small changes

slides/bibliography.bib

@@ -80,16 +80,19 @@
   numpages   = {38}
 }
 
-@article{quotienting,
-  title     = {Quotienting the delay monad by weak bisimilarity},
-  volume    = {29},
-  doi       = {10.1017/S0960129517000184},
-  number    = {1},
-  journal   = {Mathematical Structures in Computer Science},
-  publisher = {Cambridge University Press},
-  author    = {CHAPMAN, JAMES and UUSTALU, TARMO and VELTRI, NICCOLÒ},
-  year      = {2019},
-  pages     = {67–92}
+@inproceedings{quotienting,
+author = {Chapman, James and Uustalu, Tarmo and Veltri, Niccol\`{o}},
+title = {Quotienting the Delay Monad by Weak Bisimilarity},
+year = {2015},
+isbn = {9783319251493},
+publisher = {Springer-Verlag},
+address = {Berlin, Heidelberg},
+url = {https://doi.org/10.1007/978-3-319-25150-9_8},
+doi = {10.1007/978-3-319-25150-9_8},
+abstract = {The delay datatype was introduced by Capretta [3] as a means to deal with partial functions as in computability theory in Martin-L\"{o}f type theory. It is a monad and it constitutes a constructive alternative to the maybe monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay monad quotiented by weak bisimilarity is still a monad. In this paper, we consider Hofmann's alternative approach [6] of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the semi-classical axiom of countable choice. We have fully formalized our results in the Agda dependently typed programming language.},
+booktitle = {Proceedings of the 12th International Colloquium on Theoretical Aspects of Computing - ICTAC 2015 - Volume 9399},
+pages = {110–125},
+numpages = {16}
 }
 
 @article{restriction,

slides/code-examples/example.hs

@@ -1,5 +1,4 @@
 hd :: [a] -> a
-hd [] = error "empty list"
 hd (x : _) = x
 
 main :: IO ()

slides/code-examples/reverse.agda

@@ -1,10 +1,6 @@
 {-# OPTIONS --guardedness #-}
 open import Codata.Musical.Notation
 open import Data.Nat
-open import Data.Nat.Show renaming (show to showℕ)
-open import Function.Base
-open import Relation.Binary.PropositionalEquality
-open import Data.String using (String; _++_)
 
 module reverse where
   data Delay (A : Set) : Set where
@@ -14,7 +10,7 @@ module reverse where
   run_for_steps : ∀ {A : Set} → Delay A → ℕ → Delay A
   run now x for n steps = now x
   run later x for zero steps = later x
-  run later x for suc n steps = later (♯ (run ♭ x for n steps))
+  run later x for suc n steps = run ♭ x for n steps
 
   never : ∀ {A : Set} → Delay A
   never = later (♯ never)
@@ -24,7 +20,7 @@ module reverse where
     _∷_ : A → ∞ (Colist A) → Colist A
 
   reverse : ∀ {A : Set} → Colist A → Delay (Colist A)
-  reverse {A} = reverseAcc []
+  reverse {A} l = reverseAcc l []
     where
       reverseAcc : Colist A → Colist A → Delay (Colist A)
       reverseAcc [] a = now a

slides/main.tex

@@ -17,8 +17,8 @@
 			%ThirdLogo = template-art/FAUWortmarkeBlau.pdf,
 			%WordMark=None,
 			aspectratio=169,
-			fontsize=11,
-			fontbaselineskip=15,
+			fontsize=11.5,
+			fontbaselineskip=14,
 			scale=1.,
 			InsertTotalFoot
 		   ]{styles/beamerthemefau}
@@ -199,4 +199,6 @@ at (#3) {#4};
 		\printbibliography[heading=none]
 	\end{frame}
 
