  Causality among events is widely recognized as a most fundamental structure
of spacetime, and causal sets have been proposed as discrete models of the
latter in the context of quantum gravity theories, notably in the Causal Set
Programme. In the rather different context of what might be called the
'Computational Universe Programme' -- one which associates the complexity of
physical phenomena to the emergent features of models such as cellular automata
-- a choice problem arises with respect to the variety of formal systems that,
in virtue of their computational universality (Turing-completeness), qualify as
equally good candidates for a computational, unified theory of physics. This
paper proposes Causal Sets as the only objects of physical significance and
relevance to be considered under the 'computational universe' perspective, and
as the appropriate abstraction for shielding the unessential details of the
many different computationally universal candidate models. At the same time, we
propose a fully deterministic, radical alternative to the probabilistic
techniques currently considered in the Causal Set Programme for growing
discrete spacetimes. We investigate a number of computation models by grouping
them into two broad classes, based on the support on which they operate; in one
case this is linear, like a tape or a string of symbols; in the other, it is a
two-dimensional grid or a planar graph. For each model we identify the
causality relation among computation events, implement it, and conduct a
possibly exhaustive exploration of the associated causal set space, while
examining quantitative and qualitative features such as dimensionality,
curvature, planarity, emergence of pseudo-randomness, causal set substructures
and particles.
