\documentclass[t,xcolor=dvipsnames]{beamer}
\usepackage{graphics,multimedia,mathrsfs,color}

\setbeamertemplate{navigation symbols}{}

\theoremstyle{theorem-KL}
\newtheorem{theorem-KL}[theorem]{Theorem (Kearnes, Li)}

%\theoremstyle{Problem}
%\newtheorem{problem}[theorem]{Problem}

\theoremstyle{Conjecture}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{Question}
\newtheorem{question}[theorem]{Question}

\theoremstyle{Example}
\newtheorem{exmp}[theorem]{Example}

\theoremstyle{Example1}
\newtheorem{exmp1}[theorem]{Example ($d_{M}(n)\in \Theta(n)$ if $M$ is a module)}

\theoremstyle{Example2}
\newtheorem{exmp2}[theorem]{Example ($F(1,x,2)\approx x, F(x,2,3)\approx x$)}

\theoremstyle{Conjecture2}
\newtheorem{conjecture2}[theorem]{Conjecture 2}

\theoremstyle{Theorems}
\newtheorem{theorems}[theorem]{Theorems}

\theoremstyle{Lemmas}
\newtheorem{lemmas}[theorem]{Lemmas}


% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.

% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice. 

\mode<presentation>
{


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\usefonttheme{serif}

%  \setbeamercovered{transparent}
  % or whatever (possibly just delete it)
}

\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever.  Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.


%%%\title[Short Paper Title] % (optional, use only with long paper titles)
%%%{Presentation Title}
\title{Growth Functions of Finite Algebras}

%%%\subtitle
%%%{Presentation Subtitle} % (optional)

%%%\author[Author, Another] % (optional, use only with lots of authors)
%%%{F.~Author\inst{1} \and S.~Another\inst{2}}
%%%% - Use the \inst{?} command only if the authors have different
%%%%   affiliation.
\author{K. A. Kearnes}

%%%\institute[Universities of Somewhere and Elsewhere] % (optional, but mostly needed)
%%%{
%%%  \inst{1}%
%%%  Department of Computer Science\\
%%%  University of Somewhere
%%%  \and
%%%  \inst{2}%
%%%  Department of Theoretical Philosophy\\
%%%  University of Elsewhere}
%%%% - Use the \inst command only if there are several affiliations.
%%%% - Keep it simple, no one is interested in your street address.
\institute{University of Colorado}

%%%\date[Short Occasion] % (optional)
%%%{Date / Occasion}
%\date{April 28, 2006\\ University of Hawaii}
\date[]{with E. W. Kiss and \'A. Szendrei}
%76. Arbeitstagung Allgemeine Algebra\\May 22-25, 2008\\Linz, Austria}

%%%\subject{Talks}
%%%% This is only inserted into the PDF information catalog. Can be left
%%%% out. 



%%%% If you have a file called "university-logo-filename.xxx", where xxx
%%%% is a graphic format that can be processed by latex or pdflatex,
%%%% resp., then you can add a logo as follows:

%%%% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
%%%% \logo{\pgfuseimage{university-logo}}



%%%% Delete this, if you do not want the table of contents to pop up at
%%%% the beginning of each subsection:
%%%\AtBeginSubsection[]
%%%{
%%%  \begin{frame}<beamer>
%%%    \frametitle{Outline}
%%%    \tableofcontents[currentsection,currentsubsection]
%%%  \end{frame}
%%%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}


\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb,amscd,mathrsfs}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%%%\begin{frame}
%%%  \frametitle{Outline}
%%%  \tableofcontents
%%%  % You might wish to add the option [pausesections]
%%%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.



