State Complexity of Overlap Assembly

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 find a tight upper bound on the state complexity of the binary operation overlap assembly on regular languages. This operation was introduced by Csuhaj-Varju, Petre, and Vaszil to model the process of self-assembly of two linear DNA strands into a longer DNA strand, provided that their ends “overlap”. We prove that the state complexity of the overlap assembly of languages \(L_m\) and \(L_n\), where \(m\ge 2\) and \(n\ge 1\), is at most \(2 (m-1) 3^{n-1} + 2^n\). Moreover, for \(m \ge 2\) and \(n \ge 3\) there exist languages \(L_m\) and \(L_n\) over an alphabet of size n whose overlap assembly meets the upper bound and this bound cannot be met with smaller alphabets.