13,755 research outputs found
Inductive Inference and Reverse Mathematics
The present work investigates inductive inference from the perspective
of reverse mathematics. Reverse mathematics is a framework which relates
the proof strength of theorems and axioms throughout many areas of
mathematics in an interdisciplinary way. The present work looks at
basic notions of learnability including Angluin\u27s tell-tale condition and its variants for learning in the limit and for conservative learning. Furthermore, the more general criterion of partial learning is investigated. These notions are studied in the reverse mathematics context for uniformly and weakly represented families of languages. The results are stated in terms of axioms referring to domination and induction strength
Mining State-Based Models from Proof Corpora
Interactive theorem provers have been used extensively to reason about
various software/hardware systems and mathematical theorems. The key challenge
when using an interactive prover is finding a suitable sequence of proof steps
that will lead to a successful proof requires a significant amount of human
intervention. This paper presents an automated technique that takes as input
examples of successful proofs and infers an Extended Finite State Machine as
output. This can in turn be used to generate proofs of new conjectures. Our
preliminary experiments show that the inferred models are generally accurate
(contain few false-positive sequences) and that representing existing proofs in
such a way can be very useful when guiding new ones.Comment: To Appear at Conferences on Intelligent Computer Mathematics 201
Conditional Random Field Autoencoders for Unsupervised Structured Prediction
We introduce a framework for unsupervised learning of structured predictors
with overlapping, global features. Each input's latent representation is
predicted conditional on the observable data using a feature-rich conditional
random field. Then a reconstruction of the input is (re)generated, conditional
on the latent structure, using models for which maximum likelihood estimation
has a closed-form. Our autoencoder formulation enables efficient learning
without making unrealistic independence assumptions or restricting the kinds of
features that can be used. We illustrate insightful connections to traditional
autoencoders, posterior regularization and multi-view learning. We show
competitive results with instantiations of the model for two canonical NLP
tasks: part-of-speech induction and bitext word alignment, and show that
training our model can be substantially more efficient than comparable
feature-rich baselines
The sources of Mill's views of ratiocination and induction
Steffen Ducheyne and John P. McCaskey (2014). âThe Sources of Millâs Views of Ratiocination and Induction,â in: Antis Loizides (ed.), John Stuart Millâs âA System of Logicâ: A Critical Guide (London, Routledge), pp. 63-8
Reverse mathematics and well-ordering principles
The paper is concerned with generally Pi^1_2 sentences of the form 'if X is well ordered then f(X) is well ordered', where f is a standard proof theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded omega-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we shall focus on the well-known psi-function which figures prominently in so-called predicative proof theory. However, the approach taken here lends itself to generalization in that the techniques we employ can be applied to many other proof-theoretic functions associated with cut elimination theorems. In this paper we show that the statement 'if X is well ordered then 'X0 is well ordered' is equivalent to ATR0. This was first proved by Friedman, Montalban and Weiermann [7] using recursion-theoretic and combinatorial methods. The proof given here is proof-theoretic, the main techniques being Schuette's method of proof search (deduction chains) [13], generalized to omega logic, and cut elimination for infinitary ramified analysis
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