State Complexity of Overlap Assembly

TL;DR

This paper establishes the exact maximum state complexity for the overlap assembly operation on regular languages, revealing tight bounds and demonstrating their attainability with specific language constructions.

Contribution

It provides a tight upper bound on the state complexity of overlap assembly and constructs languages that meet this bound, advancing understanding of this DNA-inspired operation.

Findings

Maximum state complexity of overlap assembly is 2(m-1)3^{n-1}+2^n for m≥2, n≥1.

Existence of languages meeting the upper bound over an alphabet of size n.

Tight upper bound of m+n for unary languages and exponential complexity for binary languages.

Abstract

The \emph{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 \emph{overlap assembly} on regular languages. This operation was introduced by Csuhaj-Varj\'u, 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 $n\ge1$, 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. Finally, we prove that $m+n$ is a tight upper bound on the overlap assembly of unary languages, and that there are binary languages whose overlap assembly has exponential state complexity at least $m(2^{n-1}-2)+2$.