Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...

In this paper we calibrate the exact proof--theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents proof--theoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the proof--theoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of \Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...

Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let |n | denote the binary length of a natural number n, let |n|h denote the h-times iterated binary length of n and let inv(n) be the least h such that |n|h ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0,..., αn 〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the i-th term αi is bounded by K + |i | · |i|h. As a supplement to this (refined Friedman style) independence result we further show that e.g. primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0,..., αn 〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the i-th term αi is bounded by K +|i|· inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polya-style enumerations. Using results from Otter and from Matouˇsek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter’s tree constant 2.9557652856.... ∗ Research supported by a Heisenberg-Fellowship of the Deutsche Forschungsgemeinschaft. † The main results of this paper were obtained during the authors visit of T. Arai in

Let k be a positive integer. There is a longest finite sequence x 1 ,...,x n in k letters in which no consecutive block x i ,...,x 2i is a subsequence of any other consecutive block x j ,...,x 2j . Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). TABLE OF CONTENTS 1. Finiteness, and n(1),n(2). 2. Sequences of fixed length sequences. 3. The Main Lemma. 4. Lower bound for n(3). 5. The function n(k). 6. Related problems and computer explorations. 1. FINITENESS, AND n(1),n(2) We use Z for the set of all integers, Z + f...

this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.

this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by Nash-Williams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relations 3. Brouwer's Thesis 4. Ramsey's Theorem 5. The Finite Sequence Theorem 6. Vazsonyi's Conjecture for binary trees 7. Higman's Theorem 8. Vazsonyi's Conjecture and the Tree Theorem 9. Minimal-Bad-Sequence Arguments 10. The Principle of Open Induction 11. Concluding Remarks Except for Section 9, we will argue intuitionistically. 1 1 Dickson's Lemma

We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddablity, contains no infinite descending chain and no infinite antichain. In this paper we study the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of RCA0+Σ1 1-IND. We also prove that Fraïssé’s conjecture is equivalent, over RCA0, to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+, −}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings. While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just ATR0+Σ1 1-IND. Moreover, for all these linear orderings, L, we prove that any partial ordering, P, which does not embed L has a linearization, hyperarithmetic (or equivalently ∆1 1) in P ⊕ L, which does not embed L.

Abstract. We show that Nash-Williams ’ theorem asserting that the countable transfinite sequences of elements of a better-quasi-ordering ordered by embeddability form a better-quasi-ordering is provable in the subsystem of second order arithmetic Π 1 1-CA0 but is not equivalent to Π 1 1-CA0. We obtain some partial results towards the proof of this theorem in the weaker subsystem ATR0 and we show that the minimality lemmas typical of wqo and bqo theory imply Π 1 1-CA0 and hence cannot be used in such a proof. The most natural generalization of the notion of well-ordering to partial orderings is the concept of well-quasi-ordering or wqo. A binary relation � on a set Q is a quasi-ordering if it is reflexive and transitive (it is a partial ordering if it is also anti-symmetric) and a quasi-ordering is wqo if it contains no infinite descending sequences and no infinite anti-chains (sets of mutually incomparable elements). Ramsey’s theorem implies that a quasi-ordering (Q, �) is wqo if and only if for any function f: N → Q there exist n and m such that n < m and f(n) � f(m). For a history of the notion of wqo see [11].

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