Logic circuits from zero forcing.

Burgarth D, Giovannetti V, Hogben L, Severini S, Young M - Nat Comput (2015)

Related In: Results  -  Collection

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Fig4: The figure shows that an OR gate in which all vertices are initially white does not move the input forward
Mentions: The gadgets and have three and four vertices, respectively. By inspection on all possible combinations of white and black vertices for graphs with at most four vertices, we can observe that we have chosen the smallest possible gadgets, in terms of number of vertices and edges, realizing the two functions. One might think that the gate OR is realized also by the gadget with three vertices in Fig. 3. Although the gadget implements the OR correctly, it cannot be used as an initial or intermediate gate of a circuit, since in this gadget the color-change rule does not move forwards the output to the next gate, but it halts at vertex . See Fig. 4.

Bottom Line: Finally, we show that zero forcing can be also used to implement reversible computation.The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity.It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Physics, Aberystwyth University, Aberystwyth, SY23 3BZ UK.

ABSTRACT

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of "back forcing" as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we show that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity. Moreover, in the light of applications of zero forcing in quantum mechanics, the link with Boolean functions may suggest a new directions in quantum control theory and in the study of engineered quantum spin systems. It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits.

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