Notations [for logical primitives] Among those in current use are (highlighted ones are supported directly in Mathematica): The grouping of terms is normally inferred from precedence of operators (typically ordered  , ¬ , ⊼ , ∧ , ⊻ , ⊽ , ∨ ,  ), or explicitly indicated by parentheses, function brackets, or sometimes nested underbars or dots.

The plots below show the actual numbers of nodes reached as a function of r for the systems on pages 202 and 203 at steps 1, 10, 20, ..., 200.

[No text on this page] Examples of how images can be built up by adding together basic forms consisting of so-called two-dimensional Walsh functions.

Diagonalization arguments analogous to those on pages 1128 and 1162 show that in principle there must exist functions that can be evaluated only by computations that exceed any given bound. But actually finding examples of such functions that can readily be described as having some useful purpose has in the past seemed essentially impossible. … And with DNF Boolean expressions (see page 1096 ) functions like Xor are known to require exponentially many terms, even—as discovered in the 1980s—if any limited number of levels are allowed (see page 1096 ).

Symbolic systems [and operator systems] By introducing constants (0-argument operators) and interpreting ∘ as function application one can turn any symbolic system such as ℯ [x][y]  x[x[y]] from page 103 into an algebraic system such as ( ℯ ∘ a) ∘ b  a ∘ (a ∘ b) .

Generating causal networks If every element generated in the evolution of a generalized substitution system is assigned a unique number, then events can be represented for example by {4, 5}  {11, 12, 13} —and from a list of such events a causal network can be built up using With[{u = Map[First, list]}, MapIndexed[Function[ {e, i}, First[i]  Map[(If[# === {}, ∞ , # 〚 1, 1 〛 ] &)[ Position[u, #]]) &, Last[e]]], list]]

With these axioms one can prove results about real polynomials, but not about arbitrary mathematical functions, or integers. … This is now in practice done by Simplify and other functions in Mathematica using methods of cylindrical algebraic decomposition invented in the 1970s—which work roughly by finding a succession of points of change using Resultant .

The rules for the multiway system can then be given for example as {"AAB"  "BB", "BA"  "ABB"} The evolution of the system is given by the functions MWStep[rule_List, slist_List] := Union[Flatten[ Map[Function[s, Map[MWStep1[#, s] &, rule]], slist]]] MWStep1[p_String  q_String, s_String] := Map[StringReplacePart[s, q, #] &, StringPosition[s, p]] MWEvolveList[rule_, init_List, t_Integer] := NestList[MWStep[rule, #] &, init, t] An alternative approach uses lists instead of strings, and in effect works by tracing the internal steps that Mathematica goes through in trying out possible matchings.

But my guess is that its most important function is quite mundane: just as muscles build up lactic acid waste products, so also I suspect synapses in the brain build up waste products, and these can only safely be cleared out when the brain is not in normal use.

Probabilistic rules [for cellular automata] There appears to be a discrete transition as a function of the size of the perturbations, similar to phase transitions seen in the phenomenon of directed percolation.