\beamerdefaultoverlayspecification{<+->}
%%%\subsection[Short First Subsection Name]{First Subsection Name}
%%\subsection{Apr 3--7, 2006}


\def\m#1{\mathbf{\uppercase{#1}}}
\def\df{{\mathrm{Def}}}
\def\thm{{\mathrm{Thm}}}
\def\sd{{\mathrm{sd}}}
\def\EE{{\sf E}}
\def\UU{{\sf U}}
\def\MM{{\sf M}}
\def\PP{{\sf P}}
\def\Clo{{\mathrm{Clo}}}
\def\Pol{{\mathrm{Pol}}}
\def\Sub{{\mathrm{Sub}}}
\def\Con{{\mathrm{Con}}}
\def\Aut{{\mathrm{Aut}}}
\def\pr{{\mathrm{pr}}}
\def\var#1{{\mathcal #1}}
\def\wec#1{{\mathbf #1}}

\def\FF{{\mathfrak F}}

\begin{frame}[t]
\frametitle{Growth functions}
\pause
{\bf Question:}
For an algebra $\m A$,
what is the least size of a generating set for $\m A^n$?
\bigskip

\pause
{\bf Answer:}
$d_{\m A}(n):=\min\{g:\text{$\m A^n$ has a generating set of size $g$}\}$.
\bigskip

\end{frame}

\begin{frame}[t]
\frametitle{Examples}

\pause
\begin{enumerate}
\item If $\m a$ is a 2-element set, then 
$d_{\m a}(n)=2^n$.
\medskip

\item
If $\m b$ is a 2-element Boolean algebra, then 
$d_{\m b}(n)=\lceil \log_2(n)\rceil$.
\medskip

\item
Finite implication algebras: $\m c\in {\mathcal V}_{\text{fin}}(\langle \{0,1\}; \to\rangle)$
\begin{enumerate}
\item
If $\m c$ has a unique minimal element, then 
$d_{\m c}(n)\in\Theta(\log(n))$.
\item
Otherwise 
$d_{\m c}(n)\in 2^{\Theta(n)}$.
\end{enumerate}
\item
If $M$ is an $R$-module, then $d_M(n)\in\Theta(n)$.
\begin{enumerate}
\item
(Linear upper bound) $M^n=\oplus^n M$, \pause so
$d_{M}(n)\leq d_{M}(1)\cdot n$. \pause
\item
(Linear lower bound) $|M^n|\leq |R^{d_M(n)}|$, 
\pause so $\log_{|R|}(|M|)\cdot n\leq d_M(n)$.
\end{enumerate}
\end{enumerate}


\end{frame}

\begin{frame}[t]
\frametitle{CSP Background, I}
\pause
The Constraint Satisfaction Problem is the problem
of testing whether a relational structure satisfies 
a p.p.\ sentence
$
(\exists \bigwedge \text{atomic}).
$

\medskip
\pause
The ``algebraic approach'' to CSP 
studies the polymorphism algebra, $\m A$,
of the test structure.

\medskip
\pause
The ``Tractability and Learnability'' paper (IMMVW)
proves that the CSP is tractable 
if there is a polynomial $p$ such that every 
\underline{subalgebra}
of $\m A^n$ is generated by at most $p(n)$ elements. \pause 
($\m a$ has {\structure{polynomially generated subpowers}}.)

\medskip
\pause
The ``Few Subpowers'' paper (BIMMVW) proves that $\m a$ has
polynomially generated subpowers 
iff $\m A$ has a {\structure{cube term}}.
\end{frame}

\begin{frame}[t]
\frametitle{CSP Background, II}
\pause
The Quantified Constraint Satisfaction Problem is the problem
of testing whether a relational structure satisfies 
a sentence
$
(Q_1\ldots Q_k \bigwedge \text{atomic}).
$

\medskip
\pause
Hubie Chen showed that if there is a polynomial
$q$ such that $\m a^n$ can be generated by at most $q(n)$
elements (and there is a
pol. time
algorithm for finding the generators), 
then
there is a pol. time reduction of the QCSP
to the ordinary CSP. \pause
($\m a$ has {\structure{polynomially generated powers}}.)
\end{frame}

\begin{frame}[t]
\frametitle{Classical Algebra Background}
\pause
\begin{enumerate}
\item 
(P. Hall, 1936)
If $G$ is a finite simple nonabelian group, then $d_G(n)\leq g$ iff
\[
n \leq \frac{1}{|\Aut(G)|}\sum_{H\leq G} \mu(H)|H|^g.
\]

\item 
(J. Wiegold, 1974) If $G$ is a finite simple nonabelian group, then 
\[
d_{G}(n)\approx \log_{|G|}(n) + \log_{|G|}(|\Aut(G)|).
\]