+	\input{sections/03_app.tex}
+
 \end{document}

slides/sections/00_intro.tex

@@ -1,20 +1,17 @@
 \section{Partiality in Type Theory}
 
 \begin{frame}[t, fragile]{Partiality in Haskell}{}
-  Haskell allows users to define arbitrary partial functions
+  In Haskell we are able to define arbitrary partial functions
   \begin{itemize}
     \item<1-> Some can be spotted easily by their definition:
     \vskip 0.5cm
     \begin{minted}{haskell}
       head :: [a] -> a
-      head []     = error "empty list"
       head (x:xs) = x
     \end{minted}
-      \mycallout<2->{22, 1.5}{
+      \mycallout<2->{22.5, 1.5}{
         ghci> head []\\
-        *** Exception: empty list\\
-        CallStack (from HasCallStack):\\
-        error, called at example.hs:2:9 in main:Main
+        Exception: code-examples/example.hs:2:1-14: Non-exhaustive patterns in function head
       }
     \item<3->
     others might be more subtle:
@@ -27,7 +24,7 @@
           revAcc (x:xs) a = revAcc xs (x:a)
     \end{minted}
     
-    \mycallout<4->{22, 2}{
+    \mycallout<4->{22.5, 2}{
     ghci> ones = 1 : ones\\
     ghci> reverse ones\\
     ...
@@ -99,7 +96,7 @@
   \item Now we can define a reverse function for possibly infinite lists:
   \begin{minted}{agda}
     reverse : ∀ {A : Set} → Colist A → Delay (Colist A)
-    reverse {A} = revAcc []
+    reverse {A} l = revAcc l []
       where
         revAcc : Colist A → Colist A → Delay (Colist A)
         revAcc []       a = now a

slides/sections/01_abstracting.tex

@@ -1,25 +1,6 @@
 \section{Categorical Notions of Partiality}
 
-\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Moggi's categorical semantics~\cite{moggi}}
-Goal: interpret an effectul programming language in a category $\mathcal{C}$
-
-\begin{itemize}
-  \item<2-> Take a Monad $T$ on $\mathcal{C}$, then values are denoted by objects $A$ and computations by $TA$
-  \item<3-> Programs form a category $\mathcal{C}_T$ with $\mathcal{C}_T(X,Y) := \mathcal{C}(X, TY)$
-\end{itemize}
-
-\onslide<4->
-What properties should a monad $T$ for modelling partiality have?
-
-\begin{enumerate}
-  \item<5-> Commutativity (also entails strength)
-  \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}
+\begin{frame}[t, fragile]{Preliminaries}
   We work in category $\mathcal{C}$ that
   \begin{itemize}[<+->]
     \item has finite products
@@ -40,9 +21,28 @@ What properties should a monad $T$ for modelling partiality have?
   
 \end{frame}
 
+\begin{frame}[t, fragile]{Capturing Partiality}{Moggi's categorical semantics~\cite{moggi}}
+Goal: interpret an effectul programming language in a category $\mathcal{C}$
+
+\begin{itemize}
+  \item<2-> Take a Monad $T$ on $\mathcal{C}$, we view objects $A$ as types of values and objects $TA$ as types of computations.
+  \item<3-> Programs form a category $\mathcal{C}_T$ with $\mathcal{C}_T(X,Y) := \mathcal{C}(X, TY)$
+\end{itemize}
+
+\onslide<4->
+What properties should a monad $T$ for modelling partiality have?
+
+\begin{enumerate}
+  \item<5-> Commutativity (also entails strength)
+  \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}
+
 \newcommand{\tdom}{\text{dom}\;}
 
-\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Restriction Categories~\cite{restriction}}
+\begin{frame}[t, fragile]{Capturing Partiality}{Restriction Categories~\cite{restriction}}
 \begin{definition}<1->
   A restriction structure on $\mathcal{C}$ is a mapping $\tdom : \mathcal{C}(X,Y) \rightarrow \mathcal{C}(X,X)$ with the following properties:
   \begin{alignat}{1}
@@ -62,7 +62,7 @@ Intuitively $\tdom f$ captures the domain of definedness of $f$.
 \end{block}
 \end{frame}
 
-\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Equational Lifting Monads~\cite{eqlm}}
+\begin{frame}[t, fragile]{Capturing Partiality}{Equational Lifting Monads~\cite{eqlm}}
   The following criterion guarantees that some form of partiality is the only possible side-effect:
   \pause
   \begin{definition}
@@ -84,7 +84,7 @@ Intuitively $\tdom f$ captures the domain of definedness of $f$.
   \end{theorem}
 \end{frame}
 
