial distance, a related new means of communication, and a method of accurately determining the location of underground mineral deposits.
In the fall of 1937, after midnight one night, Tesla left the Hotel New Yorker to make his regular commute to the cathedral and the library to feed the pigeons. While crossing a street a couple of blocks from the hotel, Tesla was unable to dodge a moving taxicab and was thrown heavily to the ground. Tesla's back was severely wrenched and three of his ribs were broken in the accident (the full extent of his injuries will never be known; Tesla refused to consult a doctor—an almost lifelong custom). Tesla didn't raise any question as to who was at fault and refused medical aid, only asking to be taken to his hotel via cab. Tesla was bedridden for some months and was unable to continue feeding pigeons from his window; soon, they failed to come. In early 1938, Tesla was able to get up. He at once resumed the pigeon-feeding walks on a much more limited scale, but frequently had a messenger act for him.
Later in life, Tesla made claims concerning a "teleforce" weapon after studying the Van de Graaff generator. The press variably referred to it as a "peace ray" or death ray. Tesla described the weapon as capable of being used against ground-based infantry or for anti-aircraft purposes.
In 1937, at a luncheon in his honor concerning the death ray, Tesla stated, "But it is not an experiment ... I have built, demonstrated and used it. Only a little time will pass before I can give it to the world." His records indicate that the device is based on a narrow stream of small tungsten pellets that are accelerated via high voltage (by means akin to his magnifying transformer).
During the same year, Tesla wrote a treatise, The Art of Projecting Concentrated Non-dispersive Energy through the Natural Media, concerning charged particle beam weapons. Tesla published the document in an attempt to expound on the technical description of a "superweapon that would put an end to all war." This treatise is currently in the Nikola Tesla Museum archive in Belgrade. It describes an open-ended vacuum tube with a gas jet seal that allows particles to exit, a method of charging particles to millions of volts, and a method of creating and directing non-dispersive particle streams (through electrostatic repulsion). Tesla tried to interest the US War Department, the United Kingdom, the Soviet Union, and Yugoslavia in the device.
During the period in which the negotiations were being conducted, Tesla said that efforts had been made to steal the invention. His room had been entered and his papers had been scrutinized, but the thieves, or spies, left empty-handed. He said that there was no danger that his invention could be stolen, for he had at no time committed any part of it to paper; the blueprint for the teleforce weapon was all in his mind.
On 7 January 1943, at the age of 86, Tesla died alone in room 3327 of the New Yorker Hotel. His body was later found by maid Alice Monaghan after she had entered Tesla's room, ignoring the "do not disturb" sign that Tesla had placed on his door two days earlier. Assistant medical examiner H.W. Wembly examined the body and ruled that the cause of death had been coronary thrombosis. Tesla's remains were taken to the Frank E. Campbell Funeral Home at Madison Ave. and 81st St. A long-time friend and supporter of Tesla, Hugo Gernsback, commissioned a sculptor to create a death mask, now displayed in the Nikola Tesla Museum.
Two days later, the FBI ordered the Alien Property Custodian to seize Tesla's belongings, even though Tesla was an American citizen. Tesla's entire estate from the Hotel New Yorker and other New York City hotels was transported to the Manhattan Storage and Warehouse Company under the Office of Alien Property (OAP) seal. John G. Trump, a professor at M.I.T. and a well-known electrical engineer serving as a technical aide to the National Defense Research Committee, was called in to analyze the Tesla items in OAP custody. After a three-day investigation, Trump's report concluded that there was nothing which would constitute a hazard in unfriendly hands, stating:
On 10 January 1943, New York City mayor Fiorello La Guardia read a eulogy written by Slovene-American author Louis Adamic live over the WNYC radio while violin pieces "Ave Maria" and "Tamo daleko" were played in the background. On 12 January, two thousand people attended a state funeral for Tesla at the Cathedral of Saint John the Divine. After the funeral, Tesla's body was taken to the Ferncliff Cemetery in Ardsley, New York, where it was later cremated. The following day, a second service was conducted by prominent priests in the Trinity Chapel (today's Serbian Orthodox Cathedral of Saint Sava) in New York City.
