Laplacian of the indicator

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

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. This corresponds to the regular expression

$(R{A}^{*}\cap A^{*}S)\setminus A^{*}F{A}^{*}.$

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. A local language is 2-testable.

Examples

• Over the alphabet {a,b}

$a{a}^{*},\{ab\}.$

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

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534