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Algorithms for calculating variance
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population_variance = S / w_sum
# Bessel's correction for weighted samples
# Frequency weights
sample_frequency_variance = S / (w_sum - 1)
# Reliability weights
sample_reliability_variance = S / (w_sum - w_sum2 / w_sum) Parallel algorithm
Chan et al. note that Welford's online algorithm detailed above is a special case of an algorithm that works for combining arbitrary sets and :
.
This may be useful when, for example, multiple processing units may be assigned to discrete parts of the input.
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Chan's method for estimating the mean is numerically unstable when and both are large, because the numerical error in is not scaled down in the way that it is in the case. In such cases, prefer .
def parallel_variance(n_a, avg_a, M2_a, n_b, avg_b, M2_b):
n = n_a + n_b
delta = avg_b - avg_a
M2 = M2_a + M2_b + delta ** 2 * n_a * n_b / n
var_ab = M2 / (n - 1)
return var_ab
This can be generalized to allow parallelization with AVX, with GPUs, and computer clusters, and to covariance.
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Example
Assume that all floating point operations use standard IEEE 754 double-precision arithmetic. Consider the sample (4, 7, 13, 16) from an infinite population. Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both the naïve algorithm and two-pass algorithm compute these values correctly. Next consider the sample (, , , ), which gives rise to the same estimated variance as the first sample. The two-pass algorithm computes this variance estimate correctly, but the naïve algorithm returns 29.333333333333332 instead of 30.
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While this loss of precision may be tolerable and viewed as a minor flaw of the naïve algorithm, further increasing the offset makes the error catastrophic. Consider the sample (, , , ). Again the estimated population variance of 30 is computed correctly by the two-pass algorithm, but the naïve algorithm now computes it as −170.66666666666666. This is a serious problem with naïve algorithm and is due to catastrophic cancellation in the subtraction of two similar numbers at the final stage of the algorithm.
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Higher-order statistics
Terriberry extends Chan's formulae to calculating the third and fourth central moments, needed for example when estimating skewness and kurtosis: Here the are again the sums of powers of differences from the mean , giving For the incremental case (i.e., ), this simplifies to: By preserving the value , only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.
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An example of the online algorithm for kurtosis implemented as described is:
def online_kurtosis(data):
n = mean = M2 = M3 = M4 = 0
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for x in data:
n1 = n
n = n + 1
delta = x - mean
delta_n = delta / n
delta_n2 = delta_n * delta_n
term1 = delta * delta_n * n1
mean = mean + delta_n
M4 = M4 + term1 * delta_n2 * (n*n - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3
M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2
M2 = M2 + term1
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# Note, you may also calculate variance using M2, and skewness using M3
# Caution: If all the inputs are the same, M2 will be 0, resulting in a division by 0.
kurtosis = (n * M4) / (M2 * M2) - 3
return kurtosis Pébaÿ
further extends these results to arbitrary-order central moments, for the incremental and the pairwise cases, and subsequently Pébaÿ et al.
for weighted and compound moments. One can also find there similar formulas for covariance.
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Choi and Sweetman
offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware. A relative histogram of a random variable can be constructed in the conventional way: the range of potential values is
divided into bins and the number of occurrences within each bin are counted and plotted such that the area of each rectangle equals the portion of the sample values within that bin:
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where and represent the frequency and the relative frequency at bin and is the total area of the histogram. After this normalization, the raw moments and central moments of can be calculated from the relative histogram: where the superscript indicates the moments are calculated from the histogram. For constant bin width these two expressions can be simplified using :
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The second approach from Choi and Sweetman is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times. If sets of statistical moments are known:
for , then each can
be expressed in terms of the equivalent raw moments: where is generally taken to be the duration of the time-history, or the number of points if is constant.
