CA notes

" 'Life—like all computationally universal systems—defines the most efficient simulation of its own behavior' (Ilachisnki 2001, p. 15). This raises the important philosophical questions of the limits of the predictability of any universe—possibly our actual universe—which is itself capable, just as Life is, of producing and hosting universal computers.

"The mathematical literature on CA does not refrain from describing the Life configurations using the same imaginative vocabulary we used, that is, in terms of items that are born, that live, move around, eat other figures, die, etc. The Life zoology, though, is in an important sense in the eye of the beholder: the universe these patterns inhabit can be more neutrally described as a collection of individual cells, each of which does not directly depend on what is happening on the macro-scale. Even more neutrally, Life can be described in the simple mathematical language of matrices and discrete sequences."

"Although the variety of systems to be found in the CA literature is vast, one can generate virtually all CA by tuning the four parameters that define their structure:

"a. Discrete n-dimensional lattice of cells: We can have one-dimensional, two-dimensional, … , n-dimensional CA. The atomic components of the lattice can be differently shaped: for example, a 2D lattice can be composed of triangles, squares, or hexagons. Usually homogeneity is assumed: all cells are qualitatively identical.

"b. Discrete states: At each discrete time step, each cell is in one and only one state, σ ∈ Σ, Σ being a set of states having finite cardinality |Σ| = k.

"c. Local interactions: Each cell's behavior depends only on what happens within its local neighborhood of cells (which may or may not include the cell itself). Lattices with the same basic topology may have different definitions of neighborhood, as we will see below. It is crucial, however, that “actions at a distance” not be allowed.

"d. Discrete dynamics: At each time step, each cell updates its current state according to a deterministic transition function φ: Σn → Σ mapping neighborhood configurations (n-tuples of states of Σ) to Σ. It is also usually, though not necessarily, assumed that (i) the update is synchronous, and (ii) φ takes as input at time step t the neighborhood states at the immediately previous time step t − 1."

On C. D. Broad: "Correspondingly, there are two types of laws: (1) ‘intra-ordinal’ laws, which relate events within an order, i.e., a law connecting an aggregate of that order instantiating a property of that order at a time with some aggregate of that order instantiating some other property at a certain time; and (2) ‘trans-ordinal’ laws, which characterize the emergence of higher-level properties from lower-level ones. Emergent properties are identified by the trans-ordinal laws that they figure in; each emergent property appears in the consequent of at least one trans-ordinal law, the antecedent of which is some lower-level property...Trans-ordinal laws are what we now call ‘emergent laws,’ fundamental, irreducible laws that describe a synchronic, noncausal covariation of an emergent property and its lower-level emergent base. Emergent laws are not metaphysically necessitated by any lower-level laws, boundary conditions and any lower-level compositional principles."

On C. D. Broad: "Here we see the unpredictability element of Emergentism that is often discussed. The idea is that even the ideal theorist — Broad's mathematical archangel — with complete knowledge of the lower-level aggregates and properties will be helpless at predicting what might emerge from a specific lower-level structure with certain properties prior to observing the actual instantiation of the complex, higher-level event. This unpredictability, however, is not constitutive of emergence, but rather a consequence of the metaphysical irreducibility of the emergent properties and the trans-ordinal laws they bring in their train."

"Crucial to an account of emergence, however, is a view concerning the relationship of such levels. On this score, we find that there are, in fact, two rather different pictures of emergence, one represented by Mill and Broad, and the other represented by Alexander. For Mill and Broad, emergence involves the appearance of primitive high-level causal interactions that are additional to those of the more fundamental levels. Alexander, by contrast, is committed only to the appearance of novel qualities and associated, high-level causal patterns which cannot be directly expressed in terms of the more fundamental entities and principles. But these patterns do not supplement, much less supersede, the fundamental interactions. Rather, they are macroscopic patterns running through those very microscopic interactions. Emergent qualities are something truly new under the sun, but the world's fundamental dynamics remain unchanged."

"When we turn to the contemporary scene, easily the more popular approach to emergence descends from Alexander, not Mill and Broad. Though details differ, representatives of this approach characterize the concept of emergence strictly in terms of limits on human knowledge of complex systems. Emergence for such theorists is fundamentally an epistemological, not metaphysical, category."