-\begin{frame}[t, fragile]{Capturing Partiality Categorically}{The Maybe Monad}
+\begin{frame}[t, fragile]{Capturing Partiality}{The Maybe Monad}
 \begin{itemize}[<+->]
   \item $MX = X + 1$
   \item The maybe monad is strong and commutative:
@@ -106,7 +106,7 @@ Intuitively $\tdom f$ captures the domain of definedness of $f$.
 \end{itemize}
 \end{frame}
 
-\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Capretta's Delay Monad~\cite{delay}}
+\begin{frame}[t, fragile]{Capturing Partiality}{Capretta's Delay Monad~\cite{delay}}
 \begin{itemize}[<+->]
   \item Recall the delay codatatype:
     
@@ -125,7 +125,7 @@ Intuitively $\tdom f$ captures the domain of definedness of $f$.
 \end{itemize}
 \end{frame}
 
-\begin{frame}[t, fragile]{Capturing Partiality Categorically}{Quotienting the Delay Monad~\cite{quotienting}}
+\begin{frame}[t, fragile]{Capturing Partiality}{Quotienting the Delay Monad~\cite{quotienting}}
   Following the work by Chapman, Uustalu and Veltri we can quotient $D$ by the 'correct' kind of equality:
   \[\mprset{fraction={===}}
   \inferrule {p \downarrow c \\ q \downarrow c} {p \approx q}\hskip 2cm

slides/sections/02_goals.tex

@@ -12,28 +12,8 @@
       \item strong
       \item commutative
       \item an equational lifting monad
+      \item the initial pre-Elgot monad
     \end{itemize}
     \item Take the category of setoids and show that $K$ instantiates to $D_\approx$
   \end{itemize}
-\end{frame}
-
-\begin{frame}[t, fragile, blank]{Category Theory in Agda}{Setoid-enriched Categories}
-  \begin{minted}{agda}
-record Category (o ℓ e : Level) : Set (suc (o ⊔ ℓ ⊔ e)) where
-  field
-    Obj : Set o
-    _⇒_ : Obj → Obj → Set ℓ
-    _≈_ : ∀ {A B} → (A ⇒ B) → (A ⇒ B) → Set e
-
-    id  : ∀ {A} → (A ⇒ A)
-    _∘_ : ∀ {A B C} → (B ⇒ C) → (A ⇒ B) → (A ⇒ C)
-
-  field
-    assoc     : ∀ {A B C D} {f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} 
-      → (h ∘ g) ∘ f ≈ h ∘ (g ∘ f)
-    identityˡ : ∀ {A B} {f : A ⇒ B} → id ∘ f ≈ f
-    identityʳ : ∀ {A B} {f : A ⇒ B} → f ∘ id ≈ f
-    equiv     : ∀ {A B} → IsEquivalence (_≈_ {A} {B})
-    ∘-resp-≈  : ∀ {A B C} {f h : B ⇒ C} {g i : A ⇒ B} → f ≈ h → g ≈ i → f ∘ g ≈ h ∘ i
-  \end{minted}
 \end{frame}

slides/sections/03_app.tex

@@ -0,0 +1,21 @@
+\section*{Appendix}
+\begin{frame}[t, fragile, blank]{App: Category Theory in Agda}{Setoid-enriched Categories}
+  \begin{minted}{agda}
+record Category (o ℓ e : Level) : Set (suc (o ⊔ ℓ ⊔ e)) where
+  field
+    Obj : Set o
+    _⇒_ : Obj → Obj → Set ℓ
+    _≈_ : ∀ {A B} → (A ⇒ B) → (A ⇒ B) → Set e
+
+    id  : ∀ {A} → (A ⇒ A)
+    _∘_ : ∀ {A B C} → (B ⇒ C) → (A ⇒ B) → (A ⇒ C)
+
+  field
+    assoc     : ∀ {A B C D} {f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} 
+      → (h ∘ g) ∘ f ≈ h ∘ (g ∘ f)
+    identityˡ : ∀ {A B} {f : A ⇒ B} → id ∘ f ≈ f
+    identityʳ : ∀ {A B} {f : A ⇒ B} → f ∘ id ≈ f
+    equiv     : ∀ {A B} → IsEquivalence (_≈_