First, the matrix expressions for each part of a transformer are described. Based on the semi-tensor product (STP) of matrices, the hypervectors are reconsidered, and a linear transformation over hypervectors is constructed using projection. The properties and calculation formulas are then obtained. The framework of a dimension-free transformer (DFT) is proposed using the projection-based transformation of hypervectors (PBTH) by verifying each linear transformation in a transformer and replacing it with a proper PBTH. This allows the inputs and outputs to be of arbitrary dimensions. Due to balanced information about all entries, the DFT is expected to be more efficient in processing signals.

In this paper, we propose a transition system (TS) framework for the analysis and design of logical networks. The formula for calculating the fixed points and cycles of a TS is first provided. Then, the TS representation of Boolean networks (BNs) and Boolean control networks (BCNs) is introduced. The first kind of representation is state-based, which converts a BCN into a TS with either distinguished controls or undistinguished ones. The second representation is output-based, which is also called the simulation of the original BCN. Finally, after introducing the notion of output simulation, our method is applied to design output robust controls of logical networks.