Outfix-guided insertion

Abstract

Motivated by work on bio-operations on DNA strings, we consider an outfix-guided insertion operation that can be viewed as a generalization of the overlap assembly operation on strings studied previously. As the main result we construct a finite language L such that the outfix-guided insertion closure of L is non-regular. We consider also the closure properties of regular and (deterministic) context-free languages under the outfix-guided insertion operation and decision problems related to outfix-guided insertion. Deciding whether a language recognized by a deterministic finite automaton is closed under outfix-guided insertion can be done in polynomial time. The complexity of the corresponding question for nondeterministic finite automata remains open.

Site-Directed Insertion: Decision Problems, Maximality and Minimality

“…If y = uzv is inserted into x by matching the outfix with a substring uv of x, we say that (u, v) is an insertion guide for the operation. Note that a previous paper [2] uses the name "outfix-guided insertion" for the same operation. The site-directed insertion operation is extended in the usual way for languages by setting…”

“…Note that the alphabetic site-directed insertion will have different closure properties than the standard site-directed insertion. For example, it is not difficult to see that the context-free languages are closed under alphabetic sitedirected insertion, while the context-free languages are not closed under general site-directed insertion [2].…”

“…For a regular language L, it is decidable whether L is closed under site-directed insertion. The algorithm relies on the construction of Proposition 1 and operates in polynomial time when L is specified by a DFA [2]. Deciding whether a contextfree language is closed under site-directed insertion is undecidable [2].…”

“…The site-directed insertion (SDI) operation preserves regularity [2] (above stated as Proposition 1) and the construction can be modified to show that also alphabetic SDI preserves regularity. To conclude, we consider the nondeterministic state complexity of these operations.…”

“…Site-directed insertion (SDI) of a string y into a string x involves matching an outfix of y with a substring of x and inserting the "middle part" of y not belonging to the outfix into x. Site-directed insertion has earlier been considered under the name outfix-guided insertion [2]. The operation is an overlapping variant of the insertion operation in the same sense as the overlap assembly, a.k.a.…”