January 2011
Intermediate to advanced
192 pages
5h 34m
English
Content preview from Computability Theory
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Programs and Machines 65
Because 8 is a register-machine computable partial function, it follows that for
any fixed e, the one-place function x 7→ 8(e, x) is register-machine computable (one
program for it puts the constant e into register 2 and then runs the universal program).
Call this function [[e]]:
[[e]](x) = 8(e, x).
That is, [[e]](x) is whatever this universal program produces (if anything), given the
input he, xi.
Then, if e is the G
¨
odel number of a program that computes a one-place partial
function f , we can conclude that [[e]] = f . And if e is some other number (either
the number of a program that does not compute a partial function or
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