220,559 research outputs found
Expansion Trees with Cut
Herbrand's theorem is one of the most fundamental insights in logic. From the
syntactic point of view it suggests a compact representation of proofs in
classical first- and higher-order logic by recording the information which
instances have been chosen for which quantifiers, known in the literature as
expansion trees.
Such a representation is inherently analytic and hence corresponds to a
cut-free sequent calculus proof. Recently several extensions of such proof
representations to proofs with cut have been proposed. These extensions are
based on graphical formalisms similar to proof nets and are limited to prenex
formulas.
In this paper we present a new approach that directly extends expansion trees
by cuts and covers also non-prenex formulas. We describe a cut-elimination
procedure for our expansion trees with cut that is based on the natural
reduction steps. We prove that it is weakly normalizing using methods from the
epsilon-calculus
A History of Woodland Dynamics in the Owyhee’s: Encroachment, Stand Closure, Understory Dynamics, and Tree Biomass
Piñon and juniper woodlands in the cold desert of the Intermountain West occupy over 44.6 million acres (Miller and Tausch 2001). These woodlands are commonly associated with sagebrush communities forming a mosaic of shrub-steppe and woodland across the region. Numerous studies have documented the recent expansion (since the late 1800’s) of these woodlands that has resulted in the replacement of shrub-steppe communities. Recent debate has challenged the degree of expansion in terms of percent of new areas occupied by trees and the increase in total population of piñon and juniper since the late 1800’s. Various interest groups have become concerned over the limited scientific evidence documenting the expansion of these conifers at a broad scale (in other words, landscapes or across entire woodlands) in the Intermountain Region. The fear of many groups is historic woodlands that occupied landscapes prior to Eurasian settlement in the late 1800’s are being burned, cut, and chained in the name of restoration
On the -positivity of trees and spiders
We prove that for any tree with a vertex of degree at least six, its
chromatic symmetric function is not -positive, that is, it cannot be written
as a nonnegative linear combination of elementary symmetric functions. This
makes significant progress towards a recent conjecture of Dahlberg, She, and
van Willigenburg, who conjectured the result for all trees with a vertex of
degree at least four. We also provide a series of conditions that can identify
when the chromatic symmetric function of a spider, a tree consisting of
multiple paths identified at an end, is not -positive. These conditions also
generalize to trees and graphs with cut vertices. Finally, by applying a result
of Orellana and Scott, we provide a method to inductively calculate certain
coefficients in the elementary symmetric function expansion of the chromatic
symmetric function of a spider, leading to further -positivity conditions
for spiders.Comment: 23 pages, 2 figures. Updated with new conjectures and minor changes
in wordin
Canonical Proof nets for Classical Logic
Proof nets provide abstract counterparts to sequent proofs modulo rule
permutations; the idea being that if two proofs have the same underlying
proof-net, they are in essence the same proof. Providing a convincing proof-net
counterpart to proofs in the classical sequent calculus is thus an important
step in understanding classical sequent calculus proofs. By convincing, we mean
that (a) there should be a canonical function from sequent proofs to proof
nets, (b) it should be possible to check the correctness of a net in polynomial
time, (c) every correct net should be obtainable from a sequent calculus proof,
and (d) there should be a cut-elimination procedure which preserves
correctness. Previous attempts to give proof-net-like objects for propositional
classical logic have failed at least one of the above conditions. In [23], the
author presented a calculus of proof nets (expansion nets) satisfying (a) and
(b); the paper defined a sequent calculus corresponding to expansion nets but
gave no explicit demonstration of (c). That sequent calculus, called LK\ast in
this paper, is a novel one-sided sequent calculus with both additively and
multiplicatively formulated disjunction rules. In this paper (a self-contained
extended version of [23]), we give a full proof of (c) for expansion nets with
respect to LK\ast, and in addition give a cut-elimination procedure internal to
expansion nets - this makes expansion nets the first notion of proof-net for
classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and
Computation
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
Wilsonian renormalization, differential equations and Hopf algebras
In this paper, we present an algebraic formalism inspired by Butcher's
B-series in numerical analysis and the Connes-Kreimer approach to perturbative
renormalization. We first define power series of non linear operators and
propose several applications, among which the perturbative solution of a fixed
point equation using the non linear geometric series. Then, following
Polchinski, we show how perturbative renormalization works for a non linear
perturbation of a linear differential equation that governs the flow of
effective actions. Then, we define a general Hopf algebra of Feynman diagrams
adapted to iterations of background field effective action computations. As a
simple combinatorial illustration, we show how these techniques can be used to
recover the universality of the Tutte polynomial and its relation to the
-state Potts model. As a more sophisticated example, we use ordered diagrams
with decorations and external structures to solve the Polchinski's exact
renormalization group equation. Finally, we work out an analogous construction
for the Schwinger-Dyson equations, which yields a bijection between planar
diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given
at the conference "Combinatorics and physics" held at Max Planck Institut
fuer Mathematik in Bonn in march 2007, some misprints correcte
Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
Ordinary tensor models of rank are dominated at large by
tree-like graphs, known as melonic triangulations. We here show that
non-melonic contributions can be enhanced consistently, leading to different
types of large limits. We first study the most generic quartic model at
, with maximally enhanced non-melonic interactions. The existence of the
expansion is proved and we further characterize the dominant
triangulations. This combinatorial analysis is then used to define a
non-quartic, non-melonic class of models for which the large free energy
and the relevant expectations can be calculated explicitly. They are matched
with random matrix models which contain multi-trace invariants in their
potentials: they possess a branched polymer phase and a 2D quantum gravity
phase, and a transition between them whose entropy exponent is positive.
Finally, a non-perturbative analysis of the generic quartic model is performed,
which proves analyticity in the coupling constants in cardioid domains
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