1.
Formal language theory
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In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a language is the set of symbols, letters. The strings formed from this alphabet are called words, and the words belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is defined by means of a formal grammar such as a regular grammar or context-free grammar. The field of language theory studies primarily the purely syntactical aspects of such languages—that is. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages. The first formal language is thought to be the one used by Gottlob Frege in his Begriffsschrift, literally meaning concept writing, axel Thues early semi-Thue system, which can be used for rewriting strings, was influential on formal grammars. The elements of an alphabet are called its letters, alphabets may be infinite, however, most definitions in formal language theory specify finite alphabets, and most results only apply to them. A word over an alphabet can be any sequence of letters. The set of all words over an alphabet Σ is usually denoted by Σ*, the length of a word is the number of letters it is composed of. For any alphabet there is one word of length 0, the empty word. By concatenation one can combine two words to form a new word, whose length is the sum of the lengths of the original words, the result of concatenating a word with the empty word is the original word. A formal language L over an alphabet Σ is a subset of Σ*, that is, sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of well-formed expressions. In computer science and mathematics, which do not usually deal with natural languages, in practice, there are many languages that can be described by rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to the concept of a language. By an abuse of the definition, a formal language is often thought of as being equipped with a formal grammar that describes it. The following rules describe a formal language L over the alphabet Σ =, Every nonempty string that does not contain + or =, a string containing = is in L if and only if there is exactly one =, and it separates two valid strings of L. A string containing + but not = is in L if, no string is in L other than those implied by the previous rules
2.
String (computer science)
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In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, a string is generally understood as a data type and is often implemented as an array of bytes that stores a sequence of elements, typically characters, using some character encoding. A string may also more general arrays or other sequence data types and structures. When a string appears literally in source code, it is known as a literal or an anonymous string. In formal languages, which are used in logic and theoretical computer science. Let Σ be a non-empty finite set of symbols, called the alphabet, no assumption is made about the nature of the symbols. A string over Σ is any sequence of symbols from Σ. For example, if Σ =, then 01011 is a string over Σ, the length of a string s is the number of symbols in s and can be any non-negative integer, it is often denoted as |s|. The empty string is the string over Σ of length 0. The set of all strings over Σ of length n is denoted Σn, for example, if Σ =, then Σ2 =. Note that Σ0 = for any alphabet Σ, the set of all strings over Σ of any length is the Kleene closure of Σ and is denoted Σ*. In terms of Σn, Σ ∗ = ⋃ n ∈ N ∪ Σ n For example, if Σ =, although the set Σ* itself is countably infinite, each element of Σ* is a string of finite length. A set of strings over Σ is called a language over Σ. For example, if Σ =, the set of strings with an number of zeros, is a formal language over Σ. Concatenation is an important binary operation on Σ*, for any two strings s and t in Σ*, their concatenation is defined as the sequence of symbols in s followed by the sequence of characters in t, and is denoted st. For example, if Σ =, s = bear, and t = hug, then st = bearhug, String concatenation is an associative, but non-commutative operation. The empty string ε serves as the identity element, for any string s, therefore, the set Σ* and the concatenation operation form a monoid, the free monoid generated by Σ. In addition, the length function defines a monoid homomorphism from Σ* to the non-negative integers, a string s is said to be a substring or factor of t if there exist strings u and v such that t = usv
3.
Epsilon
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Epsilon is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel /e/. In the system of Greek numerals it has the value five and it was derived from the Phoenician letter He. Letters that arose from epsilon include the Roman E, Ë and Ɛ, in essence, the uppercase form of epsilon looks identical to Latin E. The lowercase version has two variants, both inherited from medieval Greek handwriting. One, the most common in typography and inherited from medieval minuscule. The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing, while in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols. Computer systems therefore offer distinct encodings for them, in Unicode, the character U+0一3F5 Greek lunate epsilon symbol is provided specifically for the lunate form. In TeX, \epsilon denotes the lunate form, while \varepsilon denotes the reversed-3 form, there is also a Latin epsilon or open e, which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as U+025B and U+0190 and is used as an IPA phonetic symbol, the lunate or uncial epsilon has also provided inspiration for the euro sign. The lunate epsilon is not to be confused with the set membership symbol, in addition, mathematicians have read the symbol ∈ as element of, as in 1 is an element of the natural numbers for 1 ∈ N, for example. As late as 1960, ϵ itself was used for set membership, Only gradually did a fully separate stylized symbol take the place of epsilon. In a related context, Peano also introduced the use of a backwards epsilon, ∍, for the phrase such that, the letter Ε was taken over from the Phoenician letter He when Greeks first adopted alphabetic writing. In archaic Greek writing, its shape is often identical to that of the Phoenician letter. Archaic writing often preserves the Phoenician form with a stem extending slightly below the lowest horizontal bar. In the classical era, through the influence of cursive writing styles. Besides its classical Greek sound value, the short /e/ phoneme, for instance, in early Attic before c.500 B. C. it was used also both for the long, open /ɛː/, and for the long close /eː/. In the former role, it was replaced in the classic Greek alphabet by Eta. Some dialects used yet other ways of distinguishing between various e-like sounds, in Corinth, the normal function of Ε to denote /e/ and /ɛː/ was taken by a glyph resembling a pointed B, while Ε was used only for long close /eː/
4.