In 1952, following pressure from Tesla's nephew, Sava Kosanović, Tesla's entire estate was shipped to Belgrade in 80 trunks marked N.T. In 1957, Kosanović's secretary Charlotte Muzar transported Tesla's ashes from the United States to Belgrade. The ashes are displayed in a gold-plated sphere on a marble pedestal in the Nikola Tesla Museum.
Tesla obtained around 300 patents worldwide for his inventions. Some of Tesla's patents are not accounted for, and various sources have discovered some that have lain hidden in patent archives. There are a minimum of 278 patents issued to Tesla in 26 countries that have been accounted for. Many of Tesla's patents were in the United States, Britain, and Canada, but many other patents were approved in countries around the globe.:62 Many inventions developed by Tesla were not put into patent protection.
Tesla worked every day from 9:00 a.m. until 6:00 p.m. or later, with dinner from exactly 8:10 p.m., at Delmonico's restaurant and later the Waldorf-Astoria Hotel. Tesla would telephone his dinner order to the headwaiter, who also could be the only one to serve him. "The meal was required to be ready at eight o'clock ... He dined alone, except on the rare occasions when he would give a dinner to a group to meet his social obligations. Tesla would then resume his work, often until 3:00 a.m.":283, 286
For exercise, Tesla walked between 8 to 10 miles per day. He squished his toes one hundred times for each foot every night, saying that it stimulated his brain cells.
In an interview with newspaper editor Arthur Brisbane, Tesla said that he did not believe in telepathy, stating, "Suppose I made up my mind to murder you," he said, "In a second you would know it. Now, isn't that wonderful? By what process does the mind get at all this?" In the same interview, Tesla said that he believed that all fundamental laws could be reduced to one.
Near the end of his life, Tesla walked to the park every day to feed the pigeons and even brought injured ones into his hotel room to nurse back to health. He said that he had been visited by a specific injured white pigeon daily. Tesla spent over $2,000, including building a device that comfortably supported her so her bones could heal, to fix her broken wing and leg. Tesla stated,
Tesla was 6 feet 2 inches (1.88 m) tall and weighed 142 pounds (64 kg), with almost no weight variance from 1888 to about 1926.:292 He was an elegant, stylish figure in New York City, meticulous in his grooming, clothing, and regimented in his daily activities.
Tesla read many works, memorizing complete books, and supposedly possessed a photographic memory.:33 He was a polyglot, speaking eight languages: Serbo-Croatian, Czech, English, French, German, Hungarian, Italian, and Latin.:282 Tesla related in his autobiography that he experienced detailed moments of inspiration. During his early life, Tesla was repeatedly stricken with illness. He suffered a peculiar affliction in which blinding flashes of light would appear before his eyes, often accompanied by visions.:33 Often, the visions were linked to a word or idea he might have come across; at other times they would provide the solution to a particular problem he had encountered. Just by hearing the name of an item, he would be able to envision it in realistic detail.:33 Tesla would visualize an invention in his mind with extreme precision, including all dimensions, before moving to the construction stage, a technique sometimes known as picture thinking. He typically did not make drawings by hand but worked from memory. Beginning in his childhood, Tesla had frequent flashbacks to events that had happened previously in his life.:33
During his second year of study at Graz, Tesla developed a passion for (and became very proficient at) billiards, chess and card-playing, sometimes spending more than 48 hours in a stretch at a gaming table.:43, 301 On one occasion at his laboratory, Tesla worked for a period of 84 hours without sleep or rest.:208 Kenneth Swezey, a journalist whom Tesla had befriended, confirmed that Tesla rarely slept. Swezey recalled one morning when Tesla called him at 3 a.m.: "I was sleeping in my room like one dead ... Suddenly, the telephone ring awakened me ... [Tesla] spoke animatedly, with pauses, [as he] ... work[ed] out a problem, comparing one theory to another, commenting; and when he felt he had arrived at the solution, he suddenly closed the telephone."
Tesla never married; he said his chastity was very helpful to his scientific abilities.:33 However, toward the end of his life, he told a reporter, "Sometimes I feel that by not marrying, I made too great a sacrifice to my work ..." There have been numerous accounts of women vying for Tesla's affection, even some madly in love with him.[citation needed] Tesla, though polite and soft-spoken, did not have any known relationships.