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The benefit of expressing the statistical moments in terms of is that the sets can be combined by addition, and there is no upper limit on the value of . where the subscript represents the concatenated time-history or combined . These combined values of can then be inversely transformed into raw moments representing the complete concatenated time-history Known relationships between the raw moments () and the central moments ()
are then used to compute the central moments of the concatenated time-history. Finally, the statistical moments of the concatenated history are computed from the central moments:
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Covariance
Very similar algorithms can be used to compute the covariance. Naïve algorithm
The naïve algorithm is For the algorithm above, one could use the following Python code:
def naive_covariance(data1, data2):
n = len(data1)
sum12 = 0
sum1 = sum(data1)
sum2 = sum(data2) for i1, i2 in zip(data1, data2):
sum12 += i1 * i2 covariance = (sum12 - sum1 * sum2 / n) / n
return covariance
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With estimate of the mean
As for the variance, the covariance of two random variables is also shift-invariant, so given any two constant values and it can be written: and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums. Taking the first value of each data set, the algorithm can be written as:
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def shifted_data_covariance(data_x, data_y):
n = len(data_x)
if n < 2:
return 0
kx = data_x[0]
ky = data_y[0]
Ex = Ey = Exy = 0
for ix, iy in zip(data_x, data_y):
Ex += ix - kx
Ey += iy - ky
Exy += (ix - kx) * (iy - ky)
return (Exy - Ex * Ey / n) / n
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Two-pass
The two-pass algorithm first computes the sample means, and then the covariance: The two-pass algorithm may be written as:
def two_pass_covariance(data1, data2):
n = len(data1) mean1 = sum(data1) / n
mean2 = sum(data2) / n covariance = 0 for i1, i2 in zip(data1, data2):
a = i1 - mean1
b = i2 - mean2
covariance += a * b / n
return covariance
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A slightly more accurate compensated version performs the full naive algorithm on the residuals. The final sums and should be zero, but the second pass compensates for any small error. Online A stable one-pass algorithm exists, similar to the online algorithm for computing the variance, that computes co-moment : The apparent asymmetry in that last equation is due to the fact that , so both update terms are equal to . Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals. Thus the covariance can be computed as
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def online_covariance(data1, data2):
meanx = meany = C = n = 0
for x, y in zip(data1, data2):
n += 1
dx = x - meanx
meanx += dx / n
meany += (y - meany) / n
C += dx * (y - meany) population_covar = C / n
# Bessel's correction for sample variance
sample_covar = C / (n - 1) A small modification can also be made to compute the weighted covariance:
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def online_weighted_covariance(data1, data2, data3):
meanx = meany = 0
wsum = wsum2 = 0
C = 0
for x, y, w in zip(data1, data2, data3):
wsum += w
wsum2 += w * w
dx = x - meanx
meanx += (w / wsum) * dx
meany += (w / wsum) * (y - meany)
C += w * dx * (y - meany)
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population_covar = C / wsum
# Bessel's correction for sample variance
# Frequency weights
sample_frequency_covar = C / (wsum - 1)
# Reliability weights
sample_reliability_covar = C / (wsum - wsum2 / wsum) Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation: Weighted batched version A version of the weighted online algorithm that does batched updated also exists: let
denote the weights, and write The covariance can then be computed as
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See also
Kahan summation algorithm
Squared deviations from the mean
Yamartino method References External links Statistical algorithms
Statistical deviation and dispersion
Articles with example pseudocode
Articles with example Python (programming language) code
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Algebraic number
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An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of . All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers.
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Algebraic number
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The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.
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Algebraic number
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Examples
All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a (non-zero) natural number , satisfies the above definition, because is the root of a non-zero polynomial, namely .
Quadratic irrational numbers, irrational solutions of a quadratic polynomial with integer coefficients , , and ), are algebraic numbers. If the quadratic polynomial is monic (), the roots are further qualified as quadratic integers.
Gaussian integers, complex numbers for which both and are integers, are also quadratic integers. This is because and are the two roots of the quadratic .
A constructible number can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (By designating cardinal directions for 1, −1, , and −, complex numbers such as are considered constructible.)
Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of th roots gives another algebraic number.
Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of th roots (such as the roots of ). That happens with many but not all polynomials of degree 5 or higher.
Values of trigonometric functions of rational multiples of (except when undefined): for example, , , and satisfy . This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, , , , and satisfy the irreducible polynomial , and so are conjugate algebraic integers.
Some but not all irrational numbers are algebraic:
The numbers and are algebraic since they are roots of polynomials and , respectively.
The golden ratio is algebraic since it is a root of the polynomial .
The numbers and e are not algebraic numbers (see the Lindemann–Weierstrass theorem).
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Algebraic number
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Properties
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Algebraic number
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If a polynomial with rational coefficients is multiplied through by the least common denominator, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.
Given an algebraic number, there is a unique monic polynomial with rational coefficients of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree , then the algebraic number is said to be of degree . For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.
The algebraic numbers are dense in the reals. This follows from the fact they contain the rational numbers, which are dense in the reals themselves.
The set of algebraic numbers is countable (enumerable), and therefore its Lebesgue measure as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, "almost all" real and complex numbers are transcendental.
All algebraic numbers are computable and therefore definable and arithmetical.
For real numbers and , the complex number is algebraic if and only if both and are algebraic.
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Algebraic number
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Field The sum, difference, product and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic, as can be demonstrated by using the resultant, and algebraic numbers thus form a field (sometimes denoted by , but that usually denotes the adele ring). Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. That can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it is the smallest algebraically-closed field containing the rationals and so it is called the algebraic closure of the rationals.
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Algebraic number
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The set of real algebraic numbers itself forms a field. Related fields Numbers defined by radicals
Any number that can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking (possibly complex) th roots where is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For example, the equation:
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Algebraic number
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has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations. Closed-form number
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Algebraic number
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Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as or ln 2.
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Algebraic number
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Algebraic integers An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are and Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials for all . In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers.
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Algebraic number
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The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If is a number field, its ring of integers is the subring of algebraic integers in , and is frequently denoted as . These are the prototypical examples of Dedekind domains.
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Algebraic number
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Special classes
Algebraic solution
Gaussian integer
Eisenstein integer
Quadratic irrational number
Fundamental unit
Root of unity
Gaussian period
Pisot–Vijayaraghavan number
Salem number Notes References Hardy, G. H. and Wright, E. M. 1978, 2000 (with general index) An Introduction to the Theory of Numbers: 5th Edition, Clarendon Press, Oxford UK,
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Algebraic number
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Niven, Ivan 1956. Irrational Numbers, Carus Mathematical Monograph no. 11, Mathematical Association of America.
