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Penttonen Normal Form
In formal language theory, a noncontracting grammar is in Kuroda normal form if all production rules are of the form: :''AB'' → ''CD'' or :''A'' → ''BC'' or :''A'' → ''B'' or :''A'' → ''a'' where A, B, C and D are nonterminal symbols and ''a'' is a terminal symbol. Some sources omit the ''A'' → ''B'' pattern. It is named after Sige-Yuki Kuroda, who originally called it a linear bounded grammar, a terminology that was also used by a few other authors thereafter. Every grammar in Kuroda normal form is noncontracting, and therefore, generates a context-sensitive language. Conversely, every noncontracting grammar that does not generate the empty string can be converted to Kuroda normal form. A straightforward technique attributed to György Révész transforms a grammar in Kuroda normal form to a context-sensitive grammar: ''AB'' → ''CD'' is replaced by four context-sensitive rules ''AB'' → ''AZ'', ''AZ'' → ''WZ'', ''WZ'' → ''WD'' and ''WD'' → ''CD''. This prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language Theory
In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a set called "#Definition, alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also called "words"). Words that belong to a particular formal language are sometimes called Formal language#Definition, ''well-formed words''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar. 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 meanings or semantics. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncontracting Grammar
In formal language theory, a grammar is noncontracting (or monotonic) if for all of its production rules, α → β (where α and β are strings of nonterminal and terminal symbols), it holds that , α, ≤ , β, , that is β has at least as many symbols as α. A grammar is essentially noncontracting if there may be one exception, namely, a rule ''S'' → ε where ''S'' is the start symbol and ε the empty string, and furthermore, ''S'' never occurs in the right-hand side of any rule. A context-sensitive grammar is a noncontracting grammar in which all rules are of the form αAβ → αγβ, where A is a nonterminal, and γ is a nonempty string of nonterminal and/or terminal symbols. However, some authors use the term context-sensitive grammar to refer to noncontracting grammars in general. A noncontracting grammar in which , α, 1 by 2''m'' rules:For convenience, the non-context part of left and right hand side is shown in boldface. :: ::where each ''Z'''' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonterminal
In formal languages, terminal and nonterminal symbols are parts of the ''vocabulary'' under a formal grammar. ''Vocabulary'' is a finite, nonempty set of symbols. ''Terminal symbols'' are symbols that cannot be replaced by other symbols of the vocabulary. ''Nonterminal symbols'' are symbols that can be replaced by other symbols of the vocabulary by the production rules under the same formal grammar. A formal grammar defines a formal language over the vocabulary of the grammar. In the context of formal language, the term ''vocabulary'' is more commonly known as ''alphabet''. Nonterminal symbols are also called ''syntactic variables''. Terminal symbols Terminal symbols are those symbols that can appear in the formal language defined by a formal grammar. The process of applying the production rules successively to a start symbol might not terminate, but if it terminates when there is no more production rule can be applied, the output string will consist only of terminal symb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sige-Yuki Kuroda
, also known as S.-Y. Kuroda, was Professor Emeritus and Research Professor of Linguistics at the University of California, San Diego. Although a pioneer in the application of Chomskyan generative syntax to the Japanese language, he is known for the broad range of his work across the language sciences. For instance, in formal language theory, the Kuroda normal form for context-sensitive grammars bears his name. Early life and career Kuroda was born into a prominent family of mathematicians in Japan. His grandfather, Teiji Takagi, was a student of David Hilbert. Kuroda himself received degrees in mathematics and linguistics from the University of Tokyo. In 1962, he entered MIT with the first graduating class from the new Department of Linguistics, where he wrote his seminal dissertation, ''Generative Studies in the Japanese Language'' (1965), under Chomsky's supervision. Important publications * "Classes of languages and linear-bounded automata", Information and Control, 7(2): ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-sensitive Language
In formal language theory, a context-sensitive language is a language that can be defined by a context-sensitive grammar (and equivalently by a noncontracting grammar). Context-sensitive is known as type-1 in the Chomsky hierarchy of formal languages. Computational properties Computationally, a context-sensitive language is equivalent to a linear bounded nondeterministic Turing machine, also called a linear bounded automaton. That is a non-deterministic Turing machine with a tape of only kn cells, where n is the size of the input and k is a constant associated with the machine. This means that every formal language that can be decided by such a machine is a context-sensitive language, and every context-sensitive language can be decided by such a machine. This set of languages is also known as NLINSPACE or NSPACE(''O''(''n'')), because they can be accepted using linear space on a non-deterministic Turing machine. The class LINSPACE (or DSPACE(''O''(''n''))) is defined the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty String
