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56
slides/bibliography.bib
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@ -1,3 +1,45 @@
@article{delay,
author = {Venanzio Capretta},
title = {General Recursion via Coinductive Types},
journal = {CoRR},
volume = {abs/cs/0505037},
year = {2005},
url = {http://arxiv.org/abs/cs/0505037},
eprinttype = {arXiv},
eprint = {cs/0505037},
timestamp = {Mon, 13 Aug 2018 16:46:14 +0200},
biburl = {https://dblp.org/rec/journals/corr/abs-cs-0505037.bib},
bibsource = {dblp computer science bibliography, https://dblp.org}
}
@article{elgotalgebras,
author = {Jir{\'{\i}} Ad{\'{a}}mek and
Stefan Milius and
Jir{\'{\i}} Velebil},
title = {Elgot Algebras},
journal = {CoRR},
volume = {abs/cs/0609040},
year = {2006},
url = {http://arxiv.org/abs/cs/0609040},
eprinttype = {arXiv},
eprint = {cs/0609040},
timestamp = {Mon, 04 Sep 2023 12:29:24 +0200},
biburl = {https://dblp.org/rec/journals/corr/abs-cs-0609040.bib},
bibsource = {dblp computer science bibliography, https://dblp.org}
}
@article{elgotmonad,
title = {Elgot theories: a new perspective on the equational properties of iteration},
volume = {21},
doi = {10.1017/S0960129510000496},
number = {2},
journal = {Mathematical Structures in Computer Science},
publisher = {Cambridge University Press},
author = {ADÁMEK, JIŘÍ and MILIUS, STEFAN and VELEBIL, JIŘÍ},
year = {2011},
pages = {417–480}
}
@article{eqlm,
author = {Bucalo, Anna and F\"{u}hrmann, Carsten and Simpson, Alex},
title = {An Equational Notion of Lifting Monad},
@ -65,4 +107,18 @@
month = {1},
pages = {223–259},
numpages = {37}
}
@article{uniformelgot,
author = {Sergey Goncharov},
title = {Uniform Elgot Iteration in Foundations},
journal = {CoRR},
volume = {abs/2102.11828},
year = {2021},
url = {https://arxiv.org/abs/2102.11828},
eprinttype = {arXiv},
eprint = {2102.11828},
timestamp = {Fri, 26 Feb 2021 14:31:25 +0100},
biburl = {https://dblp.org/rec/journals/corr/abs-2102-11828.bib},
bibsource = {dblp computer science bibliography, https://dblp.org}
}

3
slides/main.tex
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@ -7,7 +7,7 @@
% of the GNU Public License, version 2.
%
% ------------------------------------------------------------------------------
\documentclass[final]{beamer}
\documentclass[final, handout]{beamer}
% ========================================================================================
% Theme: inner, outer, font and colors
@ -159,6 +159,7 @@ Leon Vatthauer%\inst{1}
\usepackage{tikz}
\usepackage{tikz-cd}
\usepackage{quiver}
\usetikzlibrary{shapes.callouts}
\usepackage{mathpartir}

40
slides/quiver.sty Normal file
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@ -0,0 +1,40 @@
% *** quiver ***
% A package for drawing commutative diagrams exported from https://q.uiver.app.
%
% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ
% libraries, and defining a new TikZ style for curves of a fixed height.
%
% Version: 1.4.2
% Authors:
% - varkor (https://github.com/varkor)
% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c)
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{quiver}[2021/01/11 quiver]
% `tikz-cd` is necessary to draw commutative diagrams.
\RequirePackage{tikz-cd}
% `amssymb` is necessary for `\lrcorner` and `\ulcorner`.
\RequirePackage{amssymb}
% `calc` is necessary to draw curved arrows.
\usetikzlibrary{calc}
% `pathmorphing` is necessary to draw squiggly arrows.
\usetikzlibrary{decorations.pathmorphing}
% A TikZ style for curved arrows of a fixed height, due to AndréC.
\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
.. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
.. (\tikztotarget)\tikztonodes}},
settings/.code={\tikzset{quiver/.cd,#1}
\def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
quiver/.cd,pos/.initial=0.35,height/.initial=0}
% TikZ arrowhead/tail styles.
\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}}
\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}}
\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}}
% TikZ arrow styles.
\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}}
\endinput

