Directions for Computability Theory Beyond Pure Mathematical

Case, John

doi:10.1007/978-0-387-69245-6_2

Cited by 5 publications

(9 citation statements)

References 132 publications

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“…We thus believe that the results obtained in Gold's model can give some insights into human learning. As suggested in Case () and Heinz (), formal results in Gold's model might guide the design of meaningful experiments. We would like to see more interaction between this branch of Computability Theory, experimental Psychology, and Cognitive Science.…”

Section: Discussionmentioning

confidence: 97%

On the Necessity of U‐Shaped Learning

Carlucci

Case

2013

Topics in Cognitive Science

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A U-shaped curve in a cognitive-developmental trajectory refers to a three-step process: good performance followed by bad performance followed by good performance once again. U-shaped curves have been observed in a wide variety of cognitive-developmental and learning contexts. U-shaped learning seems to contradict the idea that learning is a monotonic, cumulative process and thus constitutes a challenge for competing theories of cognitive development and learning. U-shaped behavior in language learning (in particular in learning English past tense) has become a central topic in the Cognitive Science debate about learning models. Antagonist models (e.g., connectionism versus nativism) are often judged on their ability of modeling or accounting for U-shaped behavior. The prior literature is mostly occupied with explaining how U-shaped behavior occurs. Instead, we are interested in the necessity of this kind of apparently inefficient strategy. We present and discuss a body of results in the abstract mathematical setting of (extensions of) Gold-style computational learning theory addressing a mathematically precise version of the following question: Are there learning tasks that require U-shaped behavior? All notions considered are learning in the limit from positive data. We present results about the necessity of U-shaped learning in classical models of learning as well as in models with bounds on the memory of the learner. The pattern emerges that, for parameterized, cognitively relevant learning criteria, beyond very few initial parameter values, U-shapes are necessary for full learning power! We discuss the possible relevance of the above results for the Cognitive Science debate about learning models as well as directions for future research.

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Section: Discussionmentioning

confidence: 97%

On the Necessity of U‐Shaped Learning

Carlucci

Case

2013

Topics in Cognitive Science

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“…However, the Turing universe does not establish a clear link between Computability and Physics. In a sense, the recursion theoretic model of learning (that has something to say about deduction, induction or abduction from experiments) emphasizes some connections with Physics, although their mentors do not accept explicitly a possible non-computable reality (as stated in [8,9]). In this work, we will try to unite these views of the world, introducing the idea that the 'laws of nature' are a computable manifestation of the incomputable.…”

Section: Motivationmentioning

confidence: 99%

“…By (.) 10 :{0,1} → N we designate the operation of reading the string enclosed in brackets in decimal 8 and by (.) 2 : N →{0,1} we designate the operation of reading the natural number enclosed in brackets in binary.…”

Section: Families Of Timescalesmentioning

confidence: 99%

Incomputability at the foundations of physics (A study in the philosophy of science)

Costa

2013

Journal of Logic and Computation

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In this article, we show that once we assume that there is some uniformity in Nature, empirical laws and theories involving the concepts of Physics can be learned by the scientists. We develop a mathematical framework to explain learnability of empirical relations. Then we classify the degree of learnability according to the degree of predictability by means of timescales. We also use the same mathematical framework to make formal the claim that learnability evolves according not only with the number of observations done, but also with the precision of measurements.

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“…Finding programs for such functions (also known as sequences) is what one hopes to do in science-modelled as the search for predictive explanations. The present study does not itself involve direct applications to philosophy of science, whereas Case [7,8] surveys a number of them. Such applications essentially involve a slight extension of Turing's mechanization of mind program to apply to the (collections of) minds of human scientists (over historical time) attempting to provide predictive explanations for phenomena.…”

Section: Introductionmentioning

confidence: 99%

“…Gold [6] gave a first, simple learning criterion, later called [10,11] Ex-learning, 6 where a learner is successful if and only if (iff) it eventually stops changing its conjectures, and its final conjecture is a correct description of the input sequence. 7 Trivially, each single, describable sequence g has a suitable constant function as an Ex-learner (this learner constantly outputs an algorithmic description for g). Thus, we are interested in knowing for which sets of functions S there is a single learner h learning each member of S. We are interested herein in learning sets of (total) computable functions, and we will use (codes of) programs from a fixed programming system as possible conjectured (algorithmic) descriptions for the functions.…”

Section: Introductionmentioning

confidence: 99%

Computability-theoretic learning complexity

Case

Kötzing

2012

Phil. Trans. R. Soc. A.

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Initially discussed are some of Alan Turing's wonderfully profound and influential ideas about mind and mechanism—including regarding their connection to the main topic of the present study, which is within the field of computability-theoretic learning theory. Herein is investigated the part of this field concerned with the algorithmic, trial-and-error inference of eventually correct programs for functions from their data points. As to the main content of this study: in prior papers, beginning with the seminal work by Freivalds et al. in 1995, the notion of intrinsic complexity is used to analyse the learning complexity of sets of functions in a Gold-style learning setting. Herein are pointed out some weaknesses of this notion. Offered is an alternative based on epitomizing sets of functions—sets that are learnable under a given learning criterion, but not under other criteria that are not at least as powerful. To capture the idea of epitomizing sets, new reducibility notions are given based on robust learning (closure of learning under certain sets of computable operators). Various degrees of epitomizing sets are characterized as the sets complete with respect to corresponding reducibility notions! These characterizations also provide an easy method for showing sets to be epitomizers, and they are then employed to prove several sets to be epitomizing. Furthermore, a scheme is provided to generate easily very strong epitomizers for a multitude of learning criteria. These strong epitomizers are the so-called self-learning sets, previously applied by Case & Kötzing in 2010. These strong epitomizers can be easily generated and employed in a myriad of settings to witness with certainty the strict separation in learning power between the criteria so epitomized and other not as powerful criteria!

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Directions for Computability Theory Beyond Pure Mathematical

Cited by 5 publications

References 132 publications

On the Necessity of U‐Shaped Learning

On the Necessity of U‐Shaped Learning

Incomputability at the foundations of physics (A study in the philosophy of science)

Computability-theoretic learning complexity

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