Tesla was asocial and prone to seclude himself with his work. However, when he did engage in a social life, many people spoke very positively and admiringly of Tesla. Robert Underwood Johnson described him as attaining a "distinguished sweetness, sincerity, modesty, refinement, generosity, and force." His loyal secretary, Dorothy Skerrit, wrote: "his genial smile and nobility of bearing always denoted the gentlemanly characteristics that were so ingrained in his soul." Tesla's friend, Julian Hawthorne, wrote, "seldom did one meet a scientist or engineer who was also a poet, a philosopher, an appreciator of fine music, a linguist, and a connoisseur of food and drink.":80
Tesla was a good friend of Francis Marion Crawford, Robert Underwood Johnson, Stanford White, Fritz Lowenstein, George Scherff, and Kenneth Swezey. In middle age, Tesla became a close friend of Mark Twain; they spent a lot of time together in his lab and elsewhere. Twain notably described Tesla's induction motor invention as "the most valuable patent since the telephone." In the late 1920s, Tesla also befriended George Sylvester Viereck, a poet, writer, mystic, and later, a Nazi propagandist. Tesla occasionally attended dinner parties held by Viereck and his wife.
Tesla could be harsh at times and openly expressed disgust for overweight people, such as when he fired a secretary because of her weight.:110 He was quick to criticize clothing; on several occasions, Tesla directed a subordinate to go home and change her dress.:33
Tesla exhibited a pre-atomic understanding of physics in his writings; he disagreed with the theory of atoms being composed of smaller subatomic particles, stating there was no such thing as an electron creating an electric charge (he believed that if electrons existed at all, they were some fourth state of matter or "sub-atom" that could only exist in an experimental vacuum and that they had nothing to do with electricity).:249 Tesla believed that atoms are immutable—they could not change state or be split in any way. He was a believer in the 19th century concept of an all pervasive "ether" that transmitted electrical energy.
Tesla was generally antagonistic towards theories about the conversion of matter into energy.:247 He was also critical of Einstein's theory of relativity, saying:
Tesla claimed to have developed his own physical principle regarding matter and energy that he started working on in 1892, and in 1937, at age 81, claimed in a letter to have completed a "dynamic theory of gravity" that "[would] put an end to idle speculations and false conceptions, as that of curved space." He stated that the theory was "worked out in all details" and that he hoped to soon give it to the world. Further elucidation of his theory was never found in his writings.:309
Tesla, like many of his era, became a proponent of an imposed selective breeding version of eugenics. His opinion stemmed from the belief that humans' "pity" had interfered with the natural "ruthless workings of nature," rather than from conceptions of a "master race" or inherent superiority of one person over another. His advocacy of it was, however, to push it further. In a 1937 interview, he stated:
In 1926, Tesla commented on the ills of the social subservience of women and the struggle of women toward gender equality, and indicated that humanity's future would be run by "Queen Bees." He believed that women would become the dominant sex in the future.
Tesla made predictions about the relevant issues of a post-World War I environment in a printed article, "Science and Discovery are the great Forces which will lead to the Consummation of the War" (20 December 1914). Tesla believed that the League of Nations was not a remedy for the times and issues.[citation needed]
Tesla was raised an Orthodox Christian. Later in his life, he did not consider himself to be a "believer in the orthodox sense," and opposed religious fanaticism. Despite this, he had a profound respect for both Buddhism and Christianity.
However, his religious views remain uncertain due to other statements that he made. For example, in his article, "A Machine to End War", published in 1937, Tesla stated:
Tesla wrote a number of books and articles for magazines and journals. Among his books are My Inventions: The Autobiography of Nikola Tesla, compiled and edited by Ben Johnston; The Fantastic Inventions of Nikola Tesla, compiled and edited by David Hatcher Childress; and The Tesla Papers.
Many of Tesla's writings are freely available on the web, including the article "The Problem of Increasing Human Energy," published in The Century Magazine in 1900, and the article "Experiments With Alternate Currents Of High Potential And High Frequency," published in his book Inventions, Researches and Writings of Nikola Tesla.
Tesla's legacy has endured in books, films, radio, TV, music, live theater, comics and video games. The impact of the technologies invented or envisioned by Tesla is a recurring theme in several types of science fiction.
On Tesla's 75th birthday in 1931, Time magazine put him on its cover. The cover caption "All the world's his power house" noted his contribution to electrical power generation. He received congratulatory letters from more than 70 pioneers in science and engineering, including Albert Einstein.
Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm.
A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do.
