In this work, we consider adversarial crash faults of nodes in the network constructors model [Michail and Spirakis, 2016]. We first show that, without further assumptions, the class of graph languages that can be (stably) constructed under crash faults is nonempty but small. In particular, if an unbounded number of crash faults may occur, we prove that (i) the only constructible graph language is that of spanning cliques and (ii) a strong impossibility result holds even if the size of the graphs that the protocol outputs in populations of size n need only grow with n (the remaining nodes being waste). When there is a finite upper bound f on the number of faults, we show that it is impossible to construct any non-hereditary graph language and leave as an interesting open problem the hereditary case. On the positive side, by relaxing our requirements we prove that: (i) permitting linear waste enables to construct on

*n/(2f) − f* nodes, any graph language that is constructible in the fault-free case, (ii) partial constructibility (i.e., not having to generate all graphs in the language) allows the construction of a large class of graph languages. We then extend the original model with a minimal form of fault notifications. Our main result here is a fault-tolerant universal constructor : We develop a fault-tolerant protocol for spanning line and use it to simulate a linear-space Turing Machine M. This allows a fault-tolerant construction of any graph accepted by M in linear space, with waste

*min{n/2 + f(n), n}*, where f(n) is the number of faults in the execution. We then prove that increasing the permissible waste to

*min{2n/3+f(n), n}* allows the construction of graphs accepted by an

*O(n^2)*-space Turing Machine, which is asymptotically the maximum simulation space that we can hope for in this model. Finally, we show that logarithmic local memories can be exploited for a no-waste faulttolerant simulation of any such protocol.
Cite paper:

@inproceedings{michail2019fault,
title={Fault tolerant network constructors},
author={Michail, Othon and Spirakis, Paul G and Theofilatos, Michail},
booktitle={International Symposium on Stabilizing, Safety, and Security of Distributed Systems},
pages={243--255},
year={2019},
organization={Springer}
}