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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally e ..."
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Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
Algorithms: A quest for absolute definitions
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTurin ..."
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y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the ChurchTuring thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.
Algorithmic Complexity and Stochastic Properties of Finite Binary Sequences
, 1999
"... This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resourcebounded complexity. We also consider a new type of complexity statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexit ..."
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This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resourcebounded complexity. We also consider a new type of complexity statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexity, A. N. Kolmogorov's leading motive was developing on its basis a mathematical theory more adequately substantiating applications of probability theory, mathematical statistics and information theory. Kolmogorov wanted to deduce properties of a random object from its complexity characteristics without use of the notion of probability. In the first part of this paper we present several results in this direction. Though the subsequent development of algorithmic complexity and randomness was different, algorithmic complexity has successful applications in a traditional probabilistic framework. In the second part of the paper we consider applications to the estimation of parameters and the definition of Bernoulli sequences. All considerations have finite combinatorial character. 1.
Kolmogorov Complexity: Sources, Theory and Applications
 The Computer Journal
, 1999
"... ing applications based on different ways of approximating Kolmogorov complexity. 2. BEGINNINGS As we have already mentioned, the two main originators of the theory of Kolmogorov complexity were Ray Solomonoff (born 1926) and Andrei Nikolaevich Kolmogorov (1903 1987). The motivations behind their ..."
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ing applications based on different ways of approximating Kolmogorov complexity. 2. BEGINNINGS As we have already mentioned, the two main originators of the theory of Kolmogorov complexity were Ray Solomonoff (born 1926) and Andrei Nikolaevich Kolmogorov (1903 1987). The motivations behind their work were completely different; Solomonoff was interested in inductive inference and artificial intelligence and Kolmogorov was interested in the foundations of probability theory and, also, of information theory. They arrived, nevertheless, at the same mathematical notion, which is now known as Kolmogorov complexity. In 1964 Solomonoff published his model of inductive inference. He argued that any inference problem can be presented as a problem of extrapolating a very long sequence of binary symbols; `given a very long sequence, represented by T , what is the probability that it will be followed by a ... sequence A?'. Solomonoff assumed
In Some Curved Spaces, One Can Solve NPHard Problems in Polynomial Time
"... In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved s ..."
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In the late 1970s and the early 1980s, Yuri Matiyasevich actively used his knowledge of engineering and physical phenomena to come up with parallelized schemes for solving NPhard problems in polynomial time. In this paper, we describe one such scheme in which we use parallel computation in curved spaces. 1 Introduction and Formulation of the Problem Many practical problems are NPhard. It is well known that many important practical problems are NPhard; see, e.g., [11, 14, 27]. Under the usual hypothesis that P̸=NP, NPhardness has the following intuitive meaning: every algorithm which solves all instances of the corresponding problem requires, for
Kolmogorov Complexity Conditional to Large Integers
 Theoretical Computer Science
"... this paper the general notion of an algorithmic problem (see [7] for such discussion), as our paper is devoted to very specic problems. The plain Kolmogorov complexity, K(x), is the Kolmogorov complexity of the problem \print x". Likewise the conditional Kolmogorov complexity, dened as K(xjy) ..."
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this paper the general notion of an algorithmic problem (see [7] for such discussion), as our paper is devoted to very specic problems. The plain Kolmogorov complexity, K(x), is the Kolmogorov complexity of the problem \print x". Likewise the conditional Kolmogorov complexity, dened as K(xjy) = minfl(p) j p(y) = xg; is the complexity of the problem \given y print x"
Do stronger definitions of randomness exist?
, 2002
"... Abstract In this paper, we investigate refined definition of random sequences. Classical definitions (MartinL"of tests of randomness, uncompressibility, unpredictability, or stochasticness) make use of the notion of algorithm. We present alternative definitions based on set theory and expl ..."
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Abstract In this paper, we investigate refined definition of random sequences. Classical definitions (MartinL&quot;of tests of randomness, uncompressibility, unpredictability, or stochasticness) make use of the notion of algorithm. We present alternative definitions based on set theory and explain why they depend on the model of ZFC that is considered. We also present a possible generalisation of the definition when small infinite regularities are allowed.