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 expression^[1][4]

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

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]

Examples

• Over the alphabet {a,b,[,]}^[4]

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

Properties

• 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.^[1][7][8]