\item 
(J. Wiegold, 1987) The $d$-function of a finite semigroup $S$ is 
\begin{enumerate}
\item 
{\structure{logarithmic (=$\Theta(\log(n))$)}}
if $S$ is a perfect group ($[S,S]=S$),
\item 
{\structure{linear (=$\Theta(n)$)}}
if $S$ has a unit but is not a perfect group, and
\item 
{\structure{exponential (=$2^{\Theta(n)}$)}}
if $S$ does not have a unit.
\end{enumerate}
\item 
(H. Riedel, 1985)
A finite perfect algebra in a congruence uniform variety
has logarithmic growth (=$d$-function).
\smallskip

\item 
(M. Quick, N. Ru\v{s}kuc, 2010)
If $\m a$ is a finite ring, module, $k$-algebra or Lie algebra,
then $d_{\m a}$ is logarithmic if $\m a$ is perfect and linear
otherwise.
\end{enumerate}
\end{frame}

\begin{frame}[t]
\frametitle{Motivating Questions}

\pause
\begin{question}
(Riedel) Is it true that a finite algebra in a congruence
permutable variety has logarithmic growth if it is perfect
and linear growth if it is imperfect?
\end{question}
\smallskip

\pause
Answer: Yes.

\smallskip

\pause
\begin{question}
(Quick, Ru\v{s}kuc) Must the $d$-function
of a finite algebra be logarithmic, linear or exponential?
\end{question}

\smallskip
\pause
Answer: No.
\end{frame}

\begin{frame}[t]
\frametitle{Basic Facts}

\pause
\begin{enumerate}
\item Any monotone increasing function $f\colon \mathbb Z^+\to \mathbb Z^+$
can be the $d$-function of a countably infinite algebra.
\smallskip

\item
If $\m b$ is an expansion of $\m a$, then 
$d_{\m B}(n)\le d_{\m A}(n)$; \pause
if $\m b$ is the polynomial expansion of $\m a$, then 
$d_{\m A}(n)-d_{\m A}(1)\le d_{\m B}(n)\le d_{\m A}(n)$.
\smallskip
\pause
\item
If $\m a$ is finite, then $\lceil\log_{|A|}(n)\rceil\le d_{\m A}(n)\le |A|^{n}$.
\end{enumerate}
\end{frame}

\begin{frame}[t]
\frametitle{Classifying Algebras By Maltsev Conditions}
\begin{definition}
\begin{enumerate}
\item 
A set $\Sigma$ of identities is \emph{realizable} 
in $\m a$ 
if there is a way to interpret the function symbols 
occurring in $\Sigma$ as terms
in such a way that $\Sigma$ holds in $\m a$.
\item
A term is {\structure{basic}} if it has at most one
nonnullary function symbol. 
\item
An identity is {\structure{basic}} if the terms on both
sides are basic.
\end{enumerate}
\end{definition}

\pause
\begin{exmp}
A group realizes
\begin{enumerate}
\item $\Sigma_1 = \{p(x,y,y)\approx x, p(y,y,x)\approx x\}$.
\item $\Sigma_2 = \{1\cdot x\approx x, x\cdot 1\approx x\}$.
\end{enumerate}
\end{exmp}
\end{frame}

\begin{frame}[t]
\frametitle{Pointed Cube Terms}

\pause
\begin{exmp}
Write $\m a\models F(1,x,2)\approx x$ and 
$\m a\models F(x,y,3)\approx x$ 
as a matrix equation:
\[
\m a\models F\left(
\begin{matrix}
1&x&2\\
x&y&3
\end{matrix}
\right)\approx 
\left(
\begin{matrix}
x\\
x
\end{matrix}
\right)
\]
$F$ is a $3$-pointed, $3$-ary, $2$-cube term for $\m a$.
\end{exmp}

\begin{definition}
$F$ is a $p$-pointed, $m$-ary, $k$-cube term for $\m a$ if 
$\m a\models F(M) \approx 
\left(
\begin{matrix}
x\\
x
\end{matrix}
\right)$ 
where $M$ is a $k\times m$-matrix consisting of $p$ constants
and some variables, and every column of $M$ contains a symbol different
from $x$.
\end{definition}
\end{frame}

\begin{frame}[t]
\frametitle{Some Results}
\begin{theorem}
Let $\Sigma$ be a set of basic identities involving
only finitely many constant symbols. If $\Sigma$
does not entail the existence a pointed cube term, then
realizability of $\Sigma$ imposes no restriction 
on growth functions. \pause(Given any finite $\m b$, there is a finite
$\m a$ realizing $\Sigma$ such that $d_{\m a}=d_{\m b}$.)
\end{theorem}

