Mathematical Proof and the Reliability of DNA Evidence

Abstract

.Leonard Adleman recently found solution to a problem in graph theory using DNA technology. (See [Adleman].) However, once solution was obtained, Adleman checked that solution was correct by hand. As a result, DNA evidence served no justificatory function in this case. Perhaps this is fortunate since the mathematical establishment has often expressed its displeasure with certain types of 'proof' visual, mechanical, experimental, probabilistic [Davis 1995, p. 212]. At risk of displeasing mathematical establishment, I will suggest in this paper how DNA evidence might legitimately stand in for a mathematical proof. THE DIRECTED HAMILTONIAN PATH PROBLEM. The problem in graph theory that Adleman solved is an instance of directed Hamiltonian path problem (DHPP). A directed graph is a group of towns connected in various ways by one-way roads. One of towns is designated start town and another town is designated destination town. A path is a sequence of towns such that each town in sequence is connected by a one-way road to next town in sequence. A Hamiltonian path (HP) is a path from start town to destination town that visits every other town exactly one time. The DHPP asks if a given directed graph contains an HP and if so what HP is. The DHPP is an example of an NP-complete problem. A distinguishing feature of NP-complete problems is that while it is easy to check that a solution is correct, it is difficult to find a solution in first place. In case of DHPP, it is easy to check whether or not any particular path is an HP, but it is difficult to find a path that is an HP.