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]

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

Examples
  • Over the alphabet {a,b,[,]}[4]

Properties

References
  1. Salomaa (1981) p.97
  2. Lawson (2004) p.130
  3. Lawson (2004) p.129
  4. Sakarovitch (2009) p.228
  5. McNaughton & Papert (1971) p.14
  6. Lawson (2004) p.132
  7. McNaughton & Papert (1971) p.18
  • 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.
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