\pause
\begin{theorem}
If $\Sigma$ is a set of basic identities that
entails the existence a pointed cube term, then
any finite algebra realizing $\Sigma$ has growth
function bounded by a polynomial.
\end{theorem}
\end{frame}

\begin{frame}[t]
\frametitle{$\Sigma$ Restrictive $\Rightarrow$ Pointed Cube Term}
\pause
{\bf Assumption:}
$\Sigma$ (involving finitely many constants) entails no pointed cube term.
\medskip

\pause
{\bf Goal:} 
Given finite $\m b$, $|B|>1$, construct finite
$\m a$ realizing $\Sigma$ such that $d_{\m a}=d_{\m b}$.)
\medskip

\pause
{\bf Stage 1:} 
Show how to enlarge $\m b$ without changing 
$d_{\m b}$.
\pause
This uses the 1-point extension, 
$\m b_0 = \langle \m b\cup \{0\}; \wedge\rangle$:

\pause
\begin{picture}(100,20)
\setlength{\unitlength}{.9mm}
\put(10,-8){%
\begin{picture}(100,20)
\thicklines
\put(50,0){\circle*{1}}
\put(50,10){\oval(60,2)}
\put(53,1){\line(3,1){21}}
\put(47,1){\line(-3,1){21}}
\put(82,9){$\m b$}
\put(49,-4.5){$0$}
\end{picture}}
\end{picture}

\bigskip
\bigskip
\bigskip

\pause
{\bf Stage 2:} 
Show how to construct models of $\Sigma$ of every cardinal
larger than the number of constants in $\Sigma$.

\medskip

\pause
{\bf Stage 3:} 
Show how to merge a sufficiently large model from Stage 1
with a sufficently large model from Stage 2 to form $\m a$
with $d_{\m a}=d_{\m b}$.

\end{frame}

\begin{frame}[t]
\frametitle{Pointed Cube Term $\Rightarrow$ $\Sigma$ Restrictive}
\pause
{\bf Assumption:}
$\Sigma$ entails a pointed cube term.
\medskip

\pause
{\bf Goal:} 
Show that any finite model of $\Sigma$ has polynomially
generated powers.

\medskip

\pause
\begin{exmp2}
Idea: Split coordinates of $[{\mathbf a}]\in \m a^n$
into $k$ equal size segments, 
$\left[
\begin{matrix}
{\mathbf a}_1\\{\mathbf a}_2
\end{matrix}
\right]$. Then 
\[
F\left(
\left[
\begin{matrix}
{\mathbf 1}\\{\mathbf a}_2
\end{matrix}
\right],
\left[
\begin{matrix}
{\mathbf a}_1\\{\mathbf 2}
\end{matrix}
\right],
\left[
\begin{matrix}
{\mathbf 2}\\{\mathbf 3}
\end{matrix}
\right]
\right)=
\left[
\begin{matrix}
{\mathbf a}_1\\{\mathbf a}_2
\end{matrix}
\right]
\]
shows how to generate $[{\mathbf a}]$ from
more ``highly processed'' tuples. \pause

Must show: there are only polynomially many fully processed tuples.
\end{exmp2}
\end{frame}

\begin{frame}[t]
\frametitle{Polynomial Growth of Any Degree}

\begin{exmp}
\pause
Universe: $A = \{a_1,\ldots,a_q,1\}$ \pause

$k$-ary Partial Operation: 
$F(1,x,x,\ldots,x)=F(x,1,x,\ldots,x)=\cdots=F(x,x,x,\ldots,1)=x$ \pause

$\m p = \langle A; F, 1\rangle$ 
\pause

$\m a = \m p_0 = \langle \m p\cup \{0\}; \wedge\rangle$.

