115,531 research outputs found
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
Importing SMT and Connection proofs as expansion trees
Different automated theorem provers reason in various deductive systems and,
thus, produce proof objects which are in general not compatible. To understand
and analyze these objects, one needs to study the corresponding proof theory,
and then study the language used to represent proofs, on a prover by prover
basis. In this work we present an implementation that takes SMT and Connection
proof objects from two different provers and imports them both as expansion
trees. By representing the proofs in the same framework, all the algorithms and
tools available for expansion trees (compression, visualization, sequent
calculus proof construction, proof checking, etc.) can be employed uniformly.
The expansion proofs can also be used as a validation tool for the proof
objects produced.Comment: In Proceedings PxTP 2015, arXiv:1507.0837
FURY: Fuzzy unification and resolution based on edit distance
We present a theoretically founded framework for fuzzy
unification and resolution based on edit distance over trees.
Our framework extends classical unification and resolution
conservatively. We prove important properties of the framework
and develop the FURY system, which implements the
framework efficiently using dynamic programming. We
evaluate the framework and system on a large problem in
the bioinformatics domain, that of detecting typographical
errors in an enzyme name databas
Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning
Resolution refinements called w-resolution trees with lemmas (WRTL) and with
input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both
WRTL and WRTI when there is no regularity condition. For regular proofs, an
exponential separation between regular dag-like resolution and both regular
WRTL and regular WRTI is given.
It is proved that DLL proof search algorithms that use clause learning based
on unit propagation can be polynomially simulated by regular WRTI. More
generally, non-greedy DLL algorithms with learning by unit propagation are
equivalent to regular WRTI. A general form of clause learning, called
DLL-Learn, is defined that is equivalent to regular WRTL.
A variable extension method is used to give simulations of resolution by
regular WRTI, using a simplified form of proof trace extensions. DLL-Learn and
non-greedy DLL algorithms with learning by unit propagation can use variable
extensions to simulate general resolution without doing restarts.
Finally, an exponential lower bound for WRTL where the lemmas are restricted
to short clauses is shown
Decomposable Theories
We present in this paper a general algorithm for solving first-order formulas
in particular theories called "decomposable theories". First of all, using
special quantifiers, we give a formal characterization of decomposable theories
and show some of their properties. Then, we present a general algorithm for
solving first-order formulas in any decomposable theory "T". The algorithm is
given in the form of five rewriting rules. It transforms a first-order formula
"P", which can possibly contain free variables, into a conjunction "Q" of
solved formulas easily transformable into a Boolean combination of
existentially quantified conjunctions of atomic formulas. In particular, if "P"
has no free variables then "Q" is either the formula "true" or "false". The
correctness of our algorithm proves the completeness of the decomposable
theories.
Finally, we show that the theory "Tr" of finite or infinite trees is a
decomposable theory and give some benchmarks realized by an implementation of
our algorithm, solving formulas on two-partner games in "Tr" with more than 160
nested alternated quantifiers
A semi-analytic model comparison - gas cooling and galaxy mergers
We use stripped-down versions of three semi-analytic galaxy formation models
to study the influence of different assumptions about gas cooling and galaxy
mergers. By running the three models on identical sets of merger trees
extracted from high-resolution cosmological N-body simulations, we are able to
perform both statistical analyses and halo-by-halo comparisons. Our study
demonstrates that there is a good statistical agreement between the three
models used here, when operating on the same merger trees, reflecting a general
agreement in the underlying framework for semi-analytic models. We also show,
however, that various assumptions that are commonly adopted to treat gas
cooling and galaxy mergers can lead to significantly different results, at
least in some regimes. In particular, we find that the different models adopted
for gas cooling lead to similar results for mass scales comparable to that of
our own Galaxy. Significant differences, however, arise at larger mass scales.
These are largely (but not entirely) due to different treatments of the `rapid
cooling' regime, and different assumptions about the hot gas distribution. At
this mass regime, the predicted cooling rates can differ up to about one order
of magnitude, with important implications on the relative weight that these
models give to AGN feedback in order to counter-act excessive gas condensation
in relatively massive haloes at low redshift. Different assumptions in the
modelling of galaxy mergers can also result in significant differences in the
timings of mergers, with important consequences for the formation and evolution
of massive galaxies.Comment: 21 pages, 14 figures. Accepted for publication in MNRAS
Ultrametric spaces of branches on arborescent singularities
Let be a normal complex analytic surface singularity. We say that is
arborescent if the dual graph of any resolution of it is a tree. Whenever
are distinct branches on , we denote by their intersection
number in the sense of Mumford. If is a fixed branch, we define when and
otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of
surfaces, by proving that whenever is arborescent, then is an
ultrametric on the set of branches of different from . We compute the
maximum of , which gives an analog of a theorem of Teissier. We show that
encodes topological information about the structure of the embedded
resolutions of any finite set of branches. This generalizes a theorem of Favre
and Jonsson concerning the case when both and are smooth. We generalize
also from smooth germs to arbitrary arborescent ones their valuative
interpretation of the dual trees of the resolutions of . Our proofs are
based in an essential way on a determinantal identity of Eisenbud and Neumann.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has
a new section 4.3, accompanied by 2 new figures. Several passages were
clarified and the typos discovered in the meantime were correcte
Path ideals of rooted trees and their graded Betti numbers
Let be a rooted tree and let be a positive integer. We study
algebraic invariants and properties of the path ideal generated by monomial
corresponding to paths of length in . In particular, we give a
recursive formula to compute the graded Betti numbers, a general bound for the
regularity, an explicit computation of the linear strand, and we characterize
when this path ideal has a linear resolution.Comment: 18 page
- âŠ