Local language (formal language)

In mathematics, a local language is a formal language for which membership of a word in the language can be determined by looking at the first and last symbol and each two-symbol substring of the word.¹ Equivalently, it is a language recognised by a local automaton, a particular kind of deterministic finite automaton.²

Formally, a language L over an alphabet A is defined to be local if there are subsets R and S of A and a subset F of A×A such that a word w is in L if and only if the first letter of w is in R, the last letter of w is in S and no factor of length 2 in w is in F.³ This corresponds to the regular expression¹⁴

${\displaystyle (RA^{*}\cap A^{*}S)\setminus A^{*}FA^{*}\ .}$

More generally, a k-testable language L is one for which membership of a word w in L depends only on the prefix and suffix of length k and the set of factors of w of length k;⁵ a language is locally testable if it is k-testable for some k.⁶ A local language is 2-testable.¹

• Over the alphabet {a,b,[,]}⁴

${\displaystyle aa^{*},\ [ab]\ .}$

• The family of local languages over A is closed under intersection and Kleene star, but not complement, union or concatenation.⁴
• Every regular language not containing the empty string is the image of a local language under a strictly alphabetic morphism.¹⁷⁸