Semantic Limits of Dense Combinatorial Objects

10/19/2019
by   Leonardo N. Coregliano, et al.
0

The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as "flag algebras", while the semantic, geometric one is often associated with the name “graph limits”. The language of graph limits is generally more intuitive and expressible, but a price that one has to pay for it is that it is better suited for the case of ordinary graphs than for more general combinatorial objects. Accordingly, there have been several attempts in the literature, of varying degree of generality, to define limit objects for more complicated combinatorial structures. This paper is another attempt at a workable general theory of dense limit objects. Unlike previous efforts in this direction (with notable exception of [Ashwini Aroskar and James Cummings. Limits, regularity and removal for finite structures. Technical Report arXiv:1412.2014 [math.LO], arXiv e-print, 2014.]), we base our account on the same concepts from the first-order logic and the model theory as in the theory of flag algebras. We show how our definition naturally encompasses a host of previously considered cases (graphons, hypergraphons, digraphons, permutons, posetons, colored graphs, etc.), and we extend the fundamental properties of existence and uniqueness to this more general case. We also give an intuitive general proof of the continuous version of the Induced Removal Lemma based on the completeness theorem for propositional calculus. We capitalize on the notion of an open interpretation that often allows to transfer methods and results from one situation to another. Again, we show that some previous arguments can be quite naturally framed using this language.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

11/05/2018

Limits of Ordered Graphs and Images

The emerging theory of graph limits exhibits an interesting analytic per...
03/07/2020

Quasi-random words and limits of word sequences

Words are sequences of letters over a finite alphabet. We study two inti...
05/16/2018

Combinatorial Properties of Metrically Homogeneous Graphs

Ramsey theory looks for regularities in large objects. Model theory stud...
06/19/2018

A unifying method for the design of algorithms canonizing combinatorial objects

We devise a unified framework for the design of canonization algorithms....
12/04/2021

Order in the chaos with examples from graph theory

In randomly created structures (be they natural or artificial) very ofte...
12/27/2019

The Epistemic Landscape: a Computability Perspective

By nature, transmissible human knowledge is enumerable: every sentence, ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.