Chomsky hierarchy - Chomsky hierarchy The Chomsky hierarchy is a containment hierarchy of classes of formal grammars that generate formal languages. This hierarchy was described by Noam Chomsky in 1956. Table of contents showTocToggle("show","hide") 1 Formal grammars 2 The hierarchy 3 References Formal grammars A formal grammar consists of a finite set of terminal symbols (the letters of the words in the formal language), a finite set of nonterminal symbols, a set of production rules with a left- and a right-hand side consisting of a word of these symbols, and a start symbol. A rule may be applied to a word by replacing the left-hand side by the right-hand side. A derivation is a sequence of rule applications. Such a grammar defines the formal language of all words consisting.
Chomsky Normal Form - Chomsky Normal Form A formal grammar is in Chomsky Normal Form iff all production rules are of the form: A -> BC or A -> α where A, B and C are nonterminal symbols and α is a terminal symbol. Every grammar in Chomsky Normal is context-free, and conversely, every context-free grammar which does not generate the empty string can be transformed into an equivalent one which is in Chomsky Normal Form. The Chomsky Normal Form of a context-free grammar is important because it yields efficient algorithms. For example, the CYK algorithm which decides whether a given string can be generated by a given grammar uses the Chomsky Normal Form. The Chomsky Normal Form is named after Noam Chomsky, an US linguist that invented the Chomsky.
Noam Chomsky - Noam Chomsky Avram Noam Chomsky (born December 7, 1928) is a US professor of linguistics at the Massachusetts Institute of Technology and creator of the Chomsky hierarchy, a classification of formal languages. Outside of his linguistic work, Chomsky describes himself as a libertarian socialist and is widely known for his left-wing political writings. Noam Chomsky (photo by Donna Coveney) Table of contents showTocToggle("show","hide") 1 Short biography 2 Contributions to Linguistics 3 Contributions to Psychology 4 Criticism of post-modern views towards science 5 Political Views 5.1 Criticism of the United States government 5.2 Chomsky and Socialism 5.3 Chomsky and the Middle East 6 Accusations of anti-semitism 6.4 The Faurisson Affair 7 Other criticisms 8 Other concepts 9 Quotes regarding Chomsky 10 See also: 11 Bibliography 11.5 Linguistics 11.6.
Grammar - kind of grammar. Chinese is very context dependent. In other words, context accomplishes the same role as declension and conjugation. (Chinese does have some inflections, and had more in the past.) Latin, which is synthetic, uses affixes and inflections to accomplish the same role that Chinese does with syntax. Because Latin words are quite (though not completely) self-contained, a sentence can be made from scattered elements. In short, Latin has a complex affixion and a simple syntax, while Chinese has the opposite. Grammars of specific languages Arabic grammar Chinese grammar Dutch grammar English grammar Esperanto grammar Finnish language grammar French grammar German grammar Hebrew grammar Italian grammar Japanese grammar Latin grammar Russian grammar Spanish grammar Swedish grammar Grammatical terms adjective adjunct adverb article aspect auxiliary verb case clause closed class word.
Formal language - or λ. Note that while the alphabet is a finite set and every string has finite length, a language may very well have infinitely many member strings. Some examples of formal languages: the set of all words over {a, b}, the set { an n is a prime number }, the set of syntactically correct programs in some programming language, or the set of inputs upon which a certain Turing machine halts. A formal language can be specified in a great variety of ways, such as: Strings produced by some formal grammar (see Chomsky hierarchy) Strings produced by a regular expression Strings accepted by some automaton, such as a Turing machine or finite state automaton From a set of related YES/NO questions those ones for which the answer is YES, see.
Formal grammar - symbol. Usually such a formal grammar G is simply summarized as (N, Σ, P, S). The language of a formal grammar G = (N, Σ, P, S), denoted as L(G), is defined as all those strings over Σ that can be generated by starting with the start symbol S and then applying the production rules in P until no more nonterminal symbols are present. Example Consider, for example, the grammar G with N = {S, B}, Σ = {a, b, c}, P consisting of the following production rules 1. S -> aBSc 2. S -> abc 3. Ba -> aB 4. Bb -> bb and the nonterminal symbol S as the start symbol. Some examples of the derivation of strings in L(G) are: 'S'\ -> (2) abc S -> (1) aBSc.
