Laplacian of the indicator

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In mathematics, a local language is a formal language for which membership of a word in the language can be determined by looking at a "window" of length two.[1] Equivalently, it is a language recognised by a local automaton, a class of deterministic finite automaton.[2]

Formally, we define a language L over an alphabet A 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

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Template:Algebra-stub

  1. 1.0 1.1 1.2 1.3 Salomaa (1981) p.97
  2. Lawson (2004) p.130
  3. Lawson (2004) p.129
  4. 4.0 4.1 4.2 Sakarovitch (2009) p.228
  5. McNaughton & Papert (1971) p.14
  6. Lawson (2004) p.132
  7. McNaughton & Papert (1971) p.18