91
slides/sections/01_abstracting.tex
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@ -12,27 +12,48 @@ Goal: interpret an effectul programming language in a category $\mathcal{C}$
What properties should a monad $T$ for modelling partiality have?
\begin{enumerate}
\item<5-> Commutativity (also entails strength), i.e. the following programs should yield equal results:
\begin{multicols}{2}
\begin{minted}{haskell}
do x <- p
y <- q
return (x, y)
\end{minted}
\item<5-> Commutativity (also entails strength)
% , i.e. the following programs should yield equal results:
% \begin{multicols}{2}
% \begin{minted}{haskell}
% do x <- p
% y <- q
% return (x, y)
% \end{minted}
\begin{minted}{haskell}
do y <- q
x <- p
return (x, y)
\end{minted}
\end{multicols}
where p and q are programs.
% \begin{minted}{haskell}
% do y <- q
% x <- p
% return (x, y)
% \end{minted}
% \end{multicols}
% where p and q are programs.
\item<6-> Morphisms in $\mathcal{C}_T$ should be partial maps
\item<7-> There should be no other effect besides partiality
\end{enumerate}
\end{frame}
\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Preliminaries}
We work in category $\mathcal{C}$ that
\begin{itemize}
\item has finite products
\item has finite coproducts
\item is distributive, i.e. the following is an iso:
% https://q.uiver.app/#q=WzAsMixbMCwwLCIoWCBcXHRpbWVzIFkpICsgKFggXFx0aW1lcyBaKSJdLFs0LDAsIlggXFx0aW1lcyAoWSArIFopIl0sWzAsMSwiZHN0bF57LTF9ICA6PSBbIFxcbGFuZ2xlIGlkICwgaW5sIFxccmFuZ2xlICwgXFxsYW5nbGUgaWQgLCBpbnIgXFxyYW5nbGVdIl1d
\[\begin{tikzcd}
{(X \times Y) + (X \times Z)} &&&& {X \times (Y + Z)}
\arrow["{dstl^{-1} := [ \langle id , inl \rangle , \langle id , inr \rangle]}", from=1-1, to=1-5]
\end{tikzcd}\]
% https://q.uiver.app/#q=WzAsMixbMCwwLCJYIFxcdGltZXMgKFkgKyBaKSJdLFs0LDAsIihYIFxcdGltZXMgWSkgKyAoWCBcXHRpbWVzIFopIl0sWzAsMSwiZHN0bCJdXQ==
\[\begin{tikzcd}
{X \times (Y + Z)} &&&& {(X \times Y) + (X \times Z)}
\arrow["dstl", from=1-1, to=1-5]
\end{tikzcd}\]
\end{itemize}
\end{frame}
\newcommand{\tdom}{\text{dom}\;}
\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Restriction Categories~\cite{restriction}}
@ -48,7 +69,7 @@ What properties should a monad $T$ for modelling partiality have?
\end{definition}
\onslide<2->
Intuitively $\tdom f$ captures the domain of definiteness of $f$.
Intuitively $\tdom f$ captures the domain of definedness of $f$.
\begin{block}{Remark}<3->
Every category has a trivial restriction structure $\tdom f = id$, we call categories with a non-trivial restriction structure \textit{restriction categories}.
@ -80,7 +101,7 @@ Intuitively $\tdom f$ captures the domain of definiteness of $f$.
% write on board:
% equational lifting: do x <- p; return (return x, x) = do x <- p; return (p, x)
\begin{frame}[t, fragile]{The Maybe Monad}
\begin{frame}[t, fragile]{Capturing Partiality Categorically}{The Maybe Monad}
\begin{itemize}[<+->]
\item $MX = X + 1$
\item on a distributive category the maybe monad is strong and commutative:
@ -102,7 +123,7 @@ Intuitively $\tdom f$ captures the domain of definiteness of $f$.
\end{itemize}
\end{frame}
\begin{frame}[t, fragile]{The Delay Monad}
\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Capretta's Delay Monad~\cite{delay}}
\begin{itemize}[<+->]
\item Recall the delay codatatype:
@ -140,7 +161,7 @@ Intuitively $\tdom f$ captures the domain of definiteness of $f$.
% \end{frame}
%%%%%%%
\begin{frame}[t, fragile]{Partiality from Iteration}{Elgot Algebras}
\begin{frame}[t, fragile]{Partiality from Iteration}{Elgot Algebras~\cite{elgotalgebras}}
The following is an adaptation of Ad\'amek, Milius and Velebil's \textit{complete Elgot Algebras}:
\begin{definition}
A (unguarded) Elgot Algebra consists of:
@ -162,14 +183,38 @@ The following is an adaptation of Ad\'amek, Milius and Velebil's \textit{complet
\end{block}
\end{frame}
\begin{frame}[t, fragile]{Partiality from Iteration}{Elgot Monads~\cite{elgotmonad}}
Recall the following notion:
\begin{definition}
A monad $\mathbf{T}$ is an Elgot monad if it has an iteration operator $(f : X \rightarrow T(Y + X))^\dagger : X \rightarrow TY$ satisfying:
\begin{itemize}
\item \textbf{Fixpoint}:
\item \textbf{Naturality}:
\item \textbf{Codiagonal}:
\item \textbf{Uniformity}:
\end{itemize}
\end{definition}
\begin{block}{Remark}
Strong Elgot Monads are regarded as minimal semantic structures for interpreting effectful while-languages.
\end{block}
\end{frame}
% TODO
% maybe dont talk about elgot monads at all, give intuition for the initial pre Elgot monad.
\begin{frame}[t, fragile]{Partiality from Iteration}{pre-Elgot Monads}
TODO
\begin{frame}[t, fragile]{Partiality from Iteration}{pre-Elgot Monads~\cite{uniformelgot}}
Relaxing the requirements for Elgot monads we get the following weaker concept:
\begin{definition}
A monad $\mathbf{T}$ is called pre-Elgot if every $TX$ extends to an Elgot algebra such that Kleisli lifting is iteration preversing, i.e.
\[h^* \circ f^\# = ((h^* + id) \circ f)^\#\qquad \text{for}\; f : Z \rightarrow TX + Z, h : X \rightarrow TY\]
\end{definition}
\begin{theorem}
Every Elgot monad is pre-Elgot
\end{theorem}
\end{frame}
\begin{frame}[t, fragile]{Partiality from iteration}{The K Monad}
\begin{frame}[t, fragile]{Partiality from iteration}{The initial pre-Elgot Monad~\cite{uniformelgot}}
\begin{itemize}[<+->]
\item By defining $KX$ as the free Elgot algebra over $X$ we get a monad $K$
\item $K$ is strong and commutative
@ -178,6 +223,10 @@ The following is an adaptation of Ad\'amek, Milius and Velebil's \textit{complet
\end{itemize}
\end{frame}
\begin{frame}[t, fragile]{Partiality from iteration}{Closing the gap}
...
\end{frame}
%% TODOs:
% cite stefan