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Natural halting probabilities, partial randomness, and zeta functions
 Inform. and Comput
, 2006
"... We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and programsize complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent ..."
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Cited by 17 (8 self)
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We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and programsize complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness—which cannot be naturally characterised in terms of plain complexity—asymptotic randomness admits such a characterisation. 1
Most programs stop quickly or never halt
 Adv. Appl. Math
"... The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori compu ..."
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Cited by 9 (3 self)
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The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k> 0, we can effectively compute a time bound T such that the probability that an Nbit program will eventually halt given that it has not halted by T is smaller than 2 −k. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long ” runtimes are effectively rare. More formally, the set of times at which an Nbit program can stop after the time 2 N+constant has effectively zero density. 1
References
, 2008
"... Chaitin’s “heuristic principle”, the theorems of a finitelyspecified theory cannot be significantly more complex than the theory itself was proved for an appropriate measure of complexity in [1]. The measure δ is a computable variation of the programsize complexity H: δ(x) = H(x) − x. The theo ..."
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Chaitin’s “heuristic principle”, the theorems of a finitelyspecified theory cannot be significantly more complex than the theory itself was proved for an appropriate measure of complexity in [1]. The measure δ is a computable variation of the programsize complexity H: δ(x) = H(x) − x. The theorems of a finitelyspecified, sound, consistent theory which is strong enough to include arithmetic have bounded δcomplexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. More precisely, according to Theorem 4.6 in [1], for any finitelyspecified, sound, consistent theory strong enough to formalize arithmetic (like ZermeloFraenkel set theory with choice or Peano Arithmetic) and for any Gödel numbering g of its wellformed formulae, we can compute a bound N such that no sentence x with complexity δg(x)> N can be proved in the theory; this phenomenon is independent on the choice of the Gödel numbering. Question 1. Find other natural measures of complexity for which Chaitin’s “heuristic principle ” holds true.
AIT with Natural Complexity
, 2009
"... The project consists in developing a large part of AIT using the natural complexity ∇ [2, 3] instead of the prefix complexity H [4, 1]. All strings are binary and the set of strings is denoted by Σ ∗. The length of x is denoted by x. The logarithms are binary too. Let N = {1, 2,...} and let bin: N ..."
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The project consists in developing a large part of AIT using the natural complexity ∇ [2, 3] instead of the prefix complexity H [4, 1]. All strings are binary and the set of strings is denoted by Σ ∗. The length of x is denoted by x. The logarithms are binary too. Let N = {1, 2,...} and let bin: N → Σ ∗ be the computable bijection which associates to every n ≥ 1 its binary expansion without the leading 1, n n2 bin(n) bin(n) 1 1 λ 0 2 10 0 1
Five Questions
"... 1. Why were you initially drawn to the study of computation and randomness? I’ve always been torn between studying physics and studying computer science, but it was the theoretical aspects of both that attracted me the most. There’s clearly overlap in the computational physics arena, but I found tha ..."
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1. Why were you initially drawn to the study of computation and randomness? I’ve always been torn between studying physics and studying computer science, but it was the theoretical aspects of both that attracted me the most. There’s clearly overlap in the computational physics arena, but I found that coming up with good numerical approximations to systems wasn’t really to my taste. It was only when I began studying information theory that I came to understand that there was also an enormous overlap between the two theoretical sides of the fields. And like many others, I was intrigued with the possibility that somehow algorithmic randomness lies at the root of quantum randomness. 2. What we have learned? One of the strange things about quantum physics is that we can describe matter, which to our senses often feels hard and unyielding, by waves! Most people get their intuition about waves from music. When a bassist plucks his strings, it’s hard to place the pitch precisely, because waves have a particularly interesting property: you can’t tell both the time a note was played and its pitch with an accuracy greater than about a quarter cycle. The low range of a bass is only tens of vibrations per second, so you only hear a couple of complete cycles during
Most programs stop quickly or never halt
, 2006
"... This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproductio ..."
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This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Most short programs halt quickly
, 2008
"... Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the pr ..."
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Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect nonhalting programs. For each program length on a given machine, there is an uncomputable “critical time ” after which no more programs of that length will halt. A quantum algorithm [7, 1] has been shown to solve the halting problem to any degree of certainty less than one and various experimental studies have proposed heuristics that apply to a majority of programs [4, 15]. Is it possible to have a classical effective way to describe this phenomenon? The aim of this paper is to provide a nonquantum analysis; our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by