In cases where the same real-world system can be modeled both by an ODE system ⅅ and a Boolean system 𝔹, it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that if 𝔹 has certain structures, consistency between ⅅ and 𝔹 is implied by sufficient separation of time scales in one class of our models. Namely, if the trajectories of 𝔹 are “one-stepping” then we prove a strong form of consistency and if 𝔹 has a certain monotonicity property then there is a weaker consistency between ⅅ and 𝔹. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.

Consider the metric space C consisting of the n-dimensional Boolean cube equipped with the Hamming distance. A weak isometry of C is a permutation of C preserving a given subset of Hamming distances. In [3] Krasin showed that in most cases preserving a single Hamming distance forces a weak isometry to be an isometry. In this article we study those weak isometries that are not automatically an isometry, providing a complete classification of weak isometries of C.

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