Decreasing Complexity in Inductive Computations

Abstract

Kolmogorov or algorithmic complexity has found applications in many areas including medicine, biology, neurophysiology, physics, economics, hardware and software engineering. Conventional Kolmogorov/algorithmic complexity and its modifications are based on application of conventional, i.e., recursive, algorithms, such as Turing machines. Inductive complexity studied in this paper is based on application of unconventional algorithms such as inductive Turing machines, which are super-recursive as they can compute much more than recursive algorithm can. It is possible to apply inductive complexity in all cases where Kolmogorov complexity is used. In particular, inductive complexity has been used in the study of mathematical problem complexity. The main goal of this work is to show how inductive algorithms can reduce complexity of programs and problems. In Sect. 8.2, we build the constructive hierarchy of inductive Turing machines and study the corresponding hierarchy of inductively computable functions. Inductive Turing machines from the constructive hierarchy are very powerful because they can build (compute) the whole arithmetical hierarchy. In Sect. 8.3, it is proved that inductive algorithms from the constructive hierarchy can essentially reduce complexity of programs and problems and the more powerful inductive algorithms are utilized the larger reduction of complexity is achievable.