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Regular Grammar
In theoretical computer science and formal language theory, a regular grammar is a grammar that is ''right-regular'' or ''left-regular''. While their exact definition varies from textbook to textbook, they all require that * all production rules have at most one non-terminal symbol; * that symbol is either always at the end or always at the start of the rule's right-hand side. Every regular grammar describes a regular language. Strictly regular grammars A right-regular grammar (also called right-linear grammar) is a formal grammar (''N'', Σ, ''P'', ''S'') in which all production rules in ''P'' are of one of the following forms: # ''A'' → ''a'' # ''A'' → ''aB'' # ''A'' → ε where ''A'', ''B'', ''S'' ∈ ''N'' are non-terminal symbols, ''a'' ∈ Σ is a terminal symbol, and ε denotes the empty string, i.e. the string of length 0. ''S'' is called the start symbol. In a left-regular grammar, (also called left-linear grammar), all rules obey the forms # ''A'' → '' ...
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Theoretical Computer Science
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 ACM's 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 learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and parallel distributed processing were established. I ...
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Formal Language Theory
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity t ...
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Formal Grammar
In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. A formal grammar is defined as a set of production rules for such strings in a formal language. Formal language theory, the discipline that studies formal grammars and languages, is a branch of applied mathematics. Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas. A formal grammar is a set of rules for rewriting strings, along with a "start symbol" from which rewriting starts. Therefore, a grammar is usually thought of as a language generator. However, it can also sometimes be used as the basis for a " recognizer"—a function in computing that det ...
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Production (computer Science)
A production or production rule in computer science is a ''rewrite rule'' specifying a symbol substitution that can be recursively performed to generate new symbol sequences. A finite set of productions P is the main component in the specification of a formal grammar (specifically a generative grammar). The other components are a finite set N of nonterminal symbols, a finite set (known as an alphabet) \Sigma of terminal symbols that is disjoint from N and a distinguished symbol S \in N that is the ''start symbol''. In an unrestricted grammar, a production is of the form u \to v, where u and v are arbitrary strings of terminals and nonterminals, and u may not be the empty string. If v is the empty string, this is denoted by the symbol \epsilon, or \lambda (rather than leave the right-hand side blank). So productions are members of the cartesian product :V^*NV^* \times V^* = (V^*\setminus\Sigma^*) \times V^*, where V := N \cup \Sigma is the ''vocabulary'', ^ is the Kleene star ...
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Terminal And Nonterminal Symbols
In computer science, terminal and nonterminal symbols are the lexical elements used in specifying the production rules constituting a formal grammar. ''Terminal symbols'' are the elementary symbols of the language defined by a formal grammar. ''Nonterminal symbols'' (or ''syntactic variables'') are replaced by groups of terminal symbols according to the production rules. The terminals and nonterminals of a particular grammar are two disjoint sets. Terminal symbols Terminal symbols are literal symbols that may appear in the outputs of the production rules of a formal grammar and which cannot be changed using the rules of the grammar. Applying the rules recursively to a source string of symbols will usually terminate in a final output string consisting only of terminal symbols. Consider a grammar defined by two rules. Using pictoric marks interacting with each other: # The symbol ר can become ди # The symbol ר can become д Here д is a terminal symbol because no rule ...
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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'' ...
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Linear Grammar
In computer science, a linear grammar is a context-free grammar that has at most one nonterminal in the right-hand side of each of its productions. A linear language is a language generated by some linear grammar. Example An example of a linear grammar is ''G'' with ''N'' = , Σ = , ''P'' with start symbol ''S'' and rules : S → aSb : S → ε It generates the language \. Relationship with regular grammars Two special types of linear grammars are the following: * the left-linear or left-regular grammars, in which all rules are of the form ''A → αw'' where ''α'' is either empty or a single nonterminal and ''w'' is a string of terminals; * the right-linear or right-regular grammars, in which all rules are of the form ''A → wα'' where ''w'' is a string of terminals and ''α'' is either empty or a single nonterminal. Each of these can describe exactly the regular languages. A regular grammar is a grammar that is left-linear or right-linea ...
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Empty String
In formal language theory, the empty string, or empty word, is the unique string of length zero. Formal theory Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with ε or sometimes Λ or λ. The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string. The empty string has several properties: * , ε, = 0. Its string length is zero. * ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the concatenation operation. The set of all strings forms a free monoid with respect to ⋅ and ε. * εR = ε. Reve ...
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Context-free Grammar
In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules are of the form :A\ \to\ \alpha with A a ''single'' nonterminal symbol, and \alpha a string of terminals and/or nonterminals (\alpha can be empty). A formal grammar is "context-free" if its production rules can be applied regardless of the context of a nonterminal. No matter which symbols surround it, the single nonterminal on the left hand side can always be replaced by the right hand side. This is what distinguishes it from a context-sensitive grammar. A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the first rule in the picture, :\langle\text\rangle \to \langle\text\rangle = \langle\text\rangle ; replaces \langle\text\rangle with \langle\text\rangle = \langle\text\rangle ;. There can be multiple replacement rules for a given nonterminal symbol. Th ...
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Regular Expression
A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" or "find and replace" operations on strings, or for input validation. Regular expression techniques are developed in theoretical computer science and formal language theory. The concept of regular expressions began in the 1950s, when the American mathematician Stephen Cole Kleene formalized the concept of a regular language. They came into common use with Unix text-processing utilities. Different syntaxes for writing regular expressions have existed since the 1980s, one being the POSIX standard and another, widely used, being the Perl syntax. Regular expressions are used in search engines, in search and replace dialogs of word processors and text editors, in text processing utilities such as sed and AWK, and in lexical analysis. ...
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Nondeterministic Finite Automaton
In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if * each of its transitions is ''uniquely'' determined by its source state and input symbol, and * reading an input symbol is required for each state transition. A nondeterministic finite automaton (NFA), or nondeterministic finite-state machine, does not need to obey these restrictions. In particular, every DFA is also an NFA. Sometimes the term NFA is used in a narrower sense, referring to an NFA that is ''not'' a DFA, but not in this article. Using the subset construction algorithm, each NFA can be translated to an equivalent DFA; i.e., a DFA recognizing the same formal language. Like DFAs, NFAs only recognize regular languages. NFAs were introduced in 1959 by Michael O. Rabin and Dana Scott, who also showed their equivalence to DFAs. NFAs are used in the implementation of regular expressions: Thompson's construction is an algorithm for compiling a regular expression to an NFA ...
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Linear Grammar
In computer science, a linear grammar is a context-free grammar that has at most one nonterminal in the right-hand side of each of its productions. A linear language is a language generated by some linear grammar. Example An example of a linear grammar is ''G'' with ''N'' = , Σ = , ''P'' with start symbol ''S'' and rules : S → aSb : S → ε It generates the language \. Relationship with regular grammars Two special types of linear grammars are the following: * the left-linear or left-regular grammars, in which all rules are of the form ''A → αw'' where ''α'' is either empty or a single nonterminal and ''w'' is a string of terminals; * the right-linear or right-regular grammars, in which all rules are of the form ''A → wα'' where ''w'' is a string of terminals and ''α'' is either empty or a single nonterminal. Each of these can describe exactly the regular languages. A regular grammar is a grammar that is left-linear or right-linea ...
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