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The formal relationship between two differing approaches to the description of spacetime as an intrinsically discrete mathematical structure, namely causal set theory and the Wolfram model, is studied, and it is demonstrated that the hypergraph rewriting approach of the Wolfram model can effectively be interpreted as providing an underlying algorithmic dynamics for causal set evolution. We show how causal invariance of the hypergraph rewriting system can be used to infer conformal invariance of the induced causal partial order, in a manner that is provably compatible with the measure-theoretic arguments of Bombelli, Henson and Sorkin. We then illustrate how many of the local dimension estimation algorithms developed in the context of the Wolfram model may be reformulated as generalizations of the midpoint scaling estimator on causal sets, and are compatible with the generalized Myrheim-Meyer estimators, as well as exploring how the presence of the underlying hypergraph structure yields a significantly more robust technique for estimating spacelike distances when compared against several standard distance and predistance estimator functions in causal set theory. We finally demonstrate how the Benincasa-Dowker action on causal sets can be recovered as a special case of the discrete Einstein-Hilbert action over Wolfram model systems (with ergodicity assumptions in the hypergraph replaced by Poisson distribution assumptions in the causal set), and also how both classical and quantum sequential growth dynamics can be recovered as special cases of Wolfram model multiway evolution with an appropriate choice of discrete measure.