Deterministic soliton automata with at most one cycle

摘要

Soliton valves have been proposed as molecular switching elements. Their mathematical model is the soliton graph and the soliton automaton (Dassow and Jürgensen, J. Comput. System Sci.40 (1990), 154–181). In this paper we continue the study of the logic aspects of soliton switching. There are two cases of special importance: those of deterministic and those of strongly deterministic soliton automata. The former have deterministic state transitions in the usual sense of automaton theory. The latter do not only have deterministic state transitions, but also deterministic soliton paths—a much stronger property, as it turns out. In op cit. a characterization of indecomposable, strongly deterministic soliton automata was proved and it was shown that their transition monoids are primitive groups of permutations. Roughly speaking, the main difference between deterministic and strongly deterministic soliton automata is that in the former the underlying soliton graphs may contain cycles of odd lengths while such cycles are not permitted in the soliton graphs belonging to strongly deterministic soliton automata. In the present paper, we focus on a special class of deterministic soliton automata, that of deterministic soliton automata whose underlying graphs contain at most one cycle. For this class we derive structural descriptions. Our main results concern the elimination of certain types of loops, the treatment of soliton paths with repeated edges, the structure of cycles of odd length, and the transition monoid. As an application we show that the memory element proposed in the literature (Carter, in Bioelectronics, edited by Aizawa, Research and Development Report 50, CMC Press, Denver, CO, 1984) can be transformed in into a soliton tree, thus turning a deterministic device into a logically equivalent strongly deterministic device.