Modeling Time-Bounded Prefix Kolmogorov Complexity

Abstract.

In the literature, prefix Kolmogorov complexity is defined either in terms of self-delimiting Turing machines or in terms of partial recursive prefix functions. These notions of prefix Kolmogorov complexity are equivalent because, as Chaitin showed, every partial recursive prefix function can be simulated by a self-delimiting Turing machine. However, the simulation given by Chaitin's construction is not efficient, and so questions regarding the time-bounded equivalence of these notions remained unresolved. Here we closely examine these questions.

As our main result, we show that every partial recursive prefix function can be simulated with polynomial efficiency by a self-delimiting Turing machine if and only if P = NP. Thus, it is unlikely that Chaitin's construction can be used to show the polynomial-time equivalence of these notions of prefix Kolmogorov complexity. Here we further examine the relationships between these notions of time-bounded prefix Kolmogorov complexity.