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The formal relationship between two differing approaches to the descriptionof spacetime as an intrinsically discrete mathematical structure, namely causalset theory and the Wolfram model, is studied, and it is demonstrated that thehypergraph rewriting approach of the Wolfram model can effectively beinterpreted as providing an underlying algorithmic dynamics for causal setevolution. We show how causal invariance of the hypergraph rewriting system canbe used to infer conformal invariance of the induced causal partial order, in amanner that is provably compatible with the measure-theoretic arguments ofBombelli, Henson and Sorkin. We then illustrate how many of the local dimensionestimation algorithms developed in the context of the Wolfram model may bereformulated as generalizations of the midpoint scaling estimator on causalsets, and are compatible with the generalized Myrheim-Meyer estimators, as wellas exploring how the presence of the underlying hypergraph structure yields asignificantly more robust technique for estimating spacelike distances whencompared against several standard distance and predistance estimator functionsin causal set theory. We finally demonstrate how the Benincasa-Dowker action oncausal sets can be recovered as a special case of the discrete Einstein-Hilbertaction over Wolfram model systems (with ergodicity assumptions in thehypergraph replaced by Poisson distribution assumptions in the causal set), andalso how both classical and quantum sequential growth dynamics can be recoveredas special cases of Wolfram model multiway evolution with an appropriate choiceof discrete measure.

Algorithmic Causal Sets and the Wolfram Model