This paper proposes nonlinear operator of extreme doubly stochastic quadratic operator (EDSQO) for convergence algorithm aimed at solving consensus problem (CP) of discrete-time for multi-agent systems (MAS) on n-dimensional simplex. The first part undertakes systematic review of consensus problems. Convergence was generated via extreme doubly stochastic quadratic operators (EDSQOs) in the other part. However, this work was able to formulate convergence algorithms from doubly stochastic matrices, majorization theory, graph theory and stochastic analysis. We develop two algorithms: 1) the nonlinear algorithm of extreme doubly stochastic quadratic operator (NLAEDSQO) to generate all the convergent EDSQOs and 2) the nonlinear convergence algorithm (NLCA) of EDSQOs to investigate the optimal consensus for MAS. Experimental evaluation on convergent of EDSQOs yielded an optimal consensus for MAS. Comparative analysis with the convergence of EDSQOs and DeGroot model were carried out. The comparison was based on the complexity of operators, number of iterations to converge and the time required for convergences. This research proposed algorithm on convergence which is faster than the DeGroot linear model.
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The paper considers molecular programming in the abstract Tile Assembly Model, aTAM. Using simple constructions, an interpreter for the full Combinatory Logic, CL, is formally defined in aTAM. It provides an approach for sequential programming in aTAM, and produces a DNA molecular machine. This machine lives in a suitable solution and when receives a seed that linearly encodes a CL program and an input for the program, produces a grid which encodes a computation of the program on its input. The paper considers the construction cost and some alternative approaches. Finally, as a case study in distributed programming in aTAM, the paper considers the consensus problem and shows how an aTAM program for it can be formally derived by using π-calculus.
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