Closely related fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem. More precisely, it tries to classify problems that can or cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in principle, be solved algorithmically.
A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing. The instance is a number (e.g. 15) and the solution is "yes" if the number is prime and "no" otherwise (in this case "no"). Stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input.
To further highlight the difference between a problem and an instance, consider the following instance of the decision version of the traveling salesman problem: Is there a route of at most 2000 kilometres passing through all of Germany's 15 largest cities? The quantitative answer to this particular problem instance is of little use for solving other instances of the problem, such as asking for a round trip through all sites in Milan whose total length is at most 10 km. For this reason, complexity theory addresses computational problems and not particular problem instances.
When considering computational problems, a problem instance is a string over an alphabet. Usually, the alphabet is taken to be the binary alphabet (i.e., the set {0,1}), and thus the strings are bitstrings. As in a real-world computer, mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in binary.
Decision problems are one of the central objects of study in computational complexity theory. A decision problem is a special type of computational problem whose answer is either yes or no, or alternately either 1 or 0. A decision problem can be viewed as a formal language, where the members of the language are instances whose output is yes, and the non-members are those instances whose output is no. The objective is to decide, with the aid of an algorithm, whether a given input string is a member of the formal language under consideration. If the algorithm deciding this problem returns the answer yes, the algorithm is said to accept the input string, otherwise it is said to reject the input.
An example of a decision problem is the following. The input is an arbitrary graph. The problem consists in deciding whether the given graph is connected, or not. The formal language associated with this decision problem is then the set of all connected graphs—of course, to obtain a precise definition of this language, one has to decide how graphs are encoded as binary strings.
A function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem, that is, it isn't just yes or no. Notable examples include the traveling salesman problem and the integer factorization problem.
It is tempting to think that the notion of function problems is much richer than the notion of decision problems. However, this is not really the case, since function problems can be recast as decision problems. For example, the multiplication of two integers can be expressed as the set of triples (a, b, c) such that the relation a × b = c holds. Deciding whether a given triple is a member of this set corresponds to solving the problem of multiplying two numbers.
To measure the difficulty of solving a computational problem, one may wish to see how much time the best algorithm requires to solve the problem. However, the running time may, in general, depend on the instance. In particular, larger instances will require more time to solve. Thus the time required to solve a problem (or the space required, or any measure of complexity) is calculated as a function of the size of the instance. This is usually taken to be the size of the input in bits. Complexity theory is interested in how algorithms scale with an increase in the input size. For instance, in the problem of finding whether a graph is connected, how much more time does it take to solve a problem for a graph with 2n vertices compared to the time taken for a graph with n vertices?
If the input size is n, the time taken can be expressed as a function of n. Since the time taken on different inputs of the same size can be different, the worst-case time complexity T(n) is defined to be the maximum time taken over all inputs of size n. If T(n) is a polynomial in n, then the algorithm is said to be a polynomial time algorithm. Cobham's thesis says that a problem can be solved with a feasible amount of resources if it admits a polynomial time algorithm.
A Turing machine is a mathematical model of a general computing machine. It is a theoretical device that manipulates symbols contained on a strip of tape. Turing machines are not intended as a practical computing technology, but rather as a thought experiment representing a computing machine—anything from an advanced supercomputer to a mathematician with a pencil and paper. It is believed that if a problem can be solved by an algorithm, there exists a Turing machine that solves the problem. Indeed, this is the statement of the Church–Turing thesis. Furthermore, it is known that everything that can be computed on other models of computation known to us today, such as a RAM machine, Conway's Game of Life, cellular automata or any programming language can be computed on a Turing machine. Since Turing machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing machine is the most commonly used model in complexity theory.
A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm.
Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines, probabilistic Turing machines, non-deterministic Turing machines, quantum Turing machines, symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources (such as time or space) are bounded, some of these may be more powerful than others.
Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines. Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. What all these models have in common is that the machines operate deterministically.
However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.
For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M is said to operate within time f(n), if the time required by M on each input of length n is at most f(n). A decision problem A can be solved in time f(n) if there exists a Turing machine operating in time f(n) that solves the problem. Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f(n) on a deterministic Turing machine is then denoted by DTIME(f(n)).
Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity, circuit complexity, and decision tree complexity.
The best, worst and average case complexity refer to three different ways of measuring the time complexity (or any other complexity measure) of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:
For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time.
