On the generation of circuits and minimal forbidden sets

New methods for constructing shellable simplicial complexes

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...

A Minimal-Cost Inherent-Feedback Approach for Low-Power MRF-Based Logic Gates

The Markov random field (MRF) theory has been accepted as a highly effective framework for designing noise-tolerant nanometer digital VLSI circuits. In MRF-based design, proper feedback lines are used to control noise and keep the circuits in their valid states. However, this methodology has encountered two major problems that have limited the application of highly noise immune MRF-based circui...

Enumeration of Circuits and Minimal Forbidden Sets

Permitted and Forbidden Sets in Symmetric Threshold-Linear Networks

The richness and complexity of recurrent cortical circuits is an inexhaustible source of inspiration for thinking about high-level biological computation. In past theoretical studies, constraints on the synaptic connection patterns of threshold-linear networks were found that guaranteed bounded network dynamics, convergence to attractive fixed points, and multistability, all fundamental aspects...

Number of Minimal Path Sets in a Consecutive-k-out-of-n: F System

In this paper the combinatorial problem of determining the number of minimal path sets of a consecutive-k-out-of-n: F system is considered. For the cases where k = 2, 3 the explicit formulae are given and for k ≥ 4 a recursive relation is obtained. Direct computation for determining the number of minimal path sets of a consecutive-k-out-of-n: F system for k ≥ 4 remains a difficult task. ...