A Hacker's Revenge

~. R q(~,~) ql( ~, a) ~r g ! ] not a k. L ~i(~, a) ~, ~) p%(g, a, fi). The machine prints a fl at the end. cea(~3, a, fl, ~,). The machine copies down at the end first the symbols m a r k e d a, then those m a r k e d fl, and finally those m a r k e d y; it erases the symbols a, [3, y.

8. The expression " t h e r e is a generM process for determining ... " has been used t h r o u g h o u t this section as equivalent to " t h e r e is a machine which will determine ... ". This usage can be justified if and only if we can justify our definition of " c o m p u t a b l e ". g. G(n) m i g h t m e a n " n is s a t i s f a c t o r y " or " n is the Godel representation of a provable formula "], and this is e q u i v a l e n t to c o m p u t i n g a n u m b e r w h o s e n - t h figure is 1 if G (n) is true and 0 if it is false.

A machine will be circular if it reaches a configuration from which there is no possible move, or if it goes on moving, a:nd possibly printing symbols of the second kind, b u t cannot print any more symbols of the first kind. The significance of the term "circular" will be explained in w8. Computable sequences and numbers. A sequence is said to be computable if it can be c o m p u t e d by a circle-free machine. A n u m b e r is computable if it differs by an integer from the n u m b e r c o m p u t e d by a circle-free machine.

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