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.

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Splicing, as a binary word/language operation, was inspired by the DNA recombination under the action of restriction enzymes and ligases, and was first introduced by Tom Head in 1987. Splicing systems as generative mechanisms were defined as consisting of an initial starting set of words called an axiom set, and a set of splicing rules-each encoding a splicing operation-, as their computational engine to iteratively generate new strings starting from the axiom set. Since finite splicing systems (splicing systems with a finite axiom set and a finite set of splicing rules) generate a subclass of the family of regular languages, descriptional complexity questions about splicing systems can be answered in terms of the size of the minimal deterministic finite automata that recognize their languages. In this paper we focus on a particular type of splicing systems, called simple splicing systems, where the splicing rules are of a particular form. We prove a tight state complexity bound of 2 n − 1 for (semi-)simple splicing systems with a regular initial language with state complexity n ≥ 3. We also show that the state complexity of a (semi-)simple splicing system with a finite initial language is at most 2 n−2 + 1, and that whether this bound is reachable or not depends on the size of the alphabet and the number of splicing rules.

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

The neighbourhood of a language L with respect to an additive distance consists of all strings that have distance at most the given radius from some string of L. We show that the worst case deterministic state complexity of a radius r neighbourhood of a language recognized by an n state nondeterministic finite automaton A is (r + 2) n − 1. In the case where A is deterministic we get the same lower bound for the state complexity of the neighbourhood if we use an additive quasi-distance. The lower bound constructions use an alphabet of size linear in n. We show that the worst case state complexity of the set of strings that contain a substring within distance r from a string recognized by A is (r + 2) n−2 + 1.