Prefix-like Complexities of Finite and Infinite Sequences on Generalized Turing Machines.

AbstractGeneralized Turing machines (GTMs) are a variant of non-halting Turing machines, bycomputational power similar to machines with the oracle for the halting problem. GTMs allowa definition of a kind of descriptive (Kolmogorov) complexity that is uniform for finite andinfinite sequences. There are several natural modifications of the definition (as there are sev-eral monotone complexities). This paper studies these definitions and compares complexitiesdefined with the help of GTMs and complexities defined with the help of oracle machines.Keywords: Kolmogorov complexity, generalized Turing machine, infinite sequence, non-halting computation. 1 Introduction Recently, Schmidhuber [7, 6] introduced several modifications of Turing machines, in particular,generalized Turing machines (GTMs), and defined appropriate complexities. GTMs are machinesthat never halt and are allowed to change their output with the only requirement being that eachbit eventually stabilizes. GTMs are a natural model for computability in the limit, and thus GTMsare closely related to Turing machines with the oracle for the halting problem. These two modelsare equivalent in the case of computing functions on naturals. However, the situation is different inthe case of functions on binary sequences with additional requirements such as the self-delimitingproperty. Functions of this kind are used in algorithmic information theory for defining prefix andmonotone complexity (see [2, 1], and for comprehensive presentation and more references [3]).Originally, Kolmogorov defined the complexity of an object as the minimal size of its descriptionwith respect to some effective specifying method (mode of description), i.e. mapping from the setof descriptions to the set of objects. For the specifying method, one can assume various computingdevices (as Turing machines, maybe, non-halting, supplied with an oracle, etc.), implementing thismethod. Every assumption leads to a variant of complexity. Restricting oneself to values definedup to a bounded additive term, one can speak about complexity with respect to a certain class ofmachines (containing a universal one).For a machine of a fixed computational power, one can deal with the machine input and theinput size in various ways. To give an example, let us consider an abstract (Turing) machinethat has one (infinite in one direction) input tape containing only zeros and ones, can read theinput tape bit by bit, and generates some object. There were studied at least three variants of‘description’ (actually, definitions of the machine input and its size) of a generated object withrespect to such a machine.