24,165 research outputs found
Some observations on the logical foundations of inductive theorem proving
In this paper we study the logical foundations of automated inductive theorem
proving. To that aim we first develop a theoretical model that is centered
around the difficulty of finding induction axioms which are sufficient for
proving a goal.
Based on this model, we then analyze the following aspects: the choice of a
proof shape, the choice of an induction rule and the language of the induction
formula. In particular, using model-theoretic techniques, we clarify the
relationship between notions of inductiveness that have been considered in the
literature on automated inductive theorem proving. This is a corrected version
of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017
Elliptic divisibility sequences and undecidable problems about rational points
Julia Robinson has given a first-order definition of the rational integers Z
in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0)
where the \forall-quantifiers run over a total of 8 variables, and where F is a
polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove
that a conjecture about elliptic curves provides an interpretation of Z in Q
with quantifier complexity \forall \exists, involving only one universally
quantified variable. This improves the complexity of defining Z in Q in two
ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is
undecidable (recall that Hilbert's Tenth Problem for Q is the question whether
the \Sigma_1-theory of Q is undecidable).
In short, granting the conjecture, there is a one-parameter family of
hypersurfaces over Q for which one cannot decide whether or not they all have a
rational point.
The conjecture is related to properties of elliptic divisibility sequences on
an elliptic curve and its image under rational 2-descent, namely existence of
primitive divisors in suitable residue classes, and we discuss how to prove
weaker-in-density versions of the conjecture and present some heuristics.Comment: 39 pages, uses calrsfs. 3rd version: many small changes, change of
titl
Real closed exponential fields
In an extended abstract Ressayre considered real closed exponential fields
and integer parts that respect the exponential function. He outlined a proof
that every real closed exponential field has an exponential integer part. In
the present paper, we give a detailed account of Ressayre's construction, which
becomes canonical once we fix the real closed exponential field, a residue
field section, and a well ordering of the field. The procedure is constructible
over these objects; each step looks effective, but may require many steps. We
produce an example of an exponential field with a residue field and a
well ordering such that is low and and are ,
and Ressayre's construction cannot be completed in .Comment: 24 page
Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture
Sofic and hyperlinear groups are the countable discrete groups that can be
approximated in a suitable sense by finite symmetric groups and groups of
unitary matrices. These notions turned out to be very deep and fruitful, and
stimulated in the last 15 years an impressive amount of research touching
several seemingly distant areas of mathematics including geometric group
theory, operator algebras, dynamical systems, graph theory, and more recently
even quantum information theory. Several longstanding conjectures that are
still open for arbitrary groups were settled in the case of sofic or
hyperlinear groups. These achievements aroused the interest of an increasing
number of researchers into some fundamental questions about the nature of these
approximation properties. Many of such problems are to this day still open such
as, outstandingly: Is there any countable discrete group that is not sofic or
hyperlinear? A similar pattern can be found in the study of II_1 factors. In
this case the famous conjecture due to Connes (commonly known as the Connes
embedding conjecture) that any II_1 factor can be approximated in a suitable
sense by matrix algebras inspired several breakthroughs in the understanding of
II_1 factors, and stands out today as one of the major open problems in the
field. The aim of these notes is to present in a uniform and accessible way
some cornerstone results in the study of sofic and hyperlinear groups and the
Connes embedding conjecture. The presentation is nonetheless self contained and
accessible to any student or researcher with a graduate level mathematical
background. An appendix by V. Pestov provides a pedagogically new introduction
to the concepts of ultrafilters, ultralimits, and ultraproducts for those
mathematicians who are not familiar with them, and aiming to make these
concepts appear very natural.Comment: 157 pages, with an appendix by Vladimir Pesto
Combining semantic and syntactic structure for language modeling
Structured language models for speech recognition have been shown to remedy
the weaknesses of n-gram models. All current structured language models are,
however, limited in that they do not take into account dependencies between
non-headwords. We show that non-headword dependencies contribute to
significantly improved word error rate, and that a data-oriented parsing model
trained on semantically and syntactically annotated data can exploit these
dependencies. This paper also contains the first DOP model trained by means of
a maximum likelihood reestimation procedure, which solves some of the
theoretical shortcomings of previous DOP models.Comment: 4 page
struc2vec: Learning Node Representations from Structural Identity
Structural identity is a concept of symmetry in which network nodes are
identified according to the network structure and their relationship to other
nodes. Structural identity has been studied in theory and practice over the
past decades, but only recently has it been addressed with representational
learning techniques. This work presents struc2vec, a novel and flexible
framework for learning latent representations for the structural identity of
nodes. struc2vec uses a hierarchy to measure node similarity at different
scales, and constructs a multilayer graph to encode structural similarities and
generate structural context for nodes. Numerical experiments indicate that
state-of-the-art techniques for learning node representations fail in capturing
stronger notions of structural identity, while struc2vec exhibits much superior
performance in this task, as it overcomes limitations of prior approaches. As a
consequence, numerical experiments indicate that struc2vec improves performance
on classification tasks that depend more on structural identity.Comment: 10 pages, KDD2017, Research Trac
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