\pause
$d_{\m a}(n) = \binom{n}{0}+q\binom{n}{1}+\cdots+q^{k-1}\binom{n}{k-1}$
\end{exmp}

\pause

At present, the only growth rates we know are 
logarithmic, polynomial or exponential.

\end{frame}

\begin{frame}[t]
\frametitle{Ordinary ($0$-Pointed) Cube Terms}

\pause
\begin{theorem}
If $\m a$ is a finite algebra with a cube term, then 
$d_{\m a}(n)$ is logarithmic if $\m a$ is perfect
and linear if $\m a$ is imperfect.
\end{theorem}

\pause 

\medskip
Proof is based on an analysis of maximal subuniverses.

\pause

\medskip

A maximal subuniverse $M$ of $\m b$ is {\structure{induced}}
by surjective $\varphi\colon \m b\to \m c$ if $\varphi(M)\neq C$.
\pause
(Then $M = \varphi^{-1}(\varphi(M))$.)
\medskip

\end{frame}


\begin{frame}[t]
\frametitle{Ordinary Cube Terms $\Rightarrow$ Wiegold Dichotomy: Proof}

\pause

\begin{lemma}
If $\m a$ has a $k$-cube term, then any maximal
subuniverse $M$ of $\m a^n$ is induced by either
\begin{enumerate}
\item a projection $\pi_U\colon \m a^n\to \m a^U$ for some
$U\subseteq \{1,\ldots,n\}$ satisfying $|U|<\max(3,k)$, or
\item the natural map $\nu\colon \m a^n\to (\m a/[1,1])^n$.
\end{enumerate}
\end{lemma}


\begin{lemma}
\begin{enumerate}
\item
$\m a^n$ has a subset of size $\Theta(\log(n))$ 
that is not contained in any maximal subuniverse
induced by a projection $\pi_U\colon \m a^n\to \m a^U$ for 
$U\subseteq \{1,\ldots,n\}$ satisfying $|U|<\max(3,k)$.
\item 
$\m a^n$ has a subset of size $\Theta(n)$ 
that is not contained in any maximal subuniverse
induced by the natural map $\nu\colon \m a^n\to (\m a/[1,1])^n$.
\end{enumerate}
\end{lemma}
\end{frame}

\begin{frame}[t]
\frametitle{Things We Don't Know}
\begin{enumerate}
\pause
\item
Is there a finite algebra with sublinear growth rate 
that is not logarithmic?
\pause
\item
If finite $\m a$ has a polynomial $B(x,y)$ with a 2-sided unit
($B(1,x)\approx x\approx B(x,1)$), must the growth rate of
$\m a$ be linear or logarithmic?
\pause
\item
Is there a nice description of finite algebras with 
polynomially generated powers?

\end{enumerate}
\end{frame}
\end{document}

\begin{frame}[t]
\frametitle{Avoiding Subuniverses of Small Support}
Let $A$ by a finite set of size $a$.

\pause
Let $G\subseteq A^n$, $|G|=g$, be such that its projection
onto any $k$ coordinates is surjective. \pause
How small can $g$ be?

\pause
Write $G$ as a $g\times n$-matrix of elements of $A$:
$
G = \left[
\begin{matrix}
-{\bf a}_1-\\
\vdots\\
-{\bf a}_g-\\
\end{matrix}
\right]
$

\pause
Want the rows of any $g\times k$-minor of $G$ to exhaust $A^k$.
\pause
Else the minor is a {\structure{bad minor}}.
\end{frame}
\begin{frame}[t]
\frametitle{Probabilistic Argument}
\pause
Let $S$ be a sample space of all $g\times n$-matrices of elements of $A$.

\pause
For each sequence $\sigma$,  $1\leq i_1 < \cdlots < i_k\leq n$,
let $X_{\sigma}$ be the random variable defined on $S$ that
takes the value $1$ if the $g\times k$-minor determined
by $\sigma$ is bad, else takes value $0$.

\pause


\end{frame}
\end{document}

