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Note (d) for The Content of the Principle

And in practice if there are many simple rules that are universal they cannot all be related by simple encodings.

Note (d) for Implications for Mathematics and Its Foundations

David Hilbert ): Mathematics studies formal rules that have no intrinsic meaning, but are relevant because of their applications or history. • Platonism (e.g.

Note (a) for How Do Simple Programs Behave?

(As discussed above, any cellular automaton rule can be represented as an appropriate combination of bitwise functions.)

Note (e) for Cyclic Tag Systems

History [of cyclic tag systems] Cyclic tag systems were studied by Matthew Cook in 1994 in connection with working on the rule 110 cellular automaton for this book.

Note (d) for Irreversibility and the Second Law of Thermodynamics

Periods in rule 37R With a system of size n , the maximum possible repetition period is 2 2n .

Note (e) for Implications for Mathematics and Its Foundations

With modus ponens as the rule of inference, the shortest single-axiom system that works is known to be {(a ∘ b) ∘ ((c ∘ b) ∘ (a ∘ c))} .

Note (a) for Implications for Mathematics and Its Foundations

Nand tautologies At each step every possible transformation rule in the axioms is applied wherever it can.

Note (b) for The Notion of Numbers

(Iterating the related function BitXor[i, 2i] yields numbers whose digit sequences correspond to the rule 60 cellular automaton).

Note (c) for The Sequencing of Events in the Universe

And, as it happens, in the mid-1980s I tried hard, though at the time without much success, to use updating rules for networks as the basis for a new kind of programming language intended for massively parallel computers.

Note (d) for Space as a Network

Implementation [of network properties] Networks are conveniently represented by assigning a number to each node, then having lists of rules which specify what nodes the connection from a particular node go to.