State Complexity of Pseudocatenation

Abstract

The state complexity of a regular language \(L_m\) is the number m of states in a minimal deterministic finite automaton (DFA) accepting \(L_m\). The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation, where the two operands range over all languages of state complexities \({\le }m\) and \({\le }n\), respectively. We consider the deterministic and nondeterministic state complexity of pseudocatenation. The pseudocatenation of two words x and y with respect to an antimorphic involution \(\theta \) is the set \(\{xy,x\theta (y)\}\). This operation was introduced in the context of DNA computing as the generator of pseudopowers of words (a pseudopower of a word u is a word in \(u \{u,\theta (u)\}^*\)). We prove that the state complexity of the pseudocatenation of languages \(L_m\) and \(L_n\), where \(m, n \ge 3\), is at most \((m-1)(2^{2n} - 2^{n+1} + 2) + 2^{2n-2} - 2^{n-1} + 1\). Moreover, for \(m, n \ge 3\) there exist languages \(L_m\) and \(L_n\) over an alphabet of size 4, whose pseudocatenation meets the upper bound. We also prove that the state complexity of the positive pseudocatenation closure of a regular language \(L_n\) has an upper bound of \(2^{2n-1} - 2^n +1\), and that this bound can be reached, with the witness being a language over an alphabet of size 4.