By Rey Pastor, Julio; Pi Calleja, Pedro; Trejo, Cesar A.
Read or Download Analisis matematico. Vol. III: Analisis funcional y aplicaciones. (Spanish) Mathematical analysis. Vol. III: Functional analysis and applications PDF
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Additional info for Analisis matematico. Vol. III: Analisis funcional y aplicaciones. (Spanish) Mathematical analysis. Vol. III: Functional analysis and applications
Then whether f m (m) is or is not defined, we have a contradiction: Either f m (m) is undefined, in which case (2) tells us that it is defined and has value 1; or f m (m) is defined and has a value =1, in which case (2) tells us it has value 1; or f m (m) is defined and has value 1, in which case (2) tells us it has value 2. Since we have derived a contradiction from the assumption that d appears somewhere in the list f 1 , f 2 , . . , f m , . . , we may conclude that the supposition is false.
There is a Turing machine that does the same thing as the one in the preceding example, but with p · q in place of p + q. 3. A flow chart for such a machine is shown in Figure 3-6. Figure 3-6. Parity machine. If there were 0 or 2 or 4 or . . strokes to begin with, this machine halts in state 1, scanning a blank on a blank tape; if there were 1 or 3 or 5 or . . , it halts in state 5, scanning a stroke on an otherwise blank tape. 4. The object is to erase the leftmost stroke, fill the gap between the two blocks of strokes, and halt scanning the leftmost stroke that remains on the tape.
2, the set of all such functions is nonenumerable. And on the other hand, the set of Turing machines, and therefore of Turing-computable functions, is enumerable, since the representation of a Turing machine in the form of quadruples amounts to a representation of it by a finite string of symbols from a finite alphabet; and we have seen in Chapter 1 that the set of such strings is enumerable. These considerations show us that there must exist functions that are not Turing computable, but they do not provide an explicit example of such a function.
Analisis matematico. Vol. III: Analisis funcional y aplicaciones. (Spanish) Mathematical analysis. Vol. III: Functional analysis and applications by Rey Pastor, Julio; Pi Calleja, Pedro; Trejo, Cesar A.