Lambda
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Lambda is the 11th letter of the Greek alphabet. In the system of Greek numerals lambda has a value of 30, Lambda is related to the Phoenician letter Lamed. Letters in other alphabets that stemmed from lambda include the Latin L, the ancient grammarians and dramatists give evidence to the pronunciation as in Classical Greek times. In Modern Greek the name of the letter, Λάμδα, is pronounced, in early Greek alphabets, the shape and orientation of lambda varied. Most variants consisted of two strokes, one longer than the other, connected at their ends. The angle might be in the upper-left, lower-left, or top, other variants had a vertical line with a horizontal or sloped stroke running to the right. With the general adoption of the Ionic alphabet, Greek settled on an angle at the top, the HTML4 character entity references for the Greek capital and small letter lambda are Λ, and λ, respectively. The Unicode code points for lambda are U+039B and U+03BB, the lambda particle is a type of subatomic particle in subatomic particle physics. Lambda is the set of axioms in the axiomatic method of logical deduction in first-order logic. Lambda was used as a pattern by the Spartan army. This stood for Lacedaemon, the name of the polis of the Spartans, Lambda is the von Mangoldt function in mathematical number theory. Lambda is a associated with the Identitarian movement, intended to emulate a Spartan shield pattern. In statistics, Wilkss lambda is used in analysis of variance to compare group means on a combination of dependent variables. In the spectral decomposition of matrices, lambda indicates the diagonal matrix of the eigenvalues of the matrix, in computer science, lambda is the time window over which a process is observed for determining the working memory set for a digital computers virtual memory management. In astrophysics, lambda represents the likelihood that a body will encounter a planet or a dwarf planet leading to a deflection of a significant magnitude. An object with a value of lambda is expected to have cleared its neighborhood. In crystal optics, lambda is used to represent the period of a lattice, in NATO military operations, a chevron is painted on the vehicles of this military alliance for identification. In chemistry there are Δ and Λ isomers, see, coordination complex In electrochemistry, in cosmology, lambda is the symbol for the cosmological constant, a term added to some dynamical equations to account for the acceleration of the universe
5.
Empty set
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In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
6.
Formal language
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In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a language is the set of symbols, letters. The strings formed from this alphabet are called words, and the words belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is defined by means of a formal grammar such as a regular grammar or context-free grammar. The field of language theory studies primarily the purely syntactical aspects of such languages—that is. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages. The first formal language is thought to be the one used by Gottlob Frege in his Begriffsschrift, literally meaning concept writing, axel Thues early semi-Thue system, which can be used for rewriting strings, was influential on formal grammars. The elements of an alphabet are called its letters, alphabets may be infinite, however, most definitions in formal language theory specify finite alphabets, and most results only apply to them. A word over an alphabet can be any sequence of letters. The set of all words over an alphabet Σ is usually denoted by Σ*, the length of a word is the number of letters it is composed of. For any alphabet there is one word of length 0, the empty word. By concatenation one can combine two words to form a new word, whose length is the sum of the lengths of the original words, the result of concatenating a word with the empty word is the original word. A formal language L over an alphabet Σ is a subset of Σ*, that is, sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of well-formed expressions. In computer science and mathematics, which do not usually deal with natural languages, in practice, there are many languages that can be described by rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to the concept of a language. By an abuse of the definition, a formal language is often thought of as being equipped with a formal grammar that describes it. The following rules describe a formal language L over the alphabet Σ =, Every nonempty string that does not contain + or =, a string containing = is in L if and only if there is exactly one =, and it separates two valid strings of L. A string containing + but not = is in L if, no string is in L other than those implied by the previous rules