Definitions

# Language identification in the limit

Language identification in the limit is a formal model for inductive inference. It was introduced by E. Mark Gold in his paper with the same title In this model, a learner is provided with presentation of some language. The learning is seen as an infinite process. Each time an element of the presentation is read the learner should provide a representation for the language. We say 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 a finite number of wrong representations, and therefore converge on the correct representation in a finite number of steps, without however necessarily being able to announce its correctness since a counterexample to that representation could appear as an element arbitrarily long after.

Gold defined two types of presentations:

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

## Learnability

This model is the first known attempt to capture the notion of learnability; another learnability model is the so-called Probably approximately correct learning (PAC) model.

## Learnability characterization

Dana Angluin gave the characterizations of learnability from text (positive information) in her paper

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). It is not hard to see that if we allow an ideal learner (i.e., an arbitrary function), then an indexed class of languages is learnable in the limit if each language in the class has a tell-tale (Condition 2).

## Sufficient conditions for learnability

Condition 1 in Angluin's paper is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class.

### Finite thickness

A class of languages has finite thickness if for every string s, there are only a finite number of languages in the class that are consistent with s. This is exactly Condition 3 in Angluin's paper. Angluin showed that if a class of recursive languages has finite thickness, then it is learnable in the limit.

A class with finite thickness certainly satisfies MEF-condition and MFF-condition; in other words, finite thickness implies M-finite thickness.

### Finite elasticity

A class of languages is said to have finite elasticity if for every infinite sequence of strings $s_0, s_1, ...$ and every infinite sequence of languages in the class $L_1, L_2, ...$, there exists a finite number n such that $s_nnotin L_n$ implies $L_n$ is inconsistent with $\left\{s_1,...,s_\left\{n-1\right\}\right\}$.

It is shown that a class of recursively enumerable languages is learnable in the limit if it has finite elasticity.

## Other concepts

### Infinite cross property

A language L has infinite cross property within a class of languages $mathcal\left\{L\right\}$ if there is an infinite sequence $L_i$ of distinct languages in $mathcal\left\{L\right\}$ and a sequence of finite subset $T_i$ such that:

• $T_1 sub T_2sub ...$,
• $T_i in L_i$,
• $T_\left\{i+1\right\}notin L_i$, and
• $lim_\left\{n=infty\right\}T_i=L$.

Note that L is not necessarily a member of the class of language.

It is not hard to see that if there is a language with infinite cross property within a class of languages, then that class of languages has infinite elasticity.

## Relations between concepts

• Finite thickness implies finite elasticity; the converse is not true.
• Finite elasticity and conservatively learnable implies the existence of a mind change bound.
• Finite elasticity and M-finite thickness implies the existence of a mind change bound. However, M-finite thickness alone does not imply the existence of a mind change bound; neither does the existence of a mind change bound imply M-finite thickness.
• Existence of a mind change bound implies learnability; the converse is not true.
• If we allow for noncomputable learners, then finite elasticity implies the existence of a mind change bound; the converse is not true.
• If there is no accumulation order for a class of languages, then there is a language (not necessarily in the class) that has infinite cross property within the class, which in turn implies infinite elasticity of the class.

## Open questions

• If a countable class of recursive languages has a mind change bound for noncomputable learners, does the class also have a mind change bound for computable learners, or is the class unlearnable by a computable learner?