To classify the computation time (or similar resources, such as space consumption), one is interested in proving upper and lower bounds on the minimum amount of time required by the most efficient algorithm solving a given problem. The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T(n) on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T(n). However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T(n) for a problem requires showing that no algorithm can have time complexity lower than T(n).
Upper and lower bounds are usually stated using the big O notation, which hides constant factors and smaller terms. This makes the bounds independent of the specific details of the computational model used. For instance, if T(n) = 7n2 + 15n + 40, in big O notation one would write T(n) = O(n2).
Of course, some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following:
But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx \| x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.
Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following:
Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.
For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.
The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.
Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.
The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.
This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.
If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.
The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.
The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.
It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.
The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes.
Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory.
Along the same lines, co-NP is the class containing the complement problems (i.e. problems with the yes/no answers reversed) of NP problems. It is believed that NP is not equal to co-NP; however, it has not yet been proven. It has been shown that if these two complexity classes are not equal then P is not equal to NP.
Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes.
Problems that can be solved in theory (e.g., given large but finite time), but which in practice take too long for their solutions to be useful, are known as intractable problems. In complexity theory, problems that lack polynomial-time solutions are considered to be intractable for more than the smallest inputs. In fact, the Cobham–Edmonds thesis states that only those problems that can be solved in polynomial time can be feasibly computed on some computational device. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If NP is not the same as P, then the NP-complete problems are also intractable in this sense. To see why exponential-time algorithms might be unusable in practice, consider a program that makes 2n operations before halting. For small n, say 100, and assuming for the sake of example that the computer does 1012 operations each second, the program would run for about 4 × 1010 years, which is the same order of magnitude as the age of the universe. Even with a much faster computer, the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress. Nevertheless, a polynomial time algorithm is not always practical. If its running time is, say, n15, it is unreasonable to consider it efficient and it is still useless except on small instances.
What intractability means in practice is open to debate. Saying that a problem is not in P does not imply that all large cases of the problem are hard or even that most of them are. For example, the decision problem in Presburger arithmetic has been shown not to be in P, yet algorithms have been written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of sizes in less than quadratic time and SAT solvers routinely handle large instances of the NP-complete Boolean satisfiability problem.
Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. Most influential among these was the definition of Turing machines by Alan Turing in 1936, which turned out to be a very robust and flexible simplification of a computer.
As Fortnow & Homer (2003) point out, the beginning of systematic studies in computational complexity is attributed to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965), which laid out the definitions of time and space complexity and proved the hierarchy theorems. Also, in 1965 Edmonds defined a "good" algorithm as one with running time bounded by a polynomial of the input size.
Earlier papers studying problems solvable by Turing machines with specific bounded resources include John Myhill's definition of linear bounded automata (Myhill 1960), Raymond Smullyan's study of rudimentary sets (1961), as well as Hisao Yamada's paper on real-time computations (1962). Somewhat earlier, Boris Trakhtenbrot (1956), a pioneer in the field from the USSR, studied another specific complexity measure. As he remembers:
Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.
In 1967, Manuel Blum developed an axiomatic complexity theory based on his axioms and proved an important result, the so-called, speed-up theorem. The field really began to flourish in 1971 when the US researcher Stephen Cook and, working independently, Leonid Levin in the USSR, proved that there exist practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, "Reducibility Among Combinatorial Problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete.
The role of teacher is often formal and ongoing, carried out at a school or other place of formal education. In many countries, a person who wishes to become a teacher must first obtain specified professional qualifications or credentials from a university or college. These professional qualifications may include the study of pedagogy, the science of teaching. Teachers, like other professionals, may have to continue their education after they qualify, a process known as continuing professional development. Teachers may use a lesson plan to facilitate student learning, providing a course of study which is called the curriculum.
A teacher's role may vary among cultures. Teachers may provide instruction in literacy and numeracy, craftsmanship or vocational training, the arts, religion, civics, community roles, or life skills.
In some countries, formal education can take place through home schooling. Informal learning may be assisted by a teacher occupying a transient or ongoing role, such as a family member, or by anyone with knowledge or skills in the wider community setting.
Religious and spiritual teachers, such as gurus, mullahs, rabbis, pastors/youth pastors and lamas, may teach religious texts such as the Quran, Torah or Bible.
