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Gradient descent [in constraint satisfaction]
A standard method for finding a minimum in a smooth function f[x] is to use
FixedPoint[# - a f'[#] &, x 0 ]
If there are local minima, then which one is reached will depend on the starting point x 0 . … Newton's method for finding zeros of f[x] is related and is given by
FixedPoint[# - f[#]/f'[#] &, x 0 ]

Note that the detailed pattern of evolution—though not any final fixed point reached—can depend on the fact that the combinator rules are applied at each step in Mathematica /. order.

But inevitably functions like FixedPoint , ReplaceRepeated and FullSimplify can run into undecidability—so that ultimately they have to be limited by constructs such as $IterationLimit and TimeConstraint .

Motion at a fixed speed is then the simplest case—and the one emphasized in the so-called special theory of relativity. … And the point is that in fact one can interpret slices at different angles as corresponding to motion at different fixed speeds.
… But
Graphical representation in space and time of motion at fixed speeds.

Long halting times [in symbolic systems]
Symbolic systems with rules of the form ℯ [x_][y_] Nest[x, y, r] always evolve to fixed points—though with initial conditions of size n this can take of order Nest[r # &, 0, n] steps (see above ). In general there will be symbolic systems where the number of steps to evolve to a fixed point grows arbitrarily rapidly with n (see page 1145 ), and indeed I suspect that there are even systems with quite simple rules where proving that a fixed point is always reached in a finite number of steps is beyond, for example, the axiom system for arithmetic (see page 1163 ).

And the point is that they do this regardless of how fast the source of an effect may be moving.
… But more generally one can think of following criss-crossing connections that exist in some fixed fragment of a causal network.
… The dark gray region in each picture represents a fixed amount of time for the clock—corresponding to a fixed number of sweeps of the black line.

The digits of 1/n in base b repeat with period
MultiplicativeOrder[b, FixedPoint[#/GCD[#, b] &, n]]
which is equal to MultiplicativeOrder[b, n] for prime n , and is at most n - 1 .

One can have a rule be applied only once using
Module[{i = 1}, expr /. lhs rhs /; i++ 1]
Many symbolic systems (including the one on page 103 ) have the so-called Church–Rosser property (see page 1036 ) which implies that if a fixed point is reached in the evolution of the system, this fixed point will be the same regardless of the order in which rules are applied.

If the evolution of a combinator expression reaches a fixed point, then the expression generated is always the same (Church–Rosser property). … The largest fixed point has size 41 (case (d)). s[s[s]][s][s][s][s] (case (e)) and s[s][s][s[s]][s][s] lead to expressions that grow like 2 t/2 . … At size 8, out of all 109,824 combinator expressions it appears that 49 show exponential growth, and many more show roughly linear growth. s[s][k][s[s[s]][s]][s] goes to a fixed point of size 80. s[s[s]][s][s][s][s[k]] (case (i)) increases rapidly to size 7050 but then repeats with period 3. s[s[s[s][s]]][s][s][k] (case (j)) grows to a maximum size of 1263, but then after 98 steps evolves to a fixed point of size 17.

Properties [of example symbolic system]
All initial conditions eventually evolve to expressions of the form Nest[ ℯ , ℯ , m] , which then remain fixed. … During the evolution the rule can apply only to the inner part FixedPoint[Replace[#, ℯ [x_] x] &, expr] of an expression. … It reaches a fixed point as soon as the depth reaches 0.