Data Structures and Algorithm Analysis - Computer Science at ...

558 Chap. 17 Limits to Computationx1234561 2 3 4 5f 1 (x) x f 2 (x) x f 3 (x) x f 4 (x)111111123456123456123456791113151712345615171327Figure 17.7 An illustration of assigning functions to bins.1 2 3 4 5x f 1 (x) x f 2 (x) x f 3 (x) x f 4 (x)x1 1 1 1 1 7 1 1512 1 2 2 2 9 2 123 1 3 3 3 11 3 734 1 4 4 4 13 4 1345 1 5 5 5 15 5 256 1 6 6 6 17 6 76f new (x)231214Figure 17.8 Illustration for the argument that the number of integer functions isuncountable.any bin, as follows. Take the output value for input 1 from the function in the firstbin. Call this value F 1 (1). Add 1 to it, and assign the result as the output of a newfunction for input value 1. Regardless of the remaining values assigned to our newfunction, it must be different from the first function in the table, because the twogive different outputs for input 1. Now take the output value for 2 from the secondfunction in the table (known as F 2 (2)). Add 1 to this value and assign it as theoutput for 2 in our new function. Thus, our new function must be different fromthe function of Bin 2, because they will differ at least at the second value. Continuein this manner, assigning F new (i) = F i (i) + 1 for all values i. Thus, the newfunction must be different from any function F i at least at position i. This procedurefor constructing a new function not already in the table is called diagonalization.Because the new function is different from every other function, it must not be inthe table. This is true no matter how we try to assign functions to bins, and so thenumber of integer functions is uncountable. The significance of this is that not allfunctions can possibly be assigned to programs, so there must be functions with nocorresponding program. Figure 17.8 illustrates this argument.