Computational Complexity of Synthetic Chemistry { Basic Facts

| Three negative results: 1. Given a set of allowed starting materials and allowed chemical reactions, the problem of determining whether a chemical of speciied structure is synthesizable is Turing undecidable. 2. The question of whether it is synthesizable in N steps is NP-hard. These results apply in a general abstract setting and do not prove anything about how hard the problem is with any spe-ciic set of starting materials and allowed reactions. In particular we prove nothing about the case when the starting materials are all commercially available chemicals and the reactions are all known reactions! 3. The IUPAC naming scheme for assigning names to organic compounds will assign ambiguous names to most N-atom compounds and requires great chemical expertise. We show that both these problems are avoidable with alternative naming schemes, There are unquestionably far simpler and better methods to name chemical compounds than the IUPAC scheme. All 3 of these facts are almost trivial from the viewpoint of modern computer science, but I don't think they've been mentioned in the chemical literature. Positive results: We discuss algorithms for nd-ing out how to solve the synthesis problem (of determining whether something is synthesizable, and what the best synthetic process is). An interesting generalization of the famous \shortest path problem" in directed graphs arises in which the paths become synthetic trees, the graphs become hypergraphs, and the distances become synthetic costs. This problem is soluble either by generalizing Dijkstra's algorithm or the dynamic programming algorithm. Despite the fact that these algorithms are highly eecient as a function of the size of the hypergraph, in practice because the hypergraph arising from chemistry is enormous , that is inadequate. We discuss methods for ameliorating this. 1 Undecidability and NP-completeness of synthesizability problem In this section we propose two mathematical models of synthetic chemistry. Inside these two models, it turns out that the question of whether a given chemical is syn-thesizable from a given set of allowed starting materials with a given set of allowed reactions, is Turing undecid-able and NP-complete respectively. Note: neither of these two models addresses the spe-ciic problem of whether the chemicals currently commercially available, plus the reactions currently known to chemists (even assuming it were possible to say what these are!), lead to any undecidable problems. Instead we only prove that there exist undecidable (or NP-complete) instances of this general, abstract problem: The problem poser gives us …