In formal language theory, the empty string, or empty word, is the unique String (computer science), string of length zero. Formal theory Formally, a string is a finite, ordered sequence of character (symbol), 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 (computer science)#Formal theory, string length is zero. * ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the concatenation operation. The set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-sensitive Grammar
A context-sensitive grammar (CSG) is a formal grammar in which the left-hand sides and right-hand sides of any Production (computer science), production rules may be surrounded by a context of terminal symbol, terminal and nonterminal symbols. Context-sensitive grammars are more general than context-free grammars, in the sense that there are languages that can be described by a CSG but not by a context-free grammar. Context-sensitive grammars are less general (in the same sense) than unrestricted grammars. Thus, CSGs are positioned between context-free and unrestricted grammars in the Chomsky hierarchy. A formal language that can be described by a context-sensitive grammar, or, equivalently, by a noncontracting grammar or a linear bounded automaton, is called a context-sensitive language. Some textbooks actually define CSGs as non-contracting, although this is not how Noam Chomsky defined them in 1959. This choice of definition makes no difference in terms of the languages generated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unrestricted Grammar
In automata theory, the class of unrestricted grammars (also called semi-Thue, type-0 or phrase structure grammars) is the most general class of grammars in the Chomsky hierarchy. No restrictions are made on the productions of an unrestricted grammar, other than each of their left-hand sides being non-empty. This grammar class can generate arbitrary recursively enumerable languages. Formal definition An unrestricted grammar is a formal grammar G = (N, T, P, S), where * N is a finite set of nonterminal symbols, * T is a finite set of terminal symbols with N and T disjoint,Actually, T\cap N=\emptyset is not strictly necessary since unrestricted grammars make no real distinction between the two. The designation exists purely so that one knows when to stop generating sentential forms of the grammar; more precisely, the language L(G) recognized by G is restricted to strings of terminal symbols. * P is a finite set of production rules of the form \alpha \to \beta , where \alpha an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Equivalence (formal Languages)
In formal language theory, weak equivalence of two formal grammars, grammars means they generate the same set of strings, i.e. that the formal language they generate is the same. In compiler theory the notion is distinguished from strong (or structural) equivalence, which additionally means that the two parse trees are reasonably similar in that the same semantic interpretation can be assigned to both. Vijay-Shanker and Weir (1994) demonstrates that Indexed grammar, Linear Indexed Grammars, Combinatory categorial grammar, Combinatory Categorial Grammars, Tree-adjoining grammar, Tree-adjoining Grammars, and Head grammar, Head Grammars are weakly equivalent formalisms, in that they all define the same string languages. On the other hand, if two grammars generate the same set of derivation trees (or more generally, the same set of abstract syntactic objects), then the two grammars are strongly equivalent. Chomsky (1963) introduces the notion of strong equivalence, and argues that onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chomsky Normal Form
In formal language theory, a context-free grammar, ''G'', is said to be in Chomsky normal form (first described by Noam Chomsky) if all of its production rules are of the form: : ''A'' → ''BC'', or : ''A'' → ''a'', or : ''S'' → ε, where ''A'', ''B'', and ''C'' are nonterminal symbols, the letter ''a'' is a terminal symbol (a symbol that represents a constant value), ''S'' is the start symbol, and ε denotes the empty string. Also, neither ''B'' nor ''C'' may be the start symbol, and the third production rule can only appear if ε is in ''L''(''G''), the language produced by the context-free grammar ''G''. Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent onethat is, one that produces the same language which is in Chomsky normal form and has a size no larger than the square of the original grammar's size. Converting a grammar to Chomsky normal form To convert a grammar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backus–Naur Form
In computer science, Backus–Naur form (BNF, pronounced ), also known as Backus normal form, is a notation system for defining the Syntax (programming languages), syntax of Programming language, programming languages and other Formal language, formal languages, developed by John Backus and Peter Naur. It is a metasyntax for Context-free grammar, context-free grammars, providing a precise way to outline the rules of a language's structure. It has been widely used in official specifications, manuals, and textbooks on programming language theory, as well as to describe Document format, document formats, Instruction set, instruction sets, and Communication protocol, communication protocols. Over time, variations such as extended Backus–Naur form (EBNF) and augmented Backus–Naur form (ABNF) have emerged, building on the original framework with added features. Structure BNF specifications outline how symbols are combined to form syntactically valid sequences. Each BNF consists of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
