Binary decision diagrams One can specify a Boolean function of n variables by giving a finite automaton (and thus a network) in which paths exist only for those lists of values for which the function yields True .

So this may make one wonder what features will be emphasized in the future.

In typical voice coders (vocoders) 64k bits per second of digital data are obtained by sampling the original sound waveform 8000 times per second, and assigning one of 256 possible levels to each sample.

Mod[Binomial[t, n], 2] yields a distorted pattern that is the one produced by rule 60 (see page 58 ).

But following work by Simon Plouffe and others in 1995 it became clear that it is sometimes possible to generate, at least with overwhelming probability, the n th digit without explicitly finding previous ones.

Operator representations Instead of repeatedly applying an operation to a sequence of digits one can consider forming integers (or other numbers) by performing trees of operations on a single constant.

Structures [in rule 110] The persistent structures shown can be obtained from the following {n, w} by inserting the sequences IntegerDigits[n, 2, w] between repetitions of the background block b : {{152, 8}, {183, 8}, {18472955, 25}, {732, 10}, {129643, 18}, {0, 5}, {152, 13}, {39672, 21}, {619, 15}, {44, 7}, {334900605644, 39}, {8440, 15}, {248, 9}, {760, 11}, {38, 6}} The repetition periods and distances moved in each period for the structures are respectively {{4, -2}, {12, -6}, {12, -6}, {42, -14}, {42, -14}, {15, -4}, {15, -4}, {15, -4}, {15, -4}, {30, -8}, {92, -18}, {36, -4}, {7, 0}, {10, 2}, {3, 2}} Note that the periodicity of the background forces all rule 110 structures to have periods and distances given by {4, -2} r + {3, 2} s where r and s are non-negative integers. … Extended versions of (b) and (c) can be obtained from Flatten[{IntegerDigits[1468, 2], Table[ IntegerDigits[102524348, 2], {n}], IntegerDigits[v, 2]}] where n is a non-negative integer and v is one of {1784, 801016, 410097400, 13304, 6406392, 3280778648} Note that in most cases multiple copies of the same structure can travel next to each other, as seen on page 290 .

Compressing each block into a single cell, and n steps into one, any block cellular automaton with k colors and block size n can be translated directly into an ordinary cellular automaton with k n colors and range r = n/2 .

In addition, it is now possible to use software instead of hardware to implement SETI signal-processing algorithms—both traditional ones and presumably much more general ones that can for example pick out much weaker signals.

In 1948 Jan Łukasiewicz found the single axiom version {((a  (b  a))  (((( ¬ c)  (d  ( ¬ e)))  ((c  (d  f))  ((e  d)  (e  f))))  g))  (h  g)} equivalent for example to {(( ¬ a)  (b  ( ¬ c)))  ((a  (b  d))  ((c  b)  (c  d))), a  (b  a)} It turns out to be possible to convert any axiom system that works with modus ponens (and supports the properties of  ) into a so-called equational one that works with equivalences between expressions by using Module[{a}, Join[Thread[axioms  a  a], {((a  a)  b)  b, ((a  b)  b)  (b  a)  a}]] An analog of modus ponens for Nand is {x, x, ⊼ (y ⊼ z)}  z , and with this Jean Nicod found in 1917 the single axiom {(a ⊼ (b ⊼ c)) ⊼ ((e ⊼ (e ⊼ e)) ⊼ ((d ⊼ b) ⊼ ((a ⊼ d) ⊼ (a ⊼ d))))} which was highlighted in the 1925 edition of Principia Mathematica . In 1931 Mordechaj Wajsberg found the slightly simpler {(a ⊼ (b ⊼ c)) ⊼ (((d ⊼ c) ⊼ ((a ⊼ d) ⊼ (a ⊼ d))) ⊼ (a ⊼ (a ⊼ b)))} Such an axiom system can be converted to an equational one using Module[{a}, With[{t = a ⊼ (a ⊼ a), i = #1 ⊼ (#2 ⊼ #2) &}, Join[Thread[axioms  t], {i[t ⊼ (b ⊼ c), c]  t, i[t, b]  b, i[i[a, b], b]  i[i[b, a], a]}]]] but then involves 4 axioms.