### Superbranching degrees

- Proceedings Oberwolfach 1989, Springer Verlag Lecture Notes in Mathematics
, 1990

"... Solovay ..."

### Chaitin Ω numbers and halting problems

, 2009

"... ... 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 ..."

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... 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 problem of the optimal computer for all binary inputs of length at most n. In the present paper we investigate this property from various aspects. We consider the relative computational power between the base-two expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems. It is known that the base-two expansion of Ω and the halting problem are Turing equivalent. We thus consider an elaboration of the Turing equivalence in a certain manner.

### UNIVERSALITY PROBABILITY OF A PREFIX-FREE MACHINE

"... Abstract. We study the notion of universality probability of a universal prefix-free machine, as introduced by C.S. Wallace (see [Dow08, Section 0.2.2] and [Dow11, Section 2.5]). We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its pl ..."

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Abstract. We study the notion of universality probability of a universal prefix-free machine, as introduced by C.S. Wallace (see [Dow08, Section 0.2.2] and [Dow11, Section 2.5]). We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers which are universality probabilities of universal prefix-free machines. 1.

### A New Representation of Chaitin Ω Number Based on Compressible Strings

, 2010

"... In 1975 Chaitin introduced his Ω number as a concrete example of random real. The real Ω is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be rando ..."

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In 1975 Chaitin introduced his Ω number as a concrete example of random real. The real Ω is defined based on the set of all halting inputs for an optimal prefix-free machine U, which 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 problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation Θ of Chaitin Ω number. The real Θ is defined based on the set of all compressible strings. We investigate the properties of Θ and show that Θ is random. In addition, we generalize Θ to two directions Θ(T) and Θ(T) with real T>0. We then study their properties. In particular, we show that the computability of the real Θ(T) gives a sufficient condition for a real T ∈ (0, 1) to be a fixed point on partial randomness, i.e., to satisfy the condition that the compression rate of T equals to T.

### Partial Randomness and Dimension of Recursively Enumerable Reals

, 906

"... Abstract. A real α is called recursively enumerable (“r.e. ” for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; program-size complexity, Mart ..."

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Abstract. A real α is called recursively enumerable (“r.e. ” for short) if there exists a computable, increasing sequence of rationals which converges to α. It is known that the randomness of an r.e. real α can be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin Ω number, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rationals which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real T ∈ (0, 1], where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. As a result, we present ten equivalent characterizations of the partial randomness of an r.e. real. The resultant characterizations of partial randomness are powerful and have many important applications. One of them is to present equivalent characterizations of the dimension of an individual r.e. real. The equivalence between the notion of Hausdorff dimension and compression rate by program-size complexity (or partial randomness) has been established at present by a series of works of many researchers over the last two decades. We present ten equivalent characterizations of the dimension of an individual r.e. real. Key words: algorithmic randomness, recursively enumerable real, partial

### unknown title

, 2004

"... An extension of Chaitin’s halting probability Ω to measurement operator in infinite dimensional quantum system ..."

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An extension of Chaitin’s halting probability Ω to measurement operator in infinite dimensional quantum system

### Equivalent characterizations of partial randomness for a recursively enumerable real

, 805

"... Abstract. A real number α is called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to α. The randomness of a recursively enumerable real α can be characterized in various ways using each of the notions; program-size complexity, Martin-Löf ..."

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Abstract. A real number α is called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to α. The randomness of a recursively enumerable real α can be characterized in various ways using each of the notions; program-size complexity, Martin-Löf test, Chaitin’s Ω number, the domination and Ω-likeness of α, the universality of a computable, increasing sequence of rational numbers which converges to α, and universal probability. In this paper, we generalize these characterizations of randomness over the notion of partial randomness by parameterizing each of the notions above by a real number T ∈ (0,1]. We thus present several equivalent characterizations of partial randomness for a recursively enumerable real number. Key words: algorithmic randomness, recursively enumerable real number, partial randomness, Chaitin’s Ω number, program-size complexity, universal probability 1