Sophistication Revisited

Antunes, Luís; Fortnow, Lance

doi:10.1007/s00224-007-9095-5

Cited by 26 publications

(23 citation statements)

References 12 publications

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“…Antunes and Fortnow [6] defined a variant of sophistication, called coarse sophistication, that gets rid of the significance level by incorporating it into the minimization.…”

Section: Definition 4 (Sophistication)mentioning

confidence: 99%

“…Once again, this definition is equivalent to the definition based on structural information, given in [6], up to a logarithmic additive term. We say that S is a witness to csoph(…”

Section: Definition 4 (Sophistication)mentioning

confidence: 99%

“…It is well known that sophistication and computational depth are related [4,6], as are computational depth and randomness deficiency [8]. Antunes and Fortnow [6] used busy beaver computational depth:…”

Section: Relation To Computational Depthmentioning

confidence: 99%

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Sophistication as Randomness Deficiency

Mota

Aaronson²,

Antunes

et al. 2013

Descriptional Complexity of Formal Systems

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Abstract. The sophistication of a string measures how much structural information it contains. We introduce naive sophistication, a variant of sophistication based on randomness deficiency. Naive sophistication measures the minimum number of bits needed to specify a set in which the string is a typical element. Thanks to Vereshchagin and Vitányi, we know that sophistication and naive sophistication are equivalent up to low order terms. We use this to relate sophistication to lossy compression, and to derive an alternative formulation for busy beaver computational depth.

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“…Antunes and Fortnow [6] defined a variant of sophistication, called coarse sophistication, that gets rid of the significance level by incorporating it into the minimization.…”

Section: Definition 4 (Sophistication)mentioning

confidence: 99%

“…Once again, this definition is equivalent to the definition based on structural information, given in [6], up to a logarithmic additive term. We say that S is a witness to csoph(…”

Section: Definition 4 (Sophistication)mentioning

confidence: 99%

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Sophistication as Randomness Deficiency

Mota

Aaronson²,

Antunes

et al. 2013

Descriptional Complexity of Formal Systems

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“…An unlabeled classificatory algorithmic information schema has been proposed by(Cilibrasi & Vitányi, 2005).3 The notion of degree of sophistication for computable objects was defined by Koppel(Koppel, 1991;Koppel & Atlan, 1991;Antunes & Fortnow, 2009).4 A number w is incompressible or random if K(w) |w| + c, where c is the length of the program executes the statement PRINT x. Frontiers in Artificial Intelligence | www.frontiersin.org January 2021 | Volume 3 | Article 567356…”

mentioning

confidence: 99%

Algorithmic Probability-Guided Machine Learning on Non-Differentiable Spaces

Hernández-Orozco

Zenil

Riedel

et al. 2021

Front. Artif. Intell.

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We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this model-driven approach may require less training data and can potentially be more generalizable as it shows greater resilience to random attacks. In an algorithmic space the order of its element is given by its algorithmic probability, which arises naturally from computable processes. We investigate the shape of a discrete algorithmic space when performing regression or classification using a loss function parametrized by algorithmic complexity, demonstrating that the property of differentiation is not required to achieve results similar to those obtained using differentiable programming approaches such as deep learning. In doing so we use examples which enable the two approaches to be compared (small, given the computational power required for estimations of algorithmic complexity). We find and report that 1) machine learning can successfully be performed on a non-smooth surface using algorithmic complexity; 2) that solutions can be found using an algorithmic-probability classifier, establishing a bridge between a fundamentally discrete theory of computability and a fundamentally continuous mathematical theory of optimization methods; 3) a formulation of an algorithmically directed search technique in non-smooth manifolds can be defined and conducted; 4) exploitation techniques and numerical methods for algorithmic search to navigate these discrete non-differentiable spaces can be performed; in application of the (a) identification of generative rules from data observations; (b) solutions to image classification problems more resilient against pixel attacks compared to neural networks; (c) identification of equation parameters from a small data-set in the presence of noise in continuous ODE system problem, (d) classification of Boolean NK networks by (1) network topology, (2) underlying Boolean function, and (3) number of incoming edges.

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“…Miller has suggested that we should find ways to quantify the idea that, while highly random sets have a great deal of information, it is useless information. There are existing concepts of useful information (see Antunes and Fortnow[3] for several references), but none seem to have been successfully applied yet to the context of higher order randomness.…”

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confidence: 99%

Calibrating Randomness

Downey¹,

Hirschfeldt²,

Nies³

et al. 2006

Bull. symb. log.

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We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set.

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Sophistication Revisited

Cited by 26 publications

References 12 publications

Sophistication as Randomness Deficiency

Sophistication as Randomness Deficiency

Algorithmic Probability-Guided Machine Learning on Non-Differentiable Spaces

Calibrating Randomness

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