Ore, Øystein 1948, 1988, Number Theory and Its History, Dover Publications, Inc. New York, (pbk.)
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Artificial intelligence
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Artificial intelligence (AI) is intelligence demonstrated by machines, as opposed to natural intelligence displayed by animals including humans. Leading AI textbooks define the field as the study of "intelligent agents": any system that perceives its environment and takes actions that maximize its chance of achieving its goals. Some popular accounts use the term "artificial intelligence" to describe machines that mimic "cognitive" functions that humans associate with the human mind, such as "learning" and "problem solving", however, this definition is rejected by major AI researchers.
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Artificial intelligence
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AI applications include advanced web search engines (e.g., Google), recommendation systems (used by YouTube, Amazon and Netflix), understanding human speech (such as Siri and Alexa), self-driving cars (e.g., Tesla), automated decision-making and competing at the highest level in strategic game systems (such as chess and Go).
As machines become increasingly capable, tasks considered to require "intelligence" are often removed from the definition of AI, a phenomenon known as the AI effect. For instance, optical character recognition is frequently excluded from things considered to be AI, having become a routine technology.
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Artificial intelligence
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Artificial intelligence was founded as an academic discipline in 1956, and in the years since has experienced several waves of optimism, followed by disappointment and the loss of funding (known as an "AI winter"), followed by new approaches, success and renewed funding. AI research has tried and discarded many different approaches since its founding, including simulating the brain, modeling human problem solving, formal logic, large databases of knowledge and imitating animal behavior. In the first decades of the 21st century, highly mathematical statistical machine learning has dominated the field, and this technique has proved highly successful, helping to solve many challenging problems throughout industry and academia.
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Artificial intelligence
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The various sub-fields of AI research are centered around particular goals and the use of particular tools. The traditional goals of AI research include reasoning, knowledge representation, planning, learning, natural language processing, perception, and the ability to move and manipulate objects. General intelligence (the ability to solve an arbitrary problem) is among the field's long-term goals. To solve these problems, AI researchers have adapted and integrated a wide range of problem-solving techniques—including search and mathematical optimization, formal logic, artificial neural networks, and methods based on statistics, probability and economics. AI also draws upon computer science, psychology, linguistics, philosophy, and many other fields.
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Artificial intelligence
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The field was founded on the assumption that human intelligence "can be so precisely described that a machine can be made to simulate it".
This raises philosophical arguments about the mind and the ethics of creating artificial beings endowed with human-like intelligence. These issues have been explored by myth, fiction, and philosophy since antiquity.
Science fiction and futurology have also suggested that, with its enormous potential and power, AI may become an existential risk to humanity. History
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Artificial beings with intelligence appeared as storytelling devices in antiquity,
and have been common in fiction, as in Mary Shelley's Frankenstein or Karel Čapek's R.U.R. These characters and their fates raised many of the same issues now discussed in the ethics of artificial intelligence.
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Artificial intelligence
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The study of mechanical or "formal" reasoning began with philosophers and mathematicians in antiquity. The study of mathematical logic led directly to Alan Turing's theory of computation, which suggested that a machine, by shuffling symbols as simple as "0" and "1", could simulate any conceivable act of mathematical deduction. This insight that digital computers can simulate any process of formal reasoning is known as the Church–Turing thesis.
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Artificial intelligence
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The Church-Turing thesis, along with concurrent discoveries in neurobiology, information theory and cybernetics, led researchers to consider the possibility of building an electronic brain.
The first work that is now generally recognized as AI was McCullouch and Pitts' 1943 formal design for Turing-complete "artificial neurons". When access to digital computers became possible in the mid-1950s, AI research began to explore the possibility that human intelligence could be reduced to step-by-step symbol manipulation, known as Symbolic AI or GOFAI. Approaches based on cybernetics or artificial neural networks were abandoned or pushed into the background.
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Artificial intelligence
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The field of AI research was born at a workshop at Dartmouth College in 1956.
The attendees became the founders and leaders of AI research.
They and their students produced programs that the press described as "astonishing":
computers were learning checkers strategies, solving word problems in algebra, proving logical theorems and speaking English.
By the middle of the 1960s, research in the U.S. was heavily funded by the Department of Defense
and laboratories had been established around the world.
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Artificial intelligence
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Researchers in the 1960s and the 1970s were convinced that symbolic approaches would eventually succeed in creating a machine with artificial general intelligence and considered this the goal of their field.
Herbert Simon predicted, "machines will be capable, within twenty years, of doing any work a man can do".
Marvin Minsky agreed, writing, "within a generation ... the problem of creating 'artificial intelligence' will substantially be solved".
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Artificial intelligence
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They failed to recognize the difficulty of some of the remaining tasks. Progress slowed and in 1974, in response to the criticism of Sir James Lighthill
and ongoing pressure from the US Congress to fund more productive projects, both the U.S. and British governments cut off exploratory research in AI. The next few years would later be called an "AI winter", a period when obtaining funding for AI projects was difficult.
