# A Solution to Wiehagen's Thesis

TL;DR: This work proves Wiehagen’s Thesis in Inductive Inference for a wide range of learning criteria, including many popular criteria from the literature, and shows the limitations of the thesis by giving four learning criteria for which the thesis does not hold.

Abstract: Wiehagen's Thesis in Inductive Inference (1991) essentially states that, for each learning criterion, learning can be done in a normalized, enumerative way. The thesis was not a formal statement and thus did not allow for a formal proof, but support was given by examples of a number of different learning criteria that can be learned by enumeration. Building on recent formalizations of learning criteria, we are now able to formalize Wiehagen's Thesis. We prove the thesis for a wide range of learning criteria, including many popular criteria from the literature. We also show the limitations of the thesis by giving four learning criteria for which the thesis does not hold (and, in two cases, was probably not meant to hold). Beyond the original formulation of the thesis, we also prove stronger versions which allow for many corollaries relating to strongly decisive and conservative learning.

## Summary (1 min read)

### Introduction

- In Gold-style learning [10] (also known as inductive inference) a learner tries to learn an infinite sequence, given more and more finite information about this sequence.
- Gold, in his seminal paper [10] , gave a first, simple learning criterion, later called Ex-learning 1 , where a learner is successful iff it eventually stops changing its conjectures, and its final conjecture is a correct program (computing the input sequence).
- In Theorem 3 the authors discuss four different learning criteria in which the thesis does not hold.
- From these results on learning with a semantically 1-1 enumeration the authors can derive corollaries to conclude that the learning criteria, to which the theorems apply, allow for strongly decisive and conservative learning (see Definition 1); for example, for plain Ex-learning, this proves (a stronger version of) a result from [15] (which showed that Ex-learning can be done decisively).

### Definition 1.

- The authors say that a learner exhibits a U-shape when it first outputs a correct conjecture, abandons this, and then returns to a correct conjecture.
- Forbidding these kinds of U-shapes leads to the respective non-U-shapedness restrictions SynNU, NU and SNU.
- If the authors consider forbidding returning to abandoned conjectures more generally, they get three corresponding restrictions of decisiveness.
- Note that the literature knows many more learning criteria than those constructible from the parts given in this section (see the text book [11] or the survey [19] for an overview).

### 3 Learning by Enumeration

- From the wealth of (theoretically possible) learning criteria the authors quickly see that there are learning criteria which do not allow for learning by enumeration.
- With these definitions, the authors get the follwing theorem.

### S TA C S ' 1 4

- The following learning criteria do not allow for learning by enumeration.
- The authors can see the deep power and versatility of Theorem 13 in connection with Remark 3 and the various examples of sequence acceptance criteria fulfilling the prerequisites of Theorem 13, which leads, for example, to the following corollary.
- Then RItδ allows for learning by enumeration.

### Definition 5.

- That is (by taking the contrapositive), different pre-images under e not only give different images, but even semantically different images.
- The authors say that a learning criterion I allows for learning by semantically 1-1 enumeration iff each I-learnable set S is I-learnable by semantically 1-1 enumeration.
- Let h learn by semantically 1-1 enumeration.
- In particular, for any learning criterion I allowing for learning by semantically 1-1 enumeration, every I-learnable set is I-learnable by a strongly decisive learner.

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### Cites background from "A Solution to Wiehagen's Thesis"

...In decisive learning (Dec, Osherson et al., 1982), a learner may never return to a semantically abandoned conjecture; in strongly decisive learning (SDec, Kötzing, 2014) the learner may not even return to syntactically abandoned conjectures....

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...The following definitions were first given by Kötzing (2014)....

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##### References

3,665 citations

### Additional excerpts

...If there is an f ∈ R such that ∀z : (φz) = φf (z), we call effective [20]....

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...Unintroduced notation for computability theory follows [20]....

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3,364 citations

### "A Solution to Wiehagen's Thesis" refers background or methods in this paper

...for “Explanatory”), already studied by Gold [13]....

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...Gold, in his seminal paper [13], gave a first, simple learning criterion, later called Ex-learning,1 where a learner is successful iff it eventually stops changing its conjectures, and its final conjecture is a correct program (computing the input sequence)....

[...]

...In inductive inference (as introduced by Gold [13]) a learner tries to learn an infinite sequence of function values, given more and more finite information about this sequence....

[...]

...We consider the three sequence generating operators in this paper: G (which stands for “Gold”, who first studied it [13]), corresponding to the examples of learning criteria given in the introduction; It (iterative learning, [23]); and Td (transductive learning, [10])....

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...The most important sequence acceptance criterion is denoted Ex (which stands for “Explanatory”), already studied by Gold [13]....

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1,778 citations

791 citations

### "A Solution to Wiehagen's Thesis" refers methods in this paper

...Definition 1 With Cons we denote the restriction of consistent learning [4, 6, 17] (being correct on all known data); with Conf the restriction of conformal learning [24] (being correct or divergent on known data); with Conv we denote the restriction of conservative learning [2] (never abandoning a conjecture which is correct on all known data); with Mon we denote the restriction of monotone learning [16] (conjectures make all the outputs that previous conjectures made— monotonicity in the graphs); finally, with PMon we denote the restriction of pseudo-monotone learning [25] (conjectures make all the correct outputs that previous conjectures made)....

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638 citations

### "A Solution to Wiehagen's Thesis" refers background or methods in this paper

...Thus, the above strategy for learning employed by Blum and Blum [6] is not applicable for all learning tasks....

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...Blum and Blum [6] gave the following example....

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...Thus, the above strategy for learning employed by Blum and Blum [6] is not...

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...Definition 1 With Cons we denote the restriction of consistent learning [4, 6, 17] (being correct on all known data); with Conf the restriction of conformal learning [24] (being correct or divergent on known data); with Conv we denote the restriction of conservative learning [2] (never abandoning a conjecture which is correct on all known data); with Mon we denote the restriction of monotone learning [16] (conjectures make all the outputs that previous conjectures made— monotonicity in the graphs); finally, with PMon we denote the restriction of pseudo-monotone learning [25] (conjectures make all the correct outputs that previous conjectures made)....

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