17,220 research outputs found
On the structure of acyclic binary relations
We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain)
Every countable model of set theory embeds into its own constructible universe
The main theorem of this article is that every countable model of set theory
M, including every well-founded model, is isomorphic to a submodel of its own
constructible universe. In other words, there is an embedding that
is elementary for quantifier-free assertions. The proof uses universal digraph
combinatorics, including an acyclic version of the countable random digraph,
which I call the countable random Q-graded digraph, and higher analogues
arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a
set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph,
closely connected with the surreal numbers. The proof shows that contains
a submodel that is a universal acyclic digraph of rank . The method of
proof also establishes that the countable models of set theory are linearly
pre-ordered by embeddability: for any two countable models of set theory, one
of them is isomorphic to a submodel of the other. Indeed, they are
pre-well-ordered by embedability in order-type exactly .
Specifically, the countable well-founded models are ordered by embeddability in
accordance with the heights of their ordinals; every shorter model embeds into
every taller model; every model of set theory is universal for all
countable well-founded binary relations of rank at most ; and every
ill-founded model of set theory is universal for all countable acyclic binary
relations. Finally, strengthening a classical theorem of Ressayre, the same
proof method shows that if is any nonstandard model of PA, then every
countable model of set theory---in particular, every model of ZFC---is
isomorphic to a submodel of the hereditarily finite sets of . Indeed,
is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at
http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe.
(v2 adds a reference and makes minor corrections) (v3 includes further
changes, and removes the previous theorem 15, which was incorrect.
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
Weak Pseudo-Rationalizability
This paper generalizes rationalizability of a choice function by a single acyclic binary relation to
rationalizability by a set of such relations. Rather than selecting those options in a menu that are
maximal with respect to a single binary relation, a weakly pseudo-rationalizable choice function
selects those options that are maximal with respect to at least one binary relation in a given
set. I characterize the class of weakly pseudo-rationalizable choice functions in terms of simple
functional properties. This result also generalizes Aizerman and Malishevski's characterization
of pseudo-rationalizable choice functions, that is, choice functions rationalizable by a set of total
orders
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
CP-nets: A Tool for Representing and Reasoning withConditional Ceteris Paribus Preference Statements
Information about user preferences plays a key role in automated decision
making. In many domains it is desirable to assess such preferences in a
qualitative rather than quantitative way. In this paper, we propose a
qualitative graphical representation of preferences that reflects conditional
dependence and independence of preference statements under a ceteris paribus
(all else being equal) interpretation. Such a representation is often compact
and arguably quite natural in many circumstances. We provide a formal semantics
for this model, and describe how the structure of the network can be exploited
in several inference tasks, such as determining whether one outcome dominates
(is preferred to) another, ordering a set outcomes according to the preference
relation, and constructing the best outcome subject to available evidence
Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms
Tree projections provide a unifying framework to deal with most structural
decomposition methods of constraint satisfaction problems (CSPs). Within this
framework, a CSP instance is decomposed into a number of sub-problems, called
views, whose solutions are either already available or can be computed
efficiently. The goal is to arrange portions of these views in a tree-like
structure, called tree projection, which determines an efficiently solvable CSP
instance equivalent to the original one. Deciding whether a tree projection
exists is NP-hard. Solution methods have therefore been proposed in the
literature that do not require a tree projection to be given, and that either
correctly decide whether the given CSP instance is satisfiable, or return that
a tree projection actually does not exist. These approaches had not been
generalized so far on CSP extensions for optimization problems, where the goal
is to compute a solution of maximum value/minimum cost. The paper fills the
gap, by exhibiting a fixed-parameter polynomial-time algorithm that either
disproves the existence of tree projections or computes an optimal solution,
with the parameter being the size of the expression of the objective function
to be optimized over all possible solutions (and not the size of the whole
constraint formula, used in related works). Tractability results are also
established for the problem of returning the best K solutions. Finally,
parallel algorithms for such optimization problems are proposed and analyzed.
Given that the classes of acyclic hypergraphs, hypergraphs of bounded
treewidth, and hypergraphs of bounded generalized hypertree width are all
covered as special cases of the tree projection framework, the results in this
paper directly apply to these classes. These classes are extensively considered
in the CSP setting, as well as in conjunctive database query evaluation and
optimization
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