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Artificial intelligence
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In the early 1980s, AI research was revived by the commercial success of expert systems,
a form of AI program that simulated the knowledge and analytical skills of human experts. By 1985, the market for AI had reached over a billion dollars. At the same time, Japan's fifth generation computer project inspired the U.S and British governments to restore funding for academic research.
However, beginning with the collapse of the Lisp Machine market in 1987, AI once again fell into disrepute, and a second, longer-lasting winter began.
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Artificial intelligence
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Many researchers began to doubt that the symbolic approach would be able to imitate all the processes of human cognition, especially perception, robotics, learning and pattern recognition. A number of researchers began to look into "sub-symbolic" approaches to specific AI problems. Robotics researchers, such as Rodney Brooks, rejected symbolic AI and focused on the basic engineering problems that would allow robots to move, survive, and learn their environment.
Interest in neural networks and "connectionism" was revived by Geoffrey Hinton, David Rumelhart and others in the middle of the 1980s.
Soft computing tools were developed in the 80s, such as neural networks, fuzzy systems, Grey system theory, evolutionary computation and many tools drawn from statistics or mathematical optimization.
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Artificial intelligence
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AI gradually restored its reputation in the late 1990s and early 21st century by finding specific solutions to specific problems. The narrow focus allowed researchers to produce verifiable results, exploit more mathematical methods, and collaborate with other fields (such as statistics, economics and mathematics).
By 2000, solutions developed by AI researchers were being widely used, although in the 1990s they were rarely described as "artificial intelligence".
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Artificial intelligence
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Faster computers, algorithmic improvements, and access to large amounts of data enabled advances in machine learning and perception; data-hungry deep learning methods started to dominate accuracy benchmarks around 2012.
According to Bloomberg's Jack Clark, 2015 was a landmark year for artificial intelligence, with the number of software projects that use AI within Google increased from a "sporadic usage" in 2012 to more than 2,700 projects. He attributes this to an increase in affordable neural networks, due to a rise in cloud computing infrastructure and to an increase in research tools and datasets. In a 2017 survey, one in five companies reported they had "incorporated AI in some offerings or processes". The amount of research into AI (measured by total publications) increased by 50% in the years 2015–2019.
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Artificial intelligence
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Numerous academic researchers became concerned that AI was no longer pursuing the original goal of creating versatile, fully intelligent machines. Much of current research involves statistical AI, which is overwhelmingly used to solve specific problems, even highly successful techniques such as deep learning. This concern has led to the subfield artificial general intelligence (or "AGI"), which had several well-funded institutions by the 2010s. Goals
The general problem of simulating (or creating) intelligence has been broken down into sub-problems. These consist of particular traits or capabilities that researchers expect an intelligent system to display. The traits described below have received the most attention.
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Artificial intelligence
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Reasoning, problem solving Early researchers developed algorithms that imitated step-by-step reasoning that humans use when they solve puzzles or make logical deductions.
By the late 1980s and 1990s, AI research had developed methods for dealing with uncertain or incomplete information, employing concepts from probability and economics. Many of these algorithms proved to be insufficient for solving large reasoning problems because they experienced a "combinatorial explosion": they became exponentially slower as the problems grew larger.
Even humans rarely use the step-by-step deduction that early AI research could model. They solve most of their problems using fast, intuitive judgments. Knowledge representation
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Artificial intelligence
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Knowledge representation and knowledge engineering
allow AI programs to answer questions intelligently and make deductions about real world facts.
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Artificial intelligence
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A representation of "what exists" is an ontology: the set of objects, relations, concepts, and properties formally described so that software agents can interpret them.
The most general ontologies are called upper ontologies, which attempt to provide a foundation for all other knowledge and act as mediators between domain ontologies that cover specific knowledge about a particular knowledge domain (field of interest or area of concern). A truly intelligent program would also need access to commonsense knowledge; the set of facts that an average person knows. The semantics of an ontology is typically represented in a description logic, such as the Web Ontology Language.
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Artificial intelligence
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AI research has developed tools to represent specific domains, such as: objects, properties, categories and relations between objects;
situations, events, states and time;
causes and effects;
knowledge about knowledge (what we know about what other people know);.
default reasoning (things that humans assume are true until they are told differently and will remain true even when other facts are changing);
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Artificial intelligence
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as well as other domains. Among the most difficult problems in AI are: the breadth of commonsense knowledge (the number of atomic facts that the average person knows is enormous);
and the sub-symbolic form of most commonsense knowledge (much of what people know is not represented as "facts" or "statements" that they could express verbally). Formal knowledge representations are used in content-based indexing and retrieval,
scene interpretation,
clinical decision support,
knowledge discovery (mining "interesting" and actionable inferences from large databases),
and other areas. Planning
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Artificial intelligence
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An intelligent agent that can plan makes a representation of the state of the world, makes predictions about how their actions will change it and makes choices that maximize the utility (or "value") of the available choices.
