Simulation of Soliton Circuits

Soliton circuits are among the most promising alternatives for molecular electronic devices based on the design of molecular level conventional digital circuits. In order to capture the logical and computational aspects of these circuits, a mathematical model called soliton automaton was introduced by Dassow and Jürgensen in 1990. The underlying object of a soliton automaton is a so called soliton graph, which is a finite undirected graph allowed to have loops and multiple edges. In order for the graph to act as an automaton, it must have a perfect internal matching, which is a matching covering all vertices with degree at least 2. Such vertices are called internal , while external vertices are ones with degree 1. Let G be a soliton graph, fixed for our present discussion. The graph G defines an automaton A(G), the states of which are the perfect internal matchings of G. With a slight ambiguity, we shall also say that “M is a state of G”, rather than “M is a state of A(G)”. Inputs to A(G) are pairs of external vertices of G. In state M , a possible transition on input (v1, v2) is carried out by switching along an alternating walk – called soliton walk – connecting v1 with v2. In that case the above transition is expressed by M ′ ∈ δ(M, (v1, v2)), where M ′ denotes the induced state and δ denotes the transition function of A(G). From practical point of view it is a fundamental question to develop a simulation method for soliton circuits. Translating the above problem to the language of soliton automata, we consider the following task.