by Cristian S. Calude, Michael A. Stay – Inform. and Comput. 2006

“. We introduce the zeta number, natural halting probability and natural complexity in the Turing machine and then we relate individuals to Chaitin’s Omega number, halting probability, and program-size complexity. A classification of Turing machines based on their zeta figures is suggested: divergent, convergent. “

We introduce the zeta number, natural halting probability and natural complexity in the Turing machine and then we relate individuals to Chaitin’s Omega number, halting probability, and program-size complexity. A classification of Turing machines based on their zeta figures is suggested: divergent, convergent and tuatara. We prove the presence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proven. For instance, we show the zeta amount universal tuatara machine is c.e. and random. A totally new kind of partial randomness, asymptotic randomness, is introduced. Finally we show rather of classical (algorithmic) randomness—which cannot be naturally characterised in relation to plain complexity—asymptotic randomness admits this sort of characterisation. 1

by Cristian S. Calude A, Michael A. Stay B. 2006

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by C. S. Calude, M. A. Stay, Cristian S. Calude, Michael A. Stay, Cristian S. Calude. 2005

“. We introduce the zeta number, natural halting probability and natural complexity in the Turing machine and then we relate individuals to Chaitin’s Omega number, halting probability, and program-size complexity. A classification of Turing machines based on their zeta figures is suggested: divergent, convergent. “

We introduce the zeta number, natural halting probability and natural complexity in the Turing machine and then we relate individuals to Chaitin’s Omega number, halting probability, and program-size complexity. A classification of Turing machines based on their zeta figures is suggested: divergent, convergent and tuatara. We prove the presence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proven.

For instance, we show the zeta amount universal tuatara machine is c.e. and random. A totally new kind of partial randomness, asymptotic randomness, is introduced. Finally we show rather of classical (algorithmic) randomness—which cannot be naturally characterised in relation to plain complexity—asymptotic randomness admits this sort of characterisation.