In classical planning problems, the agent can assume that it is the only system acting in the world, allowing the agent to be certain of the consequences of its actions.
However, if the agent is not the only actor, then it requires that the agent reason under uncertainty, and continuously re-assess its environment and adapt.
Multi-agent planning uses the cooperation and competition of many agents to achieve a given goal. Emergent behavior such as this is used by evolutionary algorithms and swarm intelligence.
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Learning Machine learning (ML), a fundamental concept of AI research since the field's inception,
is the study of computer algorithms that improve automatically through experience.
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Unsupervised learning finds patterns in a stream of input. Supervised learning requires a human to label the input data first, and comes in two main varieties: classification and numerical regression. Classification is used to determine what category something belongs in—the program sees a number of examples of things from several categories and will learn to classify new inputs. Regression is the attempt to produce a function that describes the relationship between inputs and outputs and predicts how the outputs should change as the inputs change. Both classifiers and regression learners can be viewed as "function approximators" trying to learn an unknown (possibly implicit) function; for example, a spam classifier can be viewed as learning a function that maps from the text of an email to one of two categories, "spam" or "not spam".
In reinforcement learning the agent is rewarded for good responses and punished for bad ones. The agent classifies its responses to form a strategy for operating in its problem space.
Transfer learning is when knowledge gained from one problem is applied to a new problem.
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Computational learning theory can assess learners by computational complexity, by sample complexity (how much data is required), or by other notions of optimization. Natural language processing Natural language processing (NLP)
allows machines to read and understand human language. A sufficiently powerful natural language processing system would enable natural-language user interfaces and the acquisition of knowledge directly from human-written sources, such as newswire texts. Some straightforward applications of NLP include information retrieval, question answering and machine translation.
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Symbolic AI used formal syntax to translate the deep structure of sentences into logic. This failed to produce useful applications, due to the intractability of logic and the breadth of commonsense knowledge. Modern statistical techniques include co-occurrence frequencies (how often one word appears near another), "Keyword spotting" (searching for a particular word to retrieve information), transformer-based deep learning (which finds patterns in text), and others. They have achieved acceptable accuracy at the page or paragraph level, and, by 2019, could generate coherent text. Perception
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Machine perception
is the ability to use input from sensors (such as cameras, microphones, wireless signals, and active lidar, sonar, radar, and tactile sensors) to deduce aspects of the world. Applications include speech recognition,
facial recognition, and object recognition.
Computer vision is the ability to analyze visual input. Motion and manipulation
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AI is heavily used in robotics.
Localization is how a robot knows its location and maps its environment. When given a small, static, and visible environment, this is easy; however, dynamic environments, such as (in endoscopy) the interior of a patient's breathing body, pose a greater challenge.
Motion planning is the process of breaking down a movement task into "primitives" such as individual joint movements. Such movement often involves compliant motion, a process where movement requires maintaining physical contact with an object. Robots can learn from experience how to move efficiently despite the presence of friction and gear slippage. Social intelligence
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Affective computing is an interdisciplinary umbrella that comprises systems which recognize, interpret, process, or simulate human feeling, emotion and mood.
For example, some virtual assistants are programmed to speak conversationally or even to banter humorously; it makes them appear more sensitive to the emotional dynamics of human interaction, or to otherwise facilitate human–computer interaction.
However, this tends to give naïve users an unrealistic conception of how intelligent existing computer agents actually are.
Moderate successes related to affective computing include textual sentiment analysis and, more recently, multimodal sentiment analysis), wherein AI classifies the affects displayed by a videotaped subject.
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General intelligence A machine with general intelligence can solve a wide variety of problems with a breadth and versatility similar to human intelligence. There are several competing ideas about how to develop artificial general intelligence. Hans Moravec and Marvin Minsky argue that work in different individual domains can be incorporated into an advanced multi-agent system or cognitive architecture with general intelligence.
Pedro Domingos hopes that there is a conceptually straightforward, but mathematically difficult, "master algorithm" that could lead to AGI.
Others believe that anthropomorphic features like an artificial brain
or simulated child development
will someday reach a critical point where general intelligence emerges. Tools
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Search and optimization Many problems in AI can be solved theoretically by intelligently searching through many possible solutions:
Reasoning can be reduced to performing a search. For example, logical proof can be viewed as searching for a path that leads from premises to conclusions, where each step is the application of an inference rule.
Planning algorithms search through trees of goals and subgoals, attempting to find a path to a target goal, a process called means-ends analysis.
Robotics algorithms for moving limbs and grasping objects use local searches in configuration space.
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Simple exhaustive searches
are rarely sufficient for most real-world problems: the search space (the number of places to search) quickly grows to astronomical numbers. The result is a search that is too slow or never completes. The solution, for many problems, is to use "heuristics" or "rules of thumb" that prioritize choices in favor of those more likely to reach a goal and to do so in a shorter number of steps. In some search methodologies, heuristics can also serve to eliminate some choices unlikely to lead to a goal (called "pruning the search tree"). Heuristics supply the program with a "best guess" for the path on which the solution lies.
Heuristics limit the search for solutions into a smaller sample size.
