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To disallow procedures say specially set up to pick out all the infinite number of 1's in a sequence Alonzo Church in 1940 suggested that only procedures corresponding to finite computations be considered.
For ordinary multiway (semi-Thue) systems, an example with an undecidable word problem is known with 2 types of elements and 5 very complicated rules—but I am quite certain that much simpler examples are possible. (1-rule multiway systems always have decidable word problems.)
The last axiom is a schema (see page 1156 ) that states the principle of mathematical induction: that if a statement is valid for a = 0 , and its validity for a = b implies its validity for a = b + 1 , then it follows that the statement must be valid for all a .
The icosahedral group A 5 defined by the rules x 2  y 3  (x y) 5  1 has 60 elements.
Still another related approach is to consider coloring the edges of a network: if there are d + 1 possible colors, all of which appear at every node, then it follows that d coordinates can consistently be assigned to each node.)
Such patterns were discussed by Euclid and later Leonardo da Vinci in connection with the theory of lunes. 1 st century BC (Celtic). … An example of 1D ornamental patterns are molding profiles.
Surgically modifying such buds when they are as small as 0.1 mm can have dramatic effects on final leaf shape, suggesting that at least some aspects of the shape are already determined at that point.
The memory consists of an array of cells, with the presence or absence of a lump of charge at gates in each cell representing a 1 or 0 value for the bit of data associated with that cell.
The iterated map x  4x (1 - x) was also known to have a similar property (see page 918 ).
(The kind of complexity discussed here has nothing directly to do with complex numbers such as √ -1 introduced into mathematics since the 1600s.)
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