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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] |
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] |
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] |
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] |
Example
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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] |
Kuroda 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 pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transforming Into Context-sensitive Grammar
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Pumping Lemma For Context-free Languages
In computer science, in particular in formal language theory, the pumping lemma for context-free languages, also known as the Bar-Hillel lemma, is a lemma that gives a property shared by all context-free languages and generalizes the pumping lemma for regular languages. The pumping lemma can be used to construct a refutation by contradiction that a specific language is ''not'' context-free. Conversely, the pumping lemma does not suffice to guarantee that a language ''is'' context-free; there are other necessary conditions, such as Ogden's lemma, or the Interchange lemma. Formal statement If a language L is context-free, then there exists some integer p \geq 1 (called a "pumping length") (Also see s, \geq p) can be written as : s = uvwxy with substrings u, v, w, x and y, such that : 1. , vx, \geq 1, : 2. , vwx, \leq p, and : 3. uv^n wx^n y \in L for all n \geq 0. Below is a formal expression of the Pumping Lemma. \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-free Language
In formal language theory, a context-free language (CFL), also called a Chomsky type-2 language, is a language generated by a context-free grammar (CFG). Context-free languages have many applications in programming languages, in particular, most arithmetic expressions are generated by context-free grammars. Background Context-free grammar Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language. Automata The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar (and thereby the corresponding language), though going the other way (producing a grammar given an automaton) is not as direct. Examples An e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Grammar
A formal grammar is a set of Terminal and nonterminal symbols, symbols and the Production (computer science), production rules for rewriting some of them into every possible string of a formal language over an Alphabet (formal languages), alphabet. A grammar does not describe the semantics, meaning of the strings — only their form. In applied mathematics, formal language theory is the discipline that studies formal grammars and languages. Its applications are found in theoretical computer science, theoretical linguistics, Formal semantics (logic), formal semantics, mathematical logic, and other areas. A formal grammar is a Set_(mathematics), 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 determines whether a given string belongs to the language or is grammatical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chomsky Hierarchy
The Chomsky hierarchy in the fields of formal language theory, computer science, and linguistics, is a containment hierarchy of classes of formal grammars. A formal grammar describes how to form strings from a formal language's alphabet that are valid according to the language's syntax. The linguist Noam Chomsky theorized that four different classes of formal grammars existed that could generate increasingly complex languages. Each class can also completely generate the language of all inferior classes (set inclusive). History The general idea of a hierarchy of grammars was first described by Noam Chomsky in "Three models for the description of language" during the formalization of transformational-generative grammar (TGG). Marcel-Paul Schützenberger also played a role in the development of the theory of formal languages; the paper "The algebraic theory of context free languages" describes the modern hierarchy, including context-free grammars. Independently, alongside linguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
