This article describes the insert sorting algorithm using macro instructions such as if-Macro (conditional branch macro instructions), for-loop macro instructions and While-Macro instructions etc. From the viewpoint of initialization, we generalize the halting and computing problem of the While-Macro. Generally speaking, it is difficult to judge whether the While-Macro is halting or not by way ...

In this paper we study the Kolmogorov complexity for non-effective computations, that is, either halting or non-halting computations on Turing machines. This complexity function is defined as the length of the shortest inputs that produce a desired output via a possibly non-halting computation. Clearly this function gives a lower bound of the classical Kolmogorov complexity. In particular, if t...

This article defines two for-loop statements for SCMPDS. One is called for-up, which corresponds to “for (i=x; i<0; i+=n) S” in C language. Another is called fordown, which corresponds to “for (i=x; i>0; i-=n) S”. Here, we do not present their unconditional halting (called parahalting) property, because we have not found that there exists a useful for-loop statement with unconditional halting, ...

Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach., vol. 22, pp. 329–340, 1975] introduced Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n...

The main topic of the present work is the relation that a set X is strongly hyperimmune-free (shif) relative to Y . Here X is shif-below Y if and only if for every partial X-recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p(a) < p(q). It is shown that between degrees not above the halting problem this ...

The true process that generated data cannot be determined when multiple explanations are possible. Prediction requires a model of the probability that a process, chosen randomly from the set of candidate explanations, generates some future observation. The best model includes all of the information contained in the minimal description of the data that is not contained in the data. It is closely...

We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory within the setting of formal mathematical theories. 2005 Elsevier Inc. All rights reserved.

Chaitin [G. 1975] introduced Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω solve the halting prob...

— In this note we prove the existence of effectively independent instances (with respect to an arbitrary recursively axiomatizable, consistent, intuitively true and sufficiently rich theory) for some well-known undecidable problems including the Emptiness Problem, the Finiteness Problem, the Totality Problem, the Halting Problem and the Post Correspondence Problem. Applications in the Theory of...