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. Equivalently, it is a language recognised by a local automaton, a particular kind of deterministic finite automaton.

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. This corresponds to the regular expression

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

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; a language is locally testable if it is k-testable for some k. A local language is 2-testable.

Examples

• Over the alphabet {a,b,[,]}

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

Properties

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

References

• Lawson, Mark V. (2004). Finite automata. Chapman and Hall/CRC. ISBN 1-58488-255-7. Zbl 1086.68074.
• McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph. Vol. 65. With an appendix by William Henneman. MIT Press. ISBN 0-262-13076-9. Zbl 0232.94024.
• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177.
• Salomaa, Arto (1981). Jewels of Formal Language Theory. Pitman Publishing. ISBN 0-273-08522-0. Zbl 0487.68064.