Computation - computer, but require such an enormously long time to compute that the solution is impractical? (See Presburger arithmetic.) Can it be harder to solve a problem than to check a given solution? (See complexity classes P and NP). In general, questions concerning the time or space requirements of given problems are investigated in complexity theory. In addition to the general computational models, some simpler computational models are useful for special, restricted applications. Regular expressions, for example, are used to specify string patterns in UNIX and in some programming languages such as Perl. Another formalism mathematically equivalent to regular expressions, Finite automata are used in circuit design and in some kinds of problem-solving. Context-free grammars are used to specify programming language syntax. Non-deterministic pushdown automata are another formalism equivalent to context-free grammars..
Context-sensitive grammar - that a rule of the form S -> ε with ε the empty string, is allowed if S does not appear on the right side of any rule. The adjective context sensitive is explained by the α and β that form then context of A and determine whether A can be replaced with γ or not. This is different from a context-free grammar where the context of a nonterminal is not taken into consideration. A formal language that can be described by a context-sensitive grammar is called a context-sensitive language. The concept of context-sensitive grammar was introduced by Noam Chomsky in the 1950s as a way to describe the syntax of natural language where it is indeed often the case that a word may or may not be appropriate in a.
Context-sensitive language - defined by a context-sensitive grammar. That is one of the four types of grammars in the Chomsky hierarchy. Of the four, this is the least often used, in both theory and practice. Computational properties Computationally the context-sensitive languages are equivalent with linear bounded non-deterministic Turing machines. 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 NLIN-SPACE, because they can be accepted using linear space on a non-deterministic Turing machine. The class LIN-SPACE is.
Context-free grammar - a's as b's, U generates all strings with more a's than b's and V generates all strings with fewer a's than b's. Derivations and Syntax trees There are basically two ways to describe how in a certain grammar a string can be derived from the start symbol. The simplest way is to list the consecutive strings of symbols, beginning with the start symbol and ending with the string, and the rules that have been applied. If we introduce a strategy such as "always replace the left-most nonterminal first" then for context-free grammars the list of applied grammar rules is by itself sufficient. This is called the leftmost derivation of a string. For example, if we take the following grammar: (1) S -> S + S (2) S -> 1 and the.
Computability theory - calculus Partial recursive functions Almost any modern programming language (when given unlimited memory), including: A language with 1 instruction, 1 parameter (see OISC and URISC here) A language with 8 instructions, no parameters (see BrainFuck) Wang tiles Recurrent neural network (finite-precision inputs/outputs/weights, infinite-precision signals initialized to zero) Cellular automaton, including: Conway's Game of Life Cellular automaton with just 1 dimension, 2 states, 3 cells per neighborhood (e.g. rule 110) Non-deterministic Turing machine Probabilistic Turing Machine Quantum computer The last three examples use a slightly different definition of accepting a language. They are said to accept a string if any computation accepts (for non-deterministic), or most computations accept (for probabilistic and quantum). Given these definitions, those machines have the same power as a Turing machine for accepting languages. What problems require more.
Counter - one below it: Deterministic or Non-deterministic FSM plus two counters Non-deterministic FSM plus one stack Non-deterministic FSM plus one counter Deterministic FSM plus one counter Deterministic or Non-deterministic FSM For the first and last, it doesn't matter whether the FSM is deterministic or non-deterministic (see determinism). They have equivalent power. The first two and the last one are levels of the Chomsky hierarchy. The first machine, an FSM plus two counters, is equivalent in power to a Turing machine. This equivalence can be shown in three steps. First, a Turing machine can be simulated by two stacks. Then, a stack can be simulated by two counters. Finally, four counters can be simulated by two counters. Step 1: A Turing machine can be simulated by two stacks. A Turing machine consists of.
Tree structure - complete binary tree, which means all nodes have exactly zero or two child nodes. The lines connecting elements are called ''branches," the elements themselves are called "nodes." Nodes without children are called "end-nodes" or "leaves." The names of relationships between nodes are modeled after family relations. In computer sciences, traditionally only names for male family members have been used. In linguistics, the names of female family members are used. It is said that this was an express counter movement to the traditional naming convention, started by the female students of linguist Noam Chomsky. However, nowadays, in computer science at least, the gender-neutral names "parent" and "child" have largely displaced the older "father" and "son" terminology. The starting node is often called the "root." A node is a "parent" of another node.
