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[1] and a journal article[2] 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.
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 arbitrarily long after.
Gold defined two types of presentations: This model is an early attempt to formally capture the notion of learnability.
Gold's journal article[3] introduces for contrast the stronger models A weaker formal model of learnability is the Probably approximately correct learning (PAC) model, introduced by Leslie Valiant in 1984.
It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about.
More formally,[7] Notes: Gold's theorem (1967) (Theorem I.8 of (Gold, 1967)) — If a language family
inductively as follows: By construction, the resulting environment
Gold's theorem is easily bypassed if negative examples are allowed.
Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper.
[8] If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates tell-tales for each language in the class (Condition 1).
[9] It is not hard to see that if an ideal learner (i.e., an arbitrary function) is allowed, then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2).
[10] The table shows which language classes are identifiable in the limit in which learning model.
Each learning model (i.e. type of presentation) can identify in the limit all classes below it.
In particular, the class of finite languages is identifiable in the limit by text presentation (cf.
Example 2 above), while the class of regular languages is not.
Pattern Languages, introduced by Dana Angluin in another 1980 paper,[12] are also identifiable by normal text presentation; they are omitted in the table, since they are above the singleton and below the primitive recursive language class, but incomparable to the classes in between.
[note 7][clarification needed] Condition 1 in Angluin's paper[9] is not always easy to verify.
Therefore, people come up with various sufficient conditions for the learnability of a language class.
A class of languages has finite thickness if every non-empty set of strings is contained in at most finitely many languages of the class.
[13] Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit.
[14] A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness.
[15] A class of languages is said to have finite elasticity if for every infinite sequence of strings
and every infinite sequence of languages in the class
[16] It is shown that a class of recursively enumerable languages is learnable in the limit if it has finite elasticity.
A bound over the number of hypothesis changes that occur before convergence.
such that: Note that L is not necessarily a member of the class of language.