When is a computer not a computer?

We recently bought an IBM-BLOKBUSTR computer, after reading Osherson’s (1985) beguiling description of its ability to do the uncomputable. We had hoped to use it in our study of the human ability to generate music. But we ran into problems that we had not anticipated. Readers of this journal who plan to use this rather unusual machine in their research, might want to profit from our experience. What makes BLOKBUSTR different from other computers is that it can do more than compute. For example, it can run a program, K-BAR, that outputs all and only the integers in the set that mathematicians call “K-bar”. K-bar is uncomputable. It consists of the indexes of Turing machines that do not halt when run on their own indexes. Since we can prove-mathematically-that this set cannot be generated by a computation, BLOKBUSTR can do something no ordinary computer can. In our lab, we feel that some human cognitive abilities require more than computing. The ability to generate and recognize good music, for example. We planned to use BLOKBUSTR’s ability to generate K-bar as a subroutine (or oracle) in a computer program that would simulate this human ability and generate music in the style of Mozart. Our aim was not to replace Mozart, who most people feel to be irreplaceable, but rather to try to characterize his style precisely. Like Stiny and Gips (1978), we feel that writing a program that generates works of art in a given style is one way to characterize that style precisely. Our plan to use K-bar was inspired by Myhill’s (1952) suggestion that enumerating a set of beautiful things might require the ability to generate a productive set and the fact that K-bar is such a set. Our choice of Mozart, rather than the more popular (among those who generate computer music) Bach, was inspired by the fact that musicians call their enumeration of Mozart’s work “K”. At least they refer to his works as “K459”, “K525” and the like. Since K-BAR enumerates the items not in the set that the mathematicians call “K”, we thought it might be apt to use it to try to enumerate the set of those of Mozart’s compositions not in the set the musicians