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A very different kind of search came to prominence in the 1990s, based on the mathematical theory of optimization. For many problems, it is possible to begin the search with some form of a guess and then refine the guess incrementally until no more refinements can be made. These algorithms can be visualized as blind hill climbing: we begin the search at a random point on the landscape, and then, by jumps or steps, we keep moving our guess uphill, until we reach the top. Other optimization algorithms are simulated annealing, beam search and random optimization.
Evolutionary computation uses a form of optimization search. For example, they may begin with a population of organisms (the guesses) and then allow them to mutate and recombine, selecting only the fittest to survive each generation (refining the guesses). Classic evolutionary algorithms include genetic algorithms, gene expression programming, and genetic programming.
Alternatively, distributed search processes can coordinate via swarm intelligence algorithms. Two popular swarm algorithms used in search are particle swarm optimization (inspired by bird flocking) and ant colony optimization (inspired by ant trails).
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Logic Logic
is used for knowledge representation and problem solving, but it can be applied to other problems as well. For example, the satplan algorithm uses logic for planning
and inductive logic programming is a method for learning.
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Several different forms of logic are used in AI research. Propositional logic involves truth functions such as "or" and "not". First-order logic
adds quantifiers and predicates, and can express facts about objects, their properties, and their relations with each other. Fuzzy logic assigns a "degree of truth" (between 0 and 1) to vague statements such as "Alice is old" (or rich, or tall, or hungry), that are too linguistically imprecise to be completely true or false.
Default logics, non-monotonic logics and circumscription are forms of logic designed to help with default reasoning and the qualification problem.
Several extensions of logic have been designed to handle specific domains of knowledge, such as: description logics;
situation calculus, event calculus and fluent calculus (for representing events and time);
causal calculus;
belief calculus (belief revision); and modal logics.
Logics to model contradictory or inconsistent statements arising in multi-agent systems have also been designed, such as paraconsistent logics.
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Probabilistic methods for uncertain reasoning
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Many problems in AI (in reasoning, planning, learning, perception, and robotics) require the agent to operate with incomplete or uncertain information. AI researchers have devised a number of powerful tools to solve these problems using methods from probability theory and economics.
Bayesian networks
are a very general tool that can be used for various problems: reasoning (using the Bayesian inference algorithm),
learning (using the expectation-maximization algorithm),
planning (using decision networks) and perception (using dynamic Bayesian networks).
Probabilistic algorithms can also be used for filtering, prediction, smoothing and finding explanations for streams of data, helping perception systems to analyze processes that occur over time (e.g., hidden Markov models or Kalman filters).
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A key concept from the science of economics is "utility": a measure of how valuable something is to an intelligent agent. Precise mathematical tools have been developed that analyze how an agent can make choices and plan, using decision theory, decision analysis,
and information value theory. These tools include models such as Markov decision processes, dynamic decision networks, game theory and mechanism design. Classifiers and statistical learning methods
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The simplest AI applications can be divided into two types: classifiers ("if shiny then diamond") and controllers ("if diamond then pick up"). Controllers do, however, also classify conditions before inferring actions, and therefore classification forms a central part of many AI systems. Classifiers are functions that use pattern matching to determine a closest match. They can be tuned according to examples, making them very attractive for use in AI. These examples are known as observations or patterns. In supervised learning, each pattern belongs to a certain predefined class. A class is a decision that has to be made. All the observations combined with their class labels are known as a data set. When a new observation is received, that observation is classified based on previous experience.
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A classifier can be trained in various ways; there are many statistical and machine learning approaches.
The decision tree is the simplest and most widely used symbolic machine learning algorithm.
K-nearest neighbor algorithm was the most widely used analogical AI until the mid-1990s.
Kernel methods such as the support vector machine (SVM) displaced k-nearest neighbor in the 1990s.
The naive Bayes classifier is reportedly the "most widely used learner" at Google, due in part to its scalability.
Neural networks are also used for classification.
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Classifier performance depends greatly on the characteristics of the data to be classified, such as the dataset size, distribution of samples across classes, the dimensionality, and the level of noise. Model-based classifiers perform well if the assumed model is an extremely good fit for the actual data. Otherwise, if no matching model is available, and if accuracy (rather than speed or scalability) is the sole concern, conventional wisdom is that discriminative classifiers (especially SVM) tend to be more accurate than model-based classifiers such as "naive Bayes" on most practical data sets. Artificial neural networks
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Neural networks
were inspired by the architecture of neurons in the human brain. A simple "neuron" N accepts input from other neurons, each of which, when activated (or "fired"), casts a weighted "vote" for or against whether neuron N should itself activate. Learning requires an algorithm to adjust these weights based on the training data; one simple algorithm (dubbed "fire together, wire together") is to increase the weight between two connected neurons when the activation of one triggers the successful activation of another. Neurons have a continuous spectrum of activation; in addition, neurons can process inputs in a nonlinear way rather than weighing straightforward votes.
