Splicing rule

Let A be an alphabet and L a language, that is, a subset of the free monoid A^∗. A splicing rule is a quadruple r = (a,b,c,d) of elements of A^∗, and the action of the rule r on L is to produce the language

${\displaystyle r(L)=\{xady:xabq,pcdy\in L\}\ .}$

If R is a set of rules then R(L) is the union of the languages produced by the rules of R. We say that R respects L if R(L) is a subset of L. The R-closure of L is the union of L and all iterates of R on L: clearly it is respected by R. A splicing language is the R-closure of a finite language.[1]

A rule set R is reflexive if (a,b,c,d) in R implies that (a,b,a,b) and (c,d,c,d) are in R. A splicing language is reflexive if it is defined by a reflexive rule set.[2]

• Let A = {a,b,c}. The rule (caba,a,cab,a) applied to the finite set {cabb,cabab,cabaab} generates the regular language caba^∗b.[3]

• All splicing languages are regular.[4]
• Not all regular languages are splicing.[5] An example is (aa)^∗ over {a,b}.[4]
• If L is a regular language on the alphabet A, and z is a letter not in A, then the language { zw : w in L } is a splicing language.[3]
• There is an algorithm to determine whether a given regular language is a reflexive splicing language.[2]
• The set of splicing rules that respect a regular language can be determined from the syntactic monoid of the language.[6]

• Anderson, James A. (2006). Automata theory with modern applications. With contributions by Tom Head. Cambridge: Cambridge University Press. ISBN 0-521-61324-8. Zbl 1127.68049.