A large class of the dynamical laws for causal sets described by a classical process of sequential growth yield a cyclic universe, whose cycles of expansion and contraction are punctuated by single 'origin elements' of the causal set. We present evidence that the effective dynamics of the immediate future of one of these origin elements, within the context of the sequential growth dynamics, yields an initial period of de Sitter like exponential expansion, and argue that the resulting picture

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... many attractive features as a model of the early universe, with the potential to solve some of the standard model puzzles without any fine tuning.