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Modern neural networks model complex relationships between inputs and outputs or and find patterns in data. They can learn continuous functions and even digital logical operations. Neural networks can be viewed a type of mathematical optimization — they perform a gradient descent on a multi-dimensional topology that was created by training the network. The most common training technique is the backpropagation algorithm.
Other learning techniques for neural networks are Hebbian learning ("fire together, wire together"), GMDH or competitive learning.
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The main categories of networks are acyclic or feedforward neural networks (where the signal passes in only one direction) and recurrent neural networks (which allow feedback and short-term memories of previous input events). Among the most popular feedforward networks are perceptrons, multi-layer perceptrons and radial basis networks.
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Deep learning
Deep learning
uses several layers of neurons between the network's inputs and outputs. The multiple layers can progressively extract higher-level features from the raw input. For example, in image processing, lower layers may identify edges, while higher layers may identify the concepts relevant to a human such as digits or letters or faces. Deep learning has drastically improved the performance of programs in many important subfields of artificial intelligence, including computer vision, speech recognition, image classification and others.
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Deep learning often uses convolutional neural networks for many or all of its layers. In a convolutional layer, each neuron receives input from only a restricted area of the previous layer called the neuron's receptive field. This can substantially reduce the number of weighted connections between neurons, and creates a hierarchy similar to the organization of the animal visual cortex.
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In a recurrent neural network the signal will propagate through a layer more than once;
thus, an RNN is an example of deep learning.
RNNs can be trained by gradient descent,
however long-term gradients which are back-propagated can "vanish" (that is, they can tend to zero) or "explode" (that is, they can tend to infinity), known as the vanishing gradient problem.
The long short term memory (LSTM) technique can prevent this in most cases. Specialized languages and hardware
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Specialized languages for artificial intelligence have been developed, such as Lisp, Prolog, TensorFlow and many others. Hardware developed for AI includes AI accelerators and neuromorphic computing. Applications AI is relevant to any intellectual task.
Modern artificial intelligence techniques are pervasive and are too numerous to list here.
Frequently, when a technique reaches mainstream use, it is no longer considered artificial intelligence; this phenomenon is described as the AI effect.
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In the 2010s, AI applications were at the heart of the most commercially successful areas of computing, and have become a ubiquitous feature of daily life. AI is used in search engines (such as Google Search),
targeting online advertisements,
recommendation systems (offered by Netflix, YouTube or Amazon),
driving internet traffic,
targeted advertising (AdSense, Facebook),
virtual assistants (such as Siri or Alexa),
autonomous vehicles (including drones and self-driving cars),
automatic language translation (Microsoft Translator, Google Translate),
facial recognition (Apple's Face ID or Microsoft's DeepFace),
image labeling (used by Facebook, Apple's iPhoto and TikTok)
and spam filtering.
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There are also thousands of successful AI applications used to solve problems for specific industries or institutions. A few examples are:
energy storage,
deepfakes,
medical diagnosis,
military logistics, or
supply chain management.
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Game playing has been a test of AI's strength since the 1950s. Deep Blue became the first computer chess-playing system to beat a reigning world chess champion, Garry Kasparov, on 11 May 1997.
In 2011, in a Jeopardy! quiz show exhibition match, IBM's question answering system, Watson, defeated the two greatest Jeopardy! champions, Brad Rutter and Ken Jennings, by a significant margin.
In March 2016, AlphaGo won 4 out of 5 games of Go in a match with Go champion Lee Sedol, becoming the first computer Go-playing system to beat a professional Go player without handicaps.
Other programs handle imperfect-information games; such as for poker at a superhuman level, Pluribus
and Cepheus.
DeepMind in the 2010s developed a "generalized artificial intelligence" that could learn many diverse Atari games on its own.
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By 2020, Natural Language Processing systems such as the enormous GPT-3 (then by far the largest artificial neural network) were matching human performance on pre-existing benchmarks, albeit without the system attaining commonsense understanding of the contents of the benchmarks.
DeepMind's AlphaFold 2 (2020) demonstrated the ability to approximate, in hours rather than months, the 3D structure of a protein.
Other applications predict the result of judicial decisions, create art (such as poetry or painting) and prove mathematical theorems. Philosophy Defining artificial intelligence Thinking vs. acting: the Turing test
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Alan Turing wrote in 1950 "I propose to consider the question 'can machines think'?"
He advised changing the question from whether a machine "thinks", to "whether or not it is possible for machinery to show intelligent behaviour".
The only thing visible is the behavior of the machine, so it does not matter if the machine is conscious, or has a mind, or whether the intelligence is merely a "simulation" and not "the real thing". He noted that we also don't know these things about other people, but that we extend a "polite convention" that they are actually "thinking". This idea forms the basis of the Turing test.
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Acting humanly vs. acting intelligently: intelligent agents AI founder John McCarthy said: "Artificial intelligence is not, by definition, simulation of human intelligence". Russell and Norvig agree and criticize the Turing test. They wrote: "Aeronautical engineering texts do not define the goal of their field as 'making machines that fly so exactly like pigeons that they can fool other pigeons. Other researchers and analysts disagree and have argued that AI should simulate natural intelligence by studying psychology or neurobiology.
