Optimal Language Learning

Case, John; Moelius, Samuel E.

doi:10.1007/978-3-540-87987-9_34

Cited by 11 publications

(5 citation statements)

References 14 publications

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“…• There exists a class of languages that can be It-identified, but that cannot be so identified strongly non-U-shapedly [CK10, Theorem 5.4] (see also [Bei84,Wie91,CM08]).…”

Section: (Proposition 14)mentioning

confidence: 99%

Learning without Coding

Moelius¹,

Zilles

2010

Lecture Notes in Computer Science

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Iterative learning is a model of language learning from positive data, due to Wiehagen. When compared to a learner in Gold's original model of language learning from positive data, an iterative learner can be thought of as memorylimited . However, an iterative learner can memorize some input elements by coding them into the syntax of its hypotheses. A main concern of this paper is: to what extent are such coding tricks necessary?One means of preventing some such coding tricks is to require that the hypothesis space used be free of redundancy, i.e., that it be 1-1. In this context, we make the following contributions. By extending a result of Lange & Zeugmann, we show that many interesting and non-trivial classes of languages can be iteratively identified using a Friedberg numbering as the hypothesis space. (Recall that a Friedberg numbering is a 1-1 effective numbering of all computably enumerable sets.) An example of such a class is the class of pattern languages over an arbitrary alphabet. On the other hand, we show that there exists an iteratively identifiable class of languages that cannot be iteratively identified using any 1-1 effective numbering as the hypothesis space.We also consider an iterative-like learning model in which the computational component of the learner is modeled as an enumeration operator , as opposed to a partial computable function. In this new model, there are no hypotheses, and, thus, no syntax in which the learner can encode what elements it has or has not yet seen. We show that there exists a class of languages that can be identified under this new model, but that cannot be iteratively identified. On the other hand, we show that there exists a class of languages that cannot be identified under this new model, but that can be iteratively identified using a Friedberg numbering as the hypothesis space.

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“…• There exists a class of languages that can be It-identified, but that cannot be so identified strongly non-U-shapedly [CK10, Theorem 5.4] (see also [Bei84,Wie91,CM08]).…”

Section: (Proposition 14)mentioning

confidence: 99%

Learning without Coding

Moelius¹,

Zilles

2010

Lecture Notes in Computer Science

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“…A very succinct and elegant proof of Theorem can be found in the recent Case and Kötzing (). The result has been strengthened in Case and Moelius () to show that all explanatory learners can be transformed in explanatory learners that are strongly non‐U‐shaped. Recall that strongly non‐U‐shaped learners are learners that stick to the first correct conjecture issued during the learning process.…”

Section: U‐shaped Learning With Full Memorymentioning

confidence: 96%

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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“…The situation changes if we require of the learner additionally to never abandon correct conjectures -either only not semantically (called non-U-shaped learning, NU, [1]) or not even syntactically (strongly non-U-shaped learning, SNU, [9]). The resulting groupings and separations are depicted in the following two diagrams.…”

Section: Differences In Countersmentioning

confidence: 98%

“…Thus, A k 1 is infinite, and W q(k 1 ) ∈ L, but h, on the text σ k 1 a k 1 with Id-counter does not converge to an index for W q(k 1 ) , a contradiction. 2 9 Note that (P (c) Proof. Let c 1 be minimal such that ¬P k 1 (c 1 ) (we know about the existence from Claim 3).…”

Section: Claimmentioning

confidence: 99%

Iterative learning from positive data and counters

Kötzing

2014

Theoretical Computer Science

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We analyze iterative learning in the limit from positive data with the additional information provided by a counter. The simplest type of counter provides the current iteration number (counting up from 0 to infinity), which is known to improve learning power over plain iterative learning. We introduce five other (weaker) counter types, for example only providing some unbounded and non-decreasing sequence of numbers. Analyzing these types allows one to understand which properties of a counter learning can benefit from. For the iterative setting, we completely characterize the relative power of the learning criteria corresponding to the counter types. In particular, for our types, the only properties improving learning power are unboundedness and strict monotonicity. Furthermore, we show that each of our types of counter improves learning power over weaker ones in some settings; to this end, we analyze transductive and non-U-shaped learning. Finally we show that, for iterative learning criteria with one of our types of counter, separations of learning criteria are necessarily witnessed by classes containing only infinite languages.

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Optimal Language Learning

Cited by 11 publications

References 14 publications

Learning without Coding

Learning without Coding

On the Necessity of U‐Shaped Learning

Iterative learning from positive data and counters

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