8/3/2019 Miklos Kresz- Simulation of soliton circuits

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Simulation of soliton circuits

Miklos Kresz

Department of Computer Science, Juhasz Gyula Teacher Training College

University of Szeged, Szeged, Hungary

kresz@jgytf.u-szeged.hu

Soliton circuits are among the most promising alternatives for molecular electronicdevices based on the design of molecular level conventional digital circuits. In order tocapture the logical and computational aspects of these circuits, a mathematical modelcalled soliton automaton was introduced by Dassow and Jurgensen 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 thegraph to act as an automaton, it must have a perfect internal matching, which is amatching 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 anautomaton A(G), the states of which are the perfect internal matchings ofG. With aslight ambiguity, we shall also say that M is a state ofG, rather than M is a stateof A(G). Inputs to A(G) are pairs of external vertices ofG. In state M, a possibletransition on input (v1, v2) is carried out by switching along an alternating walk calledsoliton walk connecting v1 with v2. In that case the above transition is expressed byM (M, (v1, v2)), where M denotes the induced state and denotes the transitionfunction ofA(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 solitonautomata, we consider the following task.

Automaton Construction Problem (ACP): Given a soliton graphG. Constructthe automatonA(G) associated withG.

In order to give a solution for ACP, first we must design a method determining theset S(G) of states ofG, then an algorithm for constructing the transition function isneeded.

The first problem can be solved by adopting an extension of the method suggestedby Itai, Rodeh and Tanimoto for bipartite graphs with perfect matchings. It is assumedthat a state M ofG has been previously found, which can be achieved by a simple mod-ification of any known matching algorithm. Our algorithm uses the straightforwardobservation that a perfect internal matching is not unique iff it contains a so-calledalternating unit by which we mean either an even-length alternating cycle or an alter-nating path connecting two external vertices.

The borrowed idea is to define the procedure NEWSTATES (G,M,,L) for anynice subgraph G ofG i.e. a subgraph having a perfect internal matching such thatevery perfect internal matching ofG can be extended to a perfect internal matchingofG , perfect internal matching M ofG, M-alternating unit and perfect internalmatching L ofG \ V(G). It finds all the additional perfect internal matchings ofG.A state ofG is obtained by adding the set of edges L to a perfect internal matching ofG.

8/3/2019 Miklos Kresz- Simulation of soliton circuits

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Working out the technical details of the above method, after a careful complexityanalysis the following result is obtained.

Theorem 1. LetG be a soliton graph, m = |E(G)| and k = |S(G)|. ThenS(G) canbe constructed inO(k m) time.

Now we are left to provide a method constructing the transition function of A(G).The basic step solving this problem is to determine the set of input pairs (v, w) for anystates M1,M2 S(G) such that M2 (M1, (v, w).

First consider the case ofM1 = M2. In that situation we can capitalize the char-acterization of the structure of the symmetric difference N(M1,M2) of M1 and M2.Making use this result, our problem can be reduced to testing the accessibility of al-ternating cycles in N(M1,M2) by M1-alternating paths starting from v. Therefore,applying an efficient alternating path procedure, we obtain the following.

Theorem 2. LetM1 andM2 be distinct states ofA(G), m = |E(G)| and let l denotethe number of external vertices. Then the set of input pairs (v, w) for which M2 (M1, (v, w)) can be constructed inO(l m) time.

Having solved the problem of transitions between distinct states, now we turn toself-transitions, i.e. transitions from a state to itself. Self-transitions can be character-ized with the help of the so-called soliton trails, i.e. external alternating walks returningto themselves only in the last step. A trail is a c-trail (l-trail) if it closes up an even-length (respectively, odd-length) cycle. An M-alternating double soliton c-trail fromexternal vertex v is a pair ofM-alternating soliton c-trails (1, 2) from v such thatthe cycles of1 and 2 are either the same or disjoint.

Now for an arbitrary external vertex v and state M of G, construct the graphG[M, v] determined by the edges traversed by anM-alternating path or an M-alternating

soliton trail starting from v. Then the key point for our algorithm is the following result.

Theorem 3. For any state M of soliton automatonA(G) and for any external vertexv ofG, M (M, (v, v)) iff one of the following conditions holds:

(a) G[M, v] is a non-bipartite graph.

(b) G[M, v] is a bipartite graph containing an M-alternating double soliton c-trailfrom v.

(c) G[M, v] is a bipartite graph not containing anM-alternating even-length cycle.

Now making use of the above theorem we can give a method deciding for any stateM and external vertex v ofG ifM (M, (v, v)) holds. For this goal we need to designefficient procedures for constructing G[M, v] and for searching alternating cycles with

certain properties in G[M, v]. Using standard algorithmic techniques such as depth-first search and breadth-first search with respect to alternating paths, an algorithm isworked out with a complexity proportional to the number of vertices and edges.

Our closing result summarizes the preceding observations.

Theorem 4. LetG be a soliton graph withn = |V(G)|, m = |E(G)|, k = |S(G)| and ldenoting the number of external vertices. Then ACP can be solved inO((k+n)(klm))time.