computer science

Language identification in the limit

Language identification in the limit is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Language identification in the limit rather than just read about it. In short: 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.

Key takeaways

Language identification in the limit belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Language identification in the limit to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Language identification in the limit from memory before moving on to harder problems.

Reference excerpt

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 arbitrarily long after. Gold defined two types of presentations:

Text (positive information): an enumeration of all strings the language consists of. Complete presentation (positive and negative information): an enumeration of all possible strings, each with a label indicating if the string belongs to the language or not.

Learnability This model is an early attempt to formally capture the notion of learnability. Gold's journal article introduces for contrast the stronger models

Finite identification (where the learner has to announce correctness after a finite number of steps), and Fixed-time identification (where correctness has to be reached after an apriori-specified number of steps). A weaker formal model of learnability is the Probably approximately correct learning (PAC) model, introduced by Leslie Valiant in 1984.

Examples

It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about.

A fictitious session to learn a regular language L over the alphabet {a,b} from text presentation:In each step, the teacher gives a string belonging to L, and the learner answers a guess for L, encoded as a regular expression. In step 3, the learner's guess is not consistent with the strings seen so far; in step 4, the teacher gives a string repeatedly. After step 6, the learner sticks to the regular expression (ab+ba)*. If this happens to be a description of the language L the teacher has in mind, it is said that the learner has learned that language.If a computer program for the learner's role would exist that was able to successfully learn each regular language, that class of languages would be identifiable in the limit. Gold has shown that this is not the case. A particular learning algorithm always guessing L to be just the union of all strings seen so far:If L is a finite language, the learner will eventually guess it correctly, however, without being able to tell when. Although the guess didn't change during step 3 to 6, the learner couldn't be sure to be correct.Gold has shown that the class of finite languages is identifiable in the limit, however, this class is neither finitely nor fixed-time identifiable. Learning from complete presentation by telling:In each step, the teacher gives a string and tells whether it belongs to L (green) or not (red, struck-out). Each possible string is eventually classified in this way by the teacher. Learning from complete presentation by request:The learner gives a query string, the teacher tells whether it belongs to L (yes) or not (no); the learner then gives a guess for L, followed by the next query string. In this example, the learner happens to query in each step just the same string as given by the teacher in example 3.In general, Gold has shown that each language class identifiable in the request-presentation setting is also identifiable in the telling-presentation setting, since the learner, instead of querying a string, just needs to wait until it is eventually given by the teacher.

Gold's theorem More formally,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Language identification in the limit

Start with the simplest possible case. Write down what Language identification in the limit claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Language identification in the limit before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Language identification in the limit ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Language identification in the limit

In research: Language identification in the limit appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Language identification in the limit in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Language identification in the limit is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational learning theory, Formal languages, so understanding it makes those chapters shorter.
In everyday life: Look for Language identification in the limit outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Language identification in the limit in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Language identification in the limit means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Language identification in the limit out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Language identification in the limit in simple terms?

Why does Language identification in the limit matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Language identification in the limit?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Language identification in the limit.