Even though standard mathematical functions are used, few of the pictures can readily be generalized to continuous values of x and y .

Implementation [of operators from axioms] Given an axiom system in the form {f[a, f[a, a]]  a, f[a, b]  f[b, a]} one can find rule numbers for the operators f[x, y] with k values for each variable that are consistent with the axiom system by using Module[{c, v}, c = Apply[Function, {v = Union[Level[axioms, {-1}]], Apply[And, axioms]}]; Select[Range[0, k k 2 - 1], With[{u = IntegerDigits[#, k, k 2 ]}, Block[{f}, f[x_, y_] := u 〚 -1 - k x - y 〛 ; Array[c, Table[k, {Length[v]}], 0, And]]] &]] For k = 4 this involves checking nearly 16 4 or 4 billion cases, though many of these can often be avoided, for example by using analogs of the so-called Davis–Putnam rules.

And from this a statement s in Peano arithmetic (with each variable explicitly quantified) can be translated to a statement in set theory by using Replace[s, { ∀ a_ b_  ∀ a (a ∈   b), ∃ a_ b_  ∃ a (a ∈  ∧ b)}, {0, ∞ }] and then adding the statements below to provide definitions (  is the set of non-negative integers, 〈 x, y, z 〉 is an ordered triple, and ↕ a determines whether each triple in a set a is of the form 〈 x, y, f[x, y] 〉 ; specifying a single-valued function).

Implementation [of constraint satisfaction] The number of squares violating the constraint used here is given by Cost[list_] := Apply[Plus, Abs[list - RotateLeft[list]]] When applied to all possible patterns, this function yields a distribution with Gaussian tails, but with a sharp point in the middle.

Since the late 1970s, it has been common to assume that the response of a cell can be modelled by derivatives of Gaussians such as those shown below, or perhaps by Gabor functions given by products of trigonometric functions and Gaussians.

Gödel in effect does this by first converting the statement to one about recursive functions and then—by using tricks of number theory such as the beta function of page 1120 —to one purely about arithmetic.

Despite their basic setup the systems discussed here are not direct analogs of standard quantum spin systems, since these normally have local Hamiltonians and non-local evolution functions, while the systems here have local evolution functions but seem always to require non-local Hamiltonians.)

What happens is that the function that determines the color of a particular cell from the colors of cells in a nearby column rapidly becomes extremely Patterns generated by rule 30 after averaging over all possible initial conditions that reproduce the arrangements of colors in the cells indicated by dots.

And as a potential example, consider setting up a quantum computer that evaluates a given Boolean function—with its initial configurations of spins encoding possible inputs to the function, and the final configuration of a particular spin representing the output from the function. One might imagine that with such a computer it would be easy to solve the NP-complete problem of satisfiability from page 768 : one would just start off with a superposition in which all 2 n possible inputs have equal amplitude, then look at whether the spin representing the output from the function has any amplitude to be in a particular configuration. … And with the setup described, even if a particular function is ultimately satisfiable the probability for a single output spin to be measured say as up can be as little as 2 -n —requiring on average 2 n trials to distinguish from 0 , just as in the classical probabilistic case.

Note that looking up Mathematica functions used in connection with some issue is often a good way to identify related issues.