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The intelligent agent paradigm
defines intelligent behavior in general, without reference to human beings. An intelligent agent is a system that perceives its environment and takes actions that maximize its chances of success. Any system that has goal-directed behavior can be analyzed as an intelligent agent: something as simple as a thermostat, as complex as a human being, as well as large systems such as firms, biomes or nations. The intelligent agent paradigm became widely accepted during the 1990s, and currently serves as the definition of the field.
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The paradigm has other advantages for AI. It provides a reliable and scientific way to test programs; researchers can directly compare or even combine different approaches to isolated problems, by asking which agent is best at maximizing a given "goal function". It also gives them a common language to communicate with other fields — such as mathematical optimization (which is defined in terms of "goals") or economics (which uses the same definition of a "rational agent").
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Evaluating approaches to AI
No established unifying theory or paradigm has guided AI research for most of its history. The unprecedented success of statistical machine learning in the 2010s eclipsed all other approaches (so much so that some sources, especially in the business world, use the term "artificial intelligence" to mean "machine learning with neural networks"). This approach is mostly sub-symbolic, neat, soft and narrow (see below). Critics argue that these questions may have to be revisited by future generations of AI researchers. Symbolic AI and its limits
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Symbolic AI (or "GOFAI") simulated the high-level conscious reasoning that people use when they solve puzzles, express legal reasoning and do mathematics. They were highly successful at "intelligent" tasks such as algebra or IQ tests. In the 1960s, Newell and Simon proposed the physical symbol systems hypothesis: "A physical symbol system has the necessary and sufficient means of general intelligent action."
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However, the symbolic approach failed dismally on many tasks that humans solve easily, such as learning, recognizing an object or commonsense reasoning. Moravec's paradox is the discovery that high-level "intelligent" tasks were easy for AI, but low level "instinctive" tasks were extremely difficult.
Philosopher Hubert Dreyfus had argued since the 1960s that human expertise depends on unconscious instinct rather than conscious symbol manipulation, and on having a "feel" for the situation, rather than explicit symbolic knowledge.
Although his arguments had been ridiculed and ignored when they were first presented, eventually AI research came to agree.
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The issue is not resolved: sub-symbolic reasoning can make many of the same inscrutable mistakes that human intuition does, such as algorithmic bias. Critics such Noam Chomsky argue continuing research into symbolic AI will still be necessary to attain general intelligence, in part because sub-symbolic AI is a move away from explainable AI: it can be difficult or impossible to understand why a modern statistical AI program made a particular decision. Neat vs. scruffy
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"Neats" hope that intelligent behavior be described using simple, elegant principles (such as logic, optimization, or neural networks). "Scruffies" expect that it necessarily requires solving a large number of unrelated problems. This issue was actively discussed in the 70s and 80s,
but in the 1990s mathematical methods and solid scientific standards became the norm, a transition that Russell and Norvig termed "the victory of the neats". Soft vs. hard computing
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Finding a provably correct or optimal solution is intractable for many important problems. Soft computing is a set of techniques, including genetic algorithms, fuzzy logic and neural networks, that are tolerant of imprecision, uncertainty, partial truth and approximation. Soft computing was introduced in the late 80s and most successful AI programs in the 21st century are examples of soft computing with neural networks. Narrow vs. general AI
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AI researchers are divided as to whether to pursue the goals of artificial general intelligence and superintelligence (general AI) directly, or to solve as many specific problems as possible (narrow AI) in hopes these solutions will lead indirectly to the field's long-term goals
General intelligence is difficult to define and difficult to measure, and modern AI has had more verifiable successes by focussing on specific problems with specific solutions. The experimental sub-field of artificial general intelligence studies this area exclusively. Machine consciousness, sentience and mind
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The philosophy of mind does not know whether a machine can have a mind, consciousness and mental states, in the same sense that human beings do. This issue considers the internal experiences of the machine, rather than its external behavior. Mainstream AI research considers this issue irrelevant, because it does not effect the goals of the field. Stuart Russell and Peter Norvig observe that most AI researchers "don't care about the [philosophy of AI] — as long as the program works, they don't care whether you call it a simulation of intelligence or real intelligence." However, the question has become central to the philosophy of mind. It is also typically the central question at issue in artificial intelligence in fiction.
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Consciousness David Chalmers identified two problems in understanding the mind, which he named the "hard" and "easy" problems of consciousness. The easy problem is understanding how the brain processes signals, makes plans and controls behavior. The hard problem is explaining how this feels or why it should feel like anything at all. Human information processing is easy to explain, however human subjective experience is difficult to explain. For example, it is easy to imagine a color blind person who has learned to identify which objects in their field of view are red, but it is not clear what would be required for the person to know what red looks like.
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Computationalism and functionalism Computationalism is the position in the philosophy of mind that the human mind is an information processing system and that thinking is a form of computing. Computationalism argues that the relationship between mind and body is similar or identical to the relationship between software and hardware and thus may be a solution to the mind-body problem. This philosophical position was inspired by the work of AI researchers and cognitive scientists in the 1960s and was originally proposed by philosophers Jerry Fodor and Hilary Putnam.
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