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Pumping Lemma For Regular Languages
In the theory of formal languages, the pumping lemma for regular languages is a Lemma (mathematics), lemma that describes an essential property of all regular languages. Informally, it says that all sufficiently long string (computer science), strings in a regular language may be ''pumped''—that is, have a middle section of the string repeated an arbitrary number of times—to produce a new string that is also part of the language. The pumping lemma is useful for proving that a specific language is not a regular language, by showing that the language does not have the property. Specifically, the pumping lemma says that for any regular language L, there exists a constant p such that any string w in L with length at least p can be split into three substrings x, y and z (w = xyz, with y being non-empty), such that the strings xz, xyz, xyyz, xyyyz, ... are also in L. The process of repeating y zero or more times is known as "pumping". Moreover, the pumping lemma guarantees that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pumping Lemma For Regular Languages Diagram
Pumping may refer to: * The operation of a pump, for moving a liquid from one location to another **The use of a breast pump for extraction of milk * Pumping (audio), a creative misuse of dynamic range compression * Pumping (computer systems), the number of times data is transmitted per clock cycle * Pumping (oil well), injecting chemicals into a wellbore * Pumping (noise reduction), an unwanted artifact of some noise reduction systems * Pumping lemma, in the theory of formal languages * Gastric lavage, cleaning the contents of the stomach * Optical pumping, in which light is used to raise electrons from a lower energy level to a higher one * Pump (skateboarding) Pumping is a skateboarding technique used to accelerate without the rider's feet leaving the board. Pumping can be done by turning or on a transition, like a ramp or quarter pipe.{{cite web , url=https://skateboarding.transworld.net/how-to/ba ..., accelerating without pushing off of the ground * "Pumping" (My Heart) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemmas , part of a neuron
{{Disambiguation ...
Lemma (from Ancient Greek ''premise'', ''assumption'', from Greek ''I take'', ''I get'') may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a part of a grass plant * Lemma (mathematics), a proven proposition used as a step in a larger proof Other uses * ''Lemma'' (album), by John Zorn (2013) See also *Analemma, a diagram showing the variation of the position of the Sun in the sky *Dilemma *Lema (other) * Lemmatisation *Neurolemma Neurilemma (also known as neurolemma, sheath of Schwann, or Schwann's sheath) is the outermost cell nucleus, nucleated cytoplasmic layer of Schwann cells (also called neurilemmocytes) that surrounds the axon of the neuron. It forms the outermost la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Languages
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] |
Pumping Lemma For Regular Tree Languages
Pumping may refer to: * The operation of a pump, for moving a liquid from one location to another **The use of a breast pump for extraction of milk * Pumping (audio), a creative misuse of dynamic range compression * Pumping (computer systems), the number of times data is transmitted per clock cycle * Pumping (oil well), injecting chemicals into a wellbore * Pumping (noise reduction), an unwanted artifact of some noise reduction systems * Pumping lemma, in the theory of formal languages * Gastric lavage, cleaning the contents of the stomach * Optical pumping, in which light is used to raise electrons from a lower energy level to a higher one * Pump (skateboarding) Pumping is a skateboarding technique used to accelerate without the rider's feet leaving the board. Pumping can be done by turning or on a transition, like a ramp or quarter pipe.{{cite web , url=https://skateboarding.transworld.net/how-to/ba ..., accelerating without pushing off of the ground * "Pumping" (My Heart) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Ogden's Lemma
In the theory of formal languages, Ogden's lemma (named after William F. Ogden) is a generalization of the pumping lemma for context-free languages. Despite Ogden's lemma being a strengthening of the pumping lemma, it is insufficient to fully characterize the class of context-free languages. This is in contrast to the Myhill–Nerode theorem, which unlike the pumping lemma for regular languages is a necessary and sufficient condition for regularity. Statement We will use underlines to indicate "marked" positions. Special cases Ogden's lemma is often stated in the following form, which can be obtained by "forgetting about" the grammar, and concentrating on the language itself: If a language is context-free, then there exists some number p\geq 1 (where may or may not be a pumping length) such that for any string of length at least in and every way of "marking" or more of the positions in , can be written as :s = uvwxy with strings and , such that # has at least one mar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Expression
A regular expression (shortened as regex or regexp), sometimes referred to as rational expression, is a sequence of characters that specifies a match 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-state Machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of ''State (computer science), states'' at any given time. The FSM can change from one state to another in response to some Input (computer science), inputs; the change from one state to another is called a ''transition''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—Deterministic finite automaton, deterministic finite-state machines and Nondeterministic finite automaton, non-deterministic finite-state machines. For any non-deterministic finite-state machine, an equivalent deterministic one can be constructed. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myhill–Nerode Theorem
In the theory of formal languages, the Myhill–Nerode theorem provides a necessary and sufficient condition for a language to be regular. The theorem is named for John Myhill and Anil Nerode, who proved it at the University of Chicago in 1957 . Statement Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that exactly one of the two strings xz and yz belongs to L. Define a relation \sim_L on strings as x\; \sim_L\ y if there is no distinguishing extension for x and y. It is easy to show that \sim_L is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes. The Myhill–Nerode theorem states that a language L is regular if and only if \sim_L has a finite number of equivalence classes, and moreover, that this number is equal to the number of states in the minimal deterministic finite automaton (DFA) accepting L. Furthermore, every minimal DFA for the language is isomorphic t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necessary And Sufficient Condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of . (Equivalently, it is impossible to have without , or the falsity of ensures the falsity of .) Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Statement
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attire, attire for semi-formal events * Informal attire, more controlled attire than casual but less than formal * Formal (university), official university dinner, ball or other event * School formal, official school dinner, ball or other event Logic and mathematics *Formal logic, or symbolic logic ** Informal logic, the complement, whose definition and scope is contentious *Formal fallacy, reasoning of invalid structure ** Informal fallacy, the complement *Informal mathematics, also called naïve mathematics *Formal cause, Aristotle's intrinsic, determining cause *Formal power series, a generalization of power series without requiring convergence, used in combinatorics *Formal calculation, a calculation which is systematic, but without a rigor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
