Applicability of feed-forward and recurrent neural networks to Boolean function complexity modeling

Understanding Boolean Function Learnability on Deep Neural Networks

“…The width of the neural network can be tradedoff by depth to alleviate the worst-case requirement for the number of neurons (Anthony 2010). Other neural networks have also been proven to universally implement boolean formulas, such as the binary pi-sigma network (Shin and Ghosh 1991), the binary product-unit network (Zhang, Yang, and Wu 2011) A body of empirical work followed the positive theoretical results on the learnability of boolean functions: (Miller 1999) shows that parity and multiplier functions are efficiently learnable, (Franco and Anthony 2004;Franco 2006;Franco and Anthony 2006) study complexity metrics that related to the the generalisation abilities of boolean functions implemented via neural networks, (Subirats et al 2006;Subirats, Jerez, and Franco 2008) and (Zhang, Ma, and Yang 2003) show algorithms for learning boolean circuits with thresholding neural networks, while (Prasad and Beg 2009) studies pre-processing techniques for using ANNs to learn boolean circuits and in (Beg, Prasad, and Beg 2008) they study approximating a boolean function's complexity using an ANN. (Pan and Srikumar 2016) showcases how neural networks with ReLU activation implement boolean functions much more compactly than with threshold linear units.…”

Input-State Approach to Boolean Networks

“…In Boolean network model, gene expression is quantized to only two levels: "T"(True) and "F"(False), or "1" and "0," respectively, denoted by , (or ). We refer to [7] for logical notations, concepts, and operators used in this paper, and refer to [2] for some related works in neural networks.…”

“…Its dynamic model is assumed to be (2) If the number of nodes in a Boolean network is , then it is obvious that the state space considers of statues, which is a finite set described as . So as a dynamic process on , there must be at least a fixed point or a cycle, and eventually a trajectory starting from any initial state must enter a cycle (a fixed point can be considered as a cycle of length one).…”