Transformational grammar - Transformational grammar [This article concentrates heavily on Chomsky and Chomsky-related aspects of this topic. This is justifiable to some degree considering his importance in the field, but it would be nice to have a more balanced view.] Transformational grammar is a broad term describing grammars (almost exclusively those of natural languages) which have been developed in a Chomskyan tradition. The term is usually synonymous with the slightly more specific transformational-generative grammar. Deep Structure and Surface Structure In the 1950s and 1960s, Chomsky developed the idea that each sentence in a language has two levels of representation - a Deep Structure and a Surface Structure. The deep structure was a direct representation of the semantics of a sentence, and was mapped onto the surface structure (which followed the phonological form of the.
Syntax - literal text of something written in a formal language or programming language, as opposed to its semantics or meaning. The analysis of programming language syntax usually entails the transformation of a linear sequence of tokens (a token is akin to an individual word or punctuation mark in a natural language) into a hierarchical syntax tree (abstract syntax trees are one convenient form of syntax tree). This process, called parsing, is in some respects analogous to syntactic analysis in linguistics; in fact, certain concepts, such as the Chomsky hierarchy and context-free grammars, are common to the study of syntax in both linguistics and computer science. However, the applications of these concepts vary widely between the two fields, and the practical resemblances are small..
Regular expression - built this notation into the editor qed, then into the Unix editor ed and eventually into grep. Ever since that time, regular expressions have been widely used in Unix and Unix-like utilities such as: expr, awk, Emacs, vim, lex, and Perl. Most tools use an implementation of the regex library built by Henry Spencer. Regular expressions correspond to the type 3 grammars of the Chomsky hierarchy and may be used to describe a regular language. Table of contents showTocToggle("show","hide") 1 Regular expressions in Formal Language Theory 2 Regular expression syntaxes 2.1 Traditional Unix regexps 2.2 POSIX modern (extended) regexps 2.3 Perl Compatible Regular Expressions (PCRE) 3.
Regular language - language; the complement of every regular language is a regular language as well. Reversing every string in a regular language yields another regular language. Concatenating two regular languages (in the sense of concatenating every string from the first language with every string from the second one) also yields a regular language. The shuffle operation, when applied to two regular languages, yields another regular language. The right quotient and the left quotient of a regular language by an arbitrary language is also regular. To locate the regular languages in the Chomsky hierarchy, one notices that every regular language is context-free. The converse is not true: for example the language consisting of all strings having the same number of a's as b's is context-free but not regular. To prove that a language such.
Regular grammar - in N and a in Σ A -> ε where A in N. The second form may also be replaced with A -> Ba. An example of a regular grammar G with N = {S, A}, Σ = {a, b, c}, P consists of the following rules S -> aS S -> bA A -> ε A -> cA and S is the start symbol. This grammar describes the same language as the regular expression a*bc*. The regular grammars describe exactly all regular languages and are in that sense equivalent with finite state automata and regular expressions. See also: Chomsky hierarchy.
Libertarian socialism - opposing what its advocates regard as illegitimate forms of authority and social hierarchy, most famously the institution of government. It has gone by various names: libertarian communism, anarcho-communism, left-anarchism, and, most commonly, anarchism. Libertarian socialists therefore believe in the abolition of privately held means of production and abolition of the state as an unnecessary and harmful institution (anarchism/libertarianism). Table of contents showTocToggle("show","hide") 1 Overview 2 Anti-capitalism 3 Opposition to the state 4 Political roots 4.1 Conflict with Marxism 5 Philosophical arguments 5.2 The importance of force 6 Historical Origins 6.3 Pre-"anarchism" libertarians 6.4 Anarchism: a new word 6.5 The spread of ideas: anarchism's influence 7 Further reading 7.6 Books 7.7 Periodicals 8 External Links Overview Libertarian socialists usually call themselves anarchists except when necessary to disambiguate or disassociate themselves with others.
List of mathematical topics - -- Algebraic number -- Algebraic number theory -- Algebraic structure -- Algebraic topology -- Algebraic variety -- Algebraically closed field -- Algebraically independent -- Algorithm -- Algorithmic information theory -- Algorithmics -- Algorithms for calculating variance -- Aliasing -- Alignments of random points -- Al-Khwarizmi -- Almost all -- Almost everywhere -- Almost perfect number -- Almost periodic function -- Almost prime -- Almost surely -- Alpha-beta pruning -- Alternating group -- Altitude (triangle) -- Amicable number -- Amortized analysis -- Amplitude -- Analysis -- Analysis of variance -- Analytic continuation -- Analytic function -- Analytic geometry -- Analytic number theory -- Analytical Society -- Anderson, Alexander -- ANCOVA -- Andreini tessellation -- Angle -- Angle excess -- Angle of attack -- Angle of incidence -- Angle of view -- Angular.
