The following generates explicit lists of n -input Boolean functions requiring successively larger numbers of Nand operations: Map[FromDigits[#, 2] &, NestWhile[Append[#, Complement[Flatten[Table[Outer[1 - Times[##] &, # 〚 i 〛 , # 〚 -i 〛 , 1], {i, Length[#]}], 2], Flatten[#, 1]]] &, {1 - Transpose[IntegerDigits[Range[2 n ] - 1, 2, n]]}, Length[Flatten[#, 1]] < 2 2 n &], {2}] The results for 2-step cellular automaton evolution in the main text were found by a recursive procedure.

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, {}}}, (# /.

Rational numbers require only division (or solving linear equations), while algebraic numbers require solving polynomial equations.

But while it was immediately clear that most cellular automata do not have the kind of reversible underlying rules assumed in traditional statistical mechanics, it still seemed initially very surprising that their overall behavior could be so elaborate—and so far from the complete orderlessness one might expect on the basis of traditional ideas of entropy maximization.

On row (a) of page 415 the parameter a varies from 1.05 to 1.65, while on row (b) b varies from 0 to 6.

Showing only the arguments to f , the pictures below illustrate how the flat functions Xor and And are confluent, while the non-flat function Implies is not.

Some it will be possible to address just by fairly straightforward but organized computer experimentation, while others will benefit from varying levels of technical skill and knowledge from existing areas of science, mathematics or elsewhere. … For while in the end it may be possible to get to something simple and elegant, it often takes huge intellectual effort to see just how this can be done.

But while this works for QED, it is only adequate for QCD in situations where the effective coupling is small. … But while there is no problem in doing this at a formal mathematical level—and indeed the expressions one gets from Feynman diagrams can always be analytically continued in this way—what general correspondence there is for actual physical processes is far from clear.

And in 1918 Hermann Weyl suggested that this could happen through local variations of scale or "gauge" in space, while in the 1920s Theodor Kaluza and Oskar Klein suggested that it could be associated with a fifth spacetime dimension of invisibly small extent. … So while it remains impossible to work out all the consequences of string theories, it is conceivable that among the representations of such theories there might be ones in which matter can be viewed as just being associated with features of space.

Thus, for example, generalized elliptic theta functions represent solutions to arbitrary polynomial equations, while multivariate hypergeometric functions represent arbitrary conformal mappings.