Teaching may be carried out informally, within the family, which is called homeschooling, or in the wider community. Formal teaching may be carried out by paid professionals. Such professionals enjoy a status in some societies on a par with physicians, lawyers, engineers, and accountants (Chartered or CPA).
A teacher's professional duties may extend beyond formal teaching. Outside of the classroom teachers may accompany students on field trips, supervise study halls, help with the organization of school functions, and serve as supervisors for extracurricular activities. In some education systems, teachers may have responsibility for student discipline.
There are a variety of bodies designed to instill, preserve and update the knowledge and professional standing of teachers. Around the world many governments operate teacher's colleges, which are generally established to serve and protect the public interest through certifying, governing and enforcing the standards of practice for the teaching profession.
The functions of the teacher's colleges may include setting out clear standards of practice, providing for the ongoing education of teachers, investigating complaints involving members, conducting hearings into allegations of professional misconduct and taking appropriate disciplinary action and accrediting teacher education programs. In many situations teachers in publicly funded schools must be members in good standing with the college, and private schools may also require their teachers to be college peoples. In other areas these roles may belong to the State Board of Education, the Superintendent of Public Instruction, the State Education Agency or other governmental bodies. In still other areas Teaching Unions may be responsible for some or all of these duties.
In education, teachers facilitate student learning, often in a school or academy or perhaps in another environment such as outdoors. A teacher who teaches on an individual basis may be described as a tutor.
The objective is typically accomplished through either an informal or formal approach to learning, including a course of study and lesson plan that teaches skills, knowledge and/or thinking skills. Different ways to teach are often referred to as pedagogy. When deciding what teaching method to use teachers consider students' background knowledge, environment, and their learning goals as well as standardized curricula as determined by the relevant authority. Many times, teachers assist in learning outside of the classroom by accompanying students on field trips. The increasing use of technology, specifically the rise of the internet over the past decade, has begun to shape the way teachers approach their roles in the classroom.
The objective is typically a course of study, lesson plan, or a practical skill. A teacher may follow standardized curricula as determined by the relevant authority. The teacher may interact with students of different ages, from infants to adults, students with different abilities and students with learning disabilities.
Teaching using pedagogy also involve assessing the educational levels of the students on particular skills. Understanding the pedagogy of the students in a classroom involves using differentiated instruction as well as supervision to meet the needs of all students in the classroom. Pedagogy can be thought of in two manners. First, teaching itself can be taught in many different ways, hence, using a pedagogy of teaching styles. Second, the pedagogy of the learners comes into play when a teacher assesses the pedagogic diversity of his/her students and differentiates for the individual students accordingly. For example, an experienced teacher and parent described the place of a teacher in learning as follows: "The real bulk of learning takes place in self-study and problem solving with a lot of feedback around that loop. The function of the teacher is to pressure the lazy, inspire the bored, deflate the cocky, encourage the timid, detect and correct individual flaws, and broaden the viewpoint of all. This function looks like that of a coach using the whole gamut of psychology to get each new class of rookies off the bench and into the game."
Perhaps the most significant difference between primary school and secondary school teaching is the relationship between teachers and children. In primary schools each class has a teacher who stays with them for most of the week and will teach them the whole curriculum. In secondary schools they will be taught by different subject specialists each session during the week and may have ten or more different teachers. The relationship between children and their teachers tends to be closer in the primary school where they act as form tutor, specialist teacher and surrogate parent during the course of the day.
This is true throughout most of the United States as well. However, alternative approaches for primary education do exist. One of these, sometimes referred to as a "platoon" system, involves placing a group of students together in one class that moves from one specialist to another for every subject. The advantage here is that students learn from teachers who specialize in one subject and who tend to be more knowledgeable in that one area than a teacher who teaches many subjects. Students still derive a strong sense of security by staying with the same group of peers for all classes.
Co-teaching has also become a new trend amongst educational institutions. Co-teaching is defined as two or more teachers working harmoniously to fulfill the needs of every student in the classroom. Co-teaching focuses the student on learning by providing a social networking support that allows them to reach their full cognitive potential. Co-teachers work in sync with one another to create a climate of learning.
Throughout the history of education the most common form of school discipline was corporal punishment. While a child was in school, a teacher was expected to act as a substitute parent, with all the normal forms of parental discipline open to them.

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