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Part of the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.

It might be assumed that, for any formal criterion, a set exists whose members are those objects (and only those objects) that satisfy the criterion; but this assumption is disproved by a set containing exactly the sets that are not members of themselves. If such a set qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and Ernst Zermelo's axiomatic set theory, the first consciously constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical ZFC set theory.

Reference:
http://en.wikipedia.org/wiki/Russell%27s_paradox

In mathematics, computer science and logic, rewriting covers a wide range of potentially non-deterministic methods of replacing subterms of a formula with other terms. What is considered are rewrite systems (also rewriting systems, or term rewriting systems, though the latter term may imply a more specific system) which, in their most basic form, consist of a set of terms, plus relations on how to transform these terms.

Term rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible substitutions that could be applied. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several declarative programming languages are based on term rewriting.

Reference:
http://en.wikipedia.org/wiki/Term_rewriting

In mathematical optimization theory, the simplex algorithm, created by the American mathematician George Dantzig in 1947, is a popular algorithm for numerical solution of the linear programming problem. The journal Computing in Science and Engineering listed it as one of the top 10 algorithms of the century.

An unrelated, but similarly named method is the Nelder-Mead method or downhill simplex method due to Nelder & Mead (1965) and is a numerical method for optimising many-dimensional unconstrained problems, belonging to the more general class of search algorithms.

In both cases, the method uses the concept of a simplex, which is a polytope of N + 1 vertices in N dimensions: a line segment in one dimension, a triangle in two dimensions, a tetrahedron in three-dimensional space and so forth.

Reference:
http://en.wikipedia.org/wiki/Simplex_algorithm

Constraint satisfaction problems or CSPs are mathematical problems where one must find states or objects that satisfy a number of constraints or criteria. CSPs are the subject of intense research in both artificial intelligence and operations research. Many CSPs require a combination of heuristics and combinatorial search methods to be solved in a reasonable time.

Examples of constraint satisfaction problems:

Eight queens puzzle
Map coloring problem
Sudoku
Boolean satisfiability
Algorithms used for solving constraint satisfaction problems include the AC-3 algorithm, Backtracking, and the min-conflicts algorithm.

Reference:
http://en.wikipedia.org/wiki/Constraint_satisfaction_problem

In both computer science and information science, an ontology is a representation of a set of concepts within a domain and the relationships between those concepts. It is used to reason about the properties of that domain, and may be used to define the domain.

Ontologies are used in artificial intelligence, the Semantic Web, software engineering, biomedical informatics, library science, and information architecture as a form of knowledge representation about the world or some part of it. Common components of ontologies include:

Individuals: instances or objects (the basic or "ground level" objects)
Classes: sets, collections, concepts or types of objects[1]
Attributes: properties, features, characteristics, or parameters that objects (and classes) can have
Relations: ways that classes and objects can be related to one another
Function terms: complex structures formed from certain relations that can be used in place of an individual term in a statement
Restrictions: formally stated descriptions of what must be true in order for some assertion to be accepted as input
Rules: statements in the form of an if-then (antecedent-consequent) sentence that describe the logical inferences that can be drawn from an assertion in a particular form
Axia: assertions (including rules) in a logical form that together comprise the overall theory that the ontology describes, for its domain of application
Events: the changing of attributes or relations
Ontologies are commonly encoded using ontology languages.

Reference:
http://en.wikipedia.org/wiki/Ontology_%28computer_science%29

An expert system, also known as a knowledge based system, is a computer program that contains the knowledge and analytical skills of one or more human experts, related to a specific subject. This class of program was first developed by researchers in artificial intelligence during the 1960s and 1970s and applied commercially throughout the 1980s.

Reference:
http://en.wikipedia.org/wiki/Expert_system

Computational linguistics is an interdisciplinary field dealing with the statistical and/or rule-based modeling of natural language from a computational perspective. This modeling is not limited to any particular field of linguistics. Traditionally, computational linguistics was usually performed by computer scientists who had specialized in the application of computers to the processing of a natural language. Recent research has shown that human language is much more complex than previously thought, so computational linguists often work as members of interdisciplinary teams, including linguists (specifically trained in linguistics), language experts (persons with some level of ability in the languages relevant to a given project), and computer scientists. In general computational linguistics draws upon the involvement of linguists, computer scientists, experts in artificial intelligence, cognitive psychologists, mathematicians, and logicians, amongst others.

Reference:
http://en.wikipedia.org/wiki/Computational_linguistics

In mathematics, linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality constraints.

Put very informally, LP problems determine the way to achieve the best outcome (such as maximum profit or lowest cost) given some list of requirements represented as linear equation.

More formally, given a polytope (for example, a polygon or a polyhedron), and a real-valued affine function

f(x_1, x_2, ..., x_n) = a_1x_1 + a_2x_2 + ... + a_nx_n + b

defined on this polytope, the goal is to find a point in the polytope where this function has the smallest (or largest) value. Such points may not exist, but if they do, searching through the polytope vertices is guaranteed to find at least one of them.

Linear programs are problems that can be expressed in canonical form:

Maximize c^Tx
Subject to Ax <= b
Where x >= 0

x represents the vector of variables, while c and b are vectors of coefficients and A is a matrix of coefficients. The expression to be maximized or minimized is called the objective function (c^Tx in this case). The equations Ax <= b are the constraints which specify a convex polyhedron over which the objective function is to be optimized.

Linear programming can be applied to various fields of study. Most extensively it is used in business and economic situations, but can also be utilized for some engineering problems. Some industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

Reference:
http://en.wikipedia.org/wiki/Linear_programming

In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set.

Reference:
http://en.wikipedia.org/wiki/Mathematical_programming

In mathematical logic and computer science, lambda calculus, also λ-calculus, is a formal system designed to investigate function definition, function application and recursion. It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s as part of a larger effort to base the foundation of mathematics upon functions rather than sets (in the hopes of avoiding obstacles like Russell's Paradox). The Kleene-Rosser paradox shows that the lambda calculus is unable to avoid set-theoretic paradoxes, but the lambda calculus emerged as a useful tool in the investigation of problems in computability or recursion theory, and forms the basis of a paradigm of computer programming called functional programming. [1]

The lambda calculus can be thought of as an idealized, minimalistic programming language. It is a close cousin of the Turing machine, another minimalist abstraction capable of expressing any algorithm. The difference between the two is that the lambda calculus takes a functional view of algorithms, while the original Turing machine takes an imperative view. That is, a Turing machine maintains 'state' - a 'notebook' of symbols that can change from one instruction to the next. The imperative paradigm can be seen in programming languages like C or BASIC. By contrast, the lambda calculus is stateless, it deals exclusively with functions which accept and return data (including other functions), but produce no side effects in 'state' and do not make alterations to incoming data (immutability.) The functional paradigm can be seen in modern languages like Lisp, Scheme and Haskell.

Church used the lambda calculus to give a negative answer to the Entscheidungsproblem and to the halting problem. The lambda calculus - and the paradigm of functional programming - is still influential, especially among the artificial intelligence community.

This article deals with the "untyped lambda calculus" as originally conceived by Church. Since then, some typed lambda calculi have been developed.

Reference:
http://en.wikipedia.org/wiki/Lambda_calculus