In this paper we study small hammock networks, more precisely the 29 hammock networks presented by Moore and Shannon in their prescient paper from 1956, where this type of network was introduced. We will first of all review the concept of the reliability polynomial, and define hammock networks, emphasizing some of their properties. We will then review the state-of-the-art from three different angles: theoretical, algorithmic, and design oriented. These will show that results obtained over many years advocate very strongly for small hammock networks. Because array-based designs (e.g., FinFETs) seem to be a perfect fit for small hammocks (as are nanoelectromecanical systems, or NEMS), we have decided to start investigating them closely. The analyses we pursue here are exact, the coefficients of the reliability polynomials being determined using our own program. The reliability polynomials for nontrivial hammock networks are reported here as functions of the probability (p) that the device is closed. As far as we know, this is the first time they have appeared in this form. Our purpose in determining exact results is a practical one, as we foresee that small hammock networks could play a key role in any arraybased design. That is why their reliability enhancements are evaluated thoroughly, using both classical and non-standard cost functions. Finally, conclusions are followed by a longer list of future directions of research, some of them practical while others more theoretical.

We show that a separable real Banach space embeds almost isometrically in a space Y with a shrinking 1-unconditional basis if and only if lim _n→∞‖ x* + x_n*‖ = lim _n→∞‖x* − x_n*‖ whenever x* ∈ X*, is a weak*-null sequence and both limits exist. If X is reflexive then Y can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.

We derive a formula for the reliability of a d-dimensional consecutive-k-out-of-n:F system, that is, a formula for the probability that an n1×⋯×nd array whose entries are (independently of each other) 0 with probability p and 1 with probability q=1-p does not include a contiguous s1×⋯×sd subarray whose every entry is 1.

Our own fresh results have suggested that the smallest size optimal sorting networks/graphs (which can be mapped onto hardware) might be the most appropriate entities (building blocks) for designing highly reliable computing systems. Sorting networks correspond to particular sorting algorithms, while their associated connectivity graphs can embody (minimal) two-terminal networks. Relying on these concepts one can associate a reliability polynomial to any sorting network. Here, we are going to extend on the work we have recently started on sorting networks by studying larger optimal sorting networks. Comparing the two-terminal reliability polynomials associated to these larger sorting connectivity graphs with those of Moore-Shannon hammocks of the same size will follow. In-depth comparisons will be done using both classical as well as novel figures-of-merit targeting the reliability enhancements of networks when performing computations. The results reported here will shed light on our previous findings, confirming that the optimal sorting network of four inputs is ideal for designing highly reliable computing systems, but also that its advantages, although of theoretical importance, are only marginal and fading quickly. In particular, the analyses of sorting networks with larger number of inputs reveals that hammocks are catching up with sorting networks at five and six inputs, and clearly overtaking sorting networks of seven inputs. All these simulations and detailed comparisons provide compelling arguments for why hammock networks are the ones we should rely upon (at more than one level) for 2D reliable computations (including the current quantum computing quest).