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Regular Language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expressions engines, which are augmented with features that allow recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by Type-3 grammars. Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: * The empty language Ø is a regular language. * For each ''a'' ∈ Σ (''a'' belongs to Σ), the singleton language is a regular language. * If ''A'' is a regular language, ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Arden's Lemma
In theoretical computer science, Arden's rule, also known as Arden's lemma, is a mathematical statement about a certain form of language equations. Background A (formal) language is simply a set of strings. Such sets can be specified by means of some language equation, which in turn is based on operations on languages. Language equations are mathematical statements that resemble numerical equations, but the variables assume values of formal languages rather than numbers. Among the most common operations on two languages ''A'' and ''B'' are the set union ''A'' ∪ ''B'', and their concatenation ''A''⋅''B''. Finally, as an operation taking a single operand, the set ''A''* denotes the Kleene star of the language ''A''. Statement of Arden's rule Arden's rule states that the set ''A''*⋅''B'' is the smallest language that is a solution for ''X'' in the linear equation ''X'' = ''A''⋅''X'' ∪ ''B'' where ''X'', ''A'', ''B'' are sets of strings. Moreover, if the set ''A'' does n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Preimage
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Syntactic Monoid
In mathematics and computer science, the syntactic monoid M(L) of a formal language L is the smallest monoid that recognizes the language L. Syntactic quotient The free monoid on a given set is the monoid whose elements are all the strings of zero or more elements from that set, with string concatenation as the monoid operation and the empty string as the identity element. Given a subset S of a free monoid M, one may define sets that consist of formal left or right inverses of elements in S. These are called quotients, and one may define right or left quotients, depending on which side one is concatenating. Thus, the right quotient of S by an element m from M is the set :S \ / \ m=\. Similarly, the left quotient is :m \setminus S=\. Syntactic equivalence The syntactic quotient induces an equivalence relation on M, called the syntactic relation, or syntactic equivalence (induced by S). The ''right syntactic equivalence'' is the equivalence relation :s \sim_S t \ \Leftri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Recognizable Set
In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some morphism to a finite monoid. Recognizable sets are useful in automata theory, formal languages and algebra. This notion is different from the notion of recognizable language. Indeed, the term "recognizable" has a different meaning in computability theory. Definition Let N be a monoid, a subset S\subseteq N is recognized by a monoid M if there exists a morphism \phi from N to M such that S=\phi^(\phi(S)), and recognizable if it is recognized by some finite monoid. This means that there exists a subset T of M (not necessarily a submonoid of M) such that the image of S is in T and the image of N \setminus S is in M \setminus T. Example Let A be an alphabet: the set A^* of words over A is a monoid, the free monoid on A. The recognizable subsets of A^* are precisely the regular languages. Indeed, such a language is recognized by the transition monoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lecture Notes In Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post- proceedings, monographs, and Festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the ...s. In addition, tutorials, state-of-the-art surveys, and "hot topics" are increasingly being included. The series is indexed by DBLP. See also *'' Monographiae Biologicae'', another monograph series published by Springer Science+Business Media *'' Lecture Notes in Physics'' *'' Lecture Notes in Mathematics'' *'' Electronic Workshops in Computing'', published by the British Computer Society References External links * Publications established in 1973 Computer science books Series of non-fiction books ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Second-order Logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence \forall P\,\forall x (Px \lor \neg Px) says that for every formula ''P'', and every individual ''x'', either ''Px'' is true or not(''Px'') is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Monadic Predicate Calculus
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form P(x), where P is a relation symbol and x is a variable. Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments. Expressiveness The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains). Löwenheim, L. (1915) "Über Möglichkeiten im Relativkalkül," ''Mathematische Annalen'' 76: 447-470. Translated as "On possibilities in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Turing Machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write and which direction to move is based on a finite table that specifies what to do for each co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prefix Grammar
A prefix is an affix which is placed before the stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy''. Particularly in the study of languages, a prefix is also called a preformative, because it alters the form of the words to which it is affixed. Prefixes, like other affixes, can be either inflectional, creating a new form of the word with the same basic meaning and same lexical category (but playing a different role in the sentence), or derivational, creating a new word with a new semantic meaning and sometimes also a different lexical category. Prefixes, like all other affixes, are usually bound morphemes. In English, there are no inflectional prefixes; English uses suffixes instead for that purpose. The word ''prefix'' is itself made up of the stem ''fix'' (meaning "attach", in this case), and the prefix ''pre-'' (meaning "before"), both of w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Two-way Finite Automaton
In computer science, in particular in automata theory, a two-way finite automaton is a finite automaton that is allowed to re-read its input. Two-way deterministic finite automaton A two-way deterministic finite automaton (2DFA) is an abstract machine, a generalized version of the deterministic finite automaton (DFA) which can revisit characters already processed. As in a DFA, there are a finite number of states with transitions between them based on the current character, but each transition is also labelled with a value indicating whether the machine will move its position in the input to the left, right, or stay at the same position. Equivalently, 2DFAs can be seen as read-only Turing machines with no work tape, only a read-only input tape. 2DFAs were introduced in a seminal 1959 paper by Rabin and Scott, who proved them to have equivalent power to one-way DFAs. That is, any formal language which can be recognized by a 2DFA can be recognized by a DFA which only examines and c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |