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Language Identification In The Limit
Language identification in the limit is a formal model for inductive inference of formal languages, mainly by computers (see machine learning and induction of regular languages). It was introduced by E. Mark Gold in a technical report and a journal article with the same title. In this model, a ''teacher'' provides to a ''learner'' some ''presentation'' (i.e. a sequence of strings) of some formal language. The learning is seen as an infinite process. Each time the learner reads an element of the presentation, it should provide a ''representation'' (e.g. a formal grammar) for the language. Gold defines that a learner can ''identify in the limit'' a class of languages if, given any presentation of any language in the class, the learner will produce only a finite number of wrong representations, and then stick with the correct representation. However, the learner need not be able to announce its correctness; and the teacher might present a counterexample to any representation arbit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inductive Inference
Inductive reasoning refers to a variety of methods of reasoning in which the conclusion of an argument is supported not with deductive certainty, but with some degree of probability. Unlike ''deductive'' reasoning (such as mathematical induction), where the conclusion is ''certain'', given the premises are correct, inductive reasoning produces conclusions that are at best ''probable'', given the evidence provided. Types The types of inductive reasoning include generalization, prediction, statistical syllogism, argument from analogy, and causal inference. There are also differences in how their results are regarded. Inductive generalization A generalization (more accurately, an ''inductive generalization'') proceeds from premises about a sample to a conclusion about the population. The observation obtained from this sample is projected onto the broader population. : The proportion Q of the sample has attribute A. : Therefore, the proportion Q of the population has attrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Set
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every natural number in a finite number of steps. A set is noncomputable (or undecidable) if it is not computable. Definition A subset S of the natural numbers is computable if there exists a total computable function f such that: :f(x)=1 if x\in S :f(x)=0 if x\notin S. In other words, the set S is computable if and only if the indicator function \mathbb_ is computable. Examples *Every recursive language is a computable. *Every finite or cofinite subset of the natural numbers is computable. **The empty set is computable. **The entire set of natural numbers is computable. **Every natural number is computable. *The subset of prime numbers is computable. *The set of Gödel numbers is computable. Non-examples *The set of Turing machines that halt is not computable. *The set of pairs of homeomorphic finite simplicial complexes is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever, but each element of S will be returned after a finite amount of time. Note that these elements do not have to be listed in a particular way, say from smallest to largest. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm can run forever, and no information is returned. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-finite Thickness
In formal language theory, in particular in algorithmic learning theory, a class ''C'' of languages has finite thickness if every string is contained in at most finitely many languages in ''C''. This condition was introduced by Dana Angluin as a 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 ... for ''C'' being identifiable in the limit. The related notion of M-finite thickness Given a language ''L'' and an indexed class ''C'' = of languages, a member language ''L''''j'' ∈ ''C'' is called a minimal concept of ''L'' within ''C'' if ''L'' ⊆ ''L''''j'', but not ''L'' ⊊ ''L''''i'' ⊆ ''L''''j'' for any ''L''''i'' ∈ ''C''. The class ''C'' is said to satisfy the MEF-condition if every finite subset ''D'' of a member language ''L''''i'' ∈ ''C'' has a minim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induction Of Regular Languages
In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction. Definitions A ''regular language'' is defined as a (finite or infinite) set of '' strings'' that can be described by one of the mathematical formalisms called "finite automaton", " regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shortest notations, it shall be introduced and used here. Given a set Σ of symbols (a.k.a. alphabet), a regular expression can be any of * ∅ (denoting the empty set of strings), * ε (denoting the singlet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pattern Language (formal Languages)
In theoretical computer science, a pattern language is a formal language that can be defined as the set of all particular instances of a string of constants and variables. Pattern Languages were introduced by Dana Angluin in the context of machine learning. Definition Given a finite set Σ of constant symbols and a countable set ''X'' of variable symbols disjoint from Σ, a pattern is a finite non-empty string of symbols from Σ∪''X''. The length of a pattern ''p'', denoted by , ''p'', , is just the number of its symbols. The set of all patterns containing exactly ''n'' distinct variables (each of which may occur several times) is denoted by ''P''''n'', the set of all patterns at all by ''P''*. A substitution is a mapping ''f'': ''P''* → ''P''* such thatAngluin's notion of substitution differs from the usual notion of string substitution. * ''f'' is a homomorphism with respect to string concatenation (⋅), formally: ∀''p'',''q''∈''P''*. ''f''(''p''⋅''q'') = ''f''(''p' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Examples
Example may refer to: * ''exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, and example.edu: second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum An exemplum (Latin for "example", exempla, ''exempli gratia'' = "for example", abbr.: ''e.g.'') is a moral anecdote, brief or extended, real or fictitious, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 expression engines, which are Regular expression#Patterns for non-regular languages, augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognised 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 regular grammar, Type-3 grammars. Formal definition The collection of regular languages over an Alphabet (formal languages), alphabet Σ is defined recursively as follows: * The empty language ∅ is a regular language. * For each ''a'' ∈ Σ (''a'' ... [...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] |
