And with the constraint of reversibility, it turns out that it is impossible to get a non-trivial phase transition in any 1D system with the kind of short-range interactions that exist in a cellular automaton.

Then the rules for the language consisting of balanced runs of parentheses (see page 939 ) can be written as {s[e]  s[e, e], s[e]  s["(", e, ")"], s[e]  s["(",")"]} Different expressions in the language can be obtained by applying different sequences of these rules, say using (this gives so-called leftmost derivations) Fold[# /. rules 〚 #2 〛 &, s[e], list] Given an expression, one can then use the following to find a list of rules that will generate it—if this exists: Parse[rules_, expr_] := Catch[Block[{t = {}}, NestWhile[ ReplaceList[#, MapIndexed[ReverseRule, rules]] &, {{expr, {}}}, (# /.

Games of chance based on wheels seem to have existed in Roman times; roulette developed in the 1700s.

And with the formulation of mechanics by Isaac Newton in 1687 space became increasingly viewed as something purely abstract, quite different in character from material objects which exist in it. … But in physics it was still assumed that space itself must have a standard fixed Euclidean form—and that everything in the universe must just exist in this space.

The idea that lichens might exist on Mars and be responsible for seasonal changes in color nevertheless became popular, especially after the discovery of atmospheric carbon dioxide in 1947.

Results in the late 1900s in astrophysics and cosmology seemed to suggest that for us to exist our universe must satisfy all sorts of constraints—and to avoid explaining this in terms of purpose the Anthropic Principle was introduced (see page 1026 ).

Continuum and cardinality Some notion of a distinction between continuous and discrete systems has existed since antiquity.

It is known that in principle there exist NP problems that are not in P, yet are not NP-complete.

A simple statement in predicate logic is ∀ x ( ∀ y x  y) ∨ ∀ x ( ∃ y ( ¬ x  y)) , where ∀ is "for all" and ∃ is "there exists" (defined in terms of ∀ on page 774 )—and this particular statement can be proved True from the axioms.

(The axiom of infinity, for example, was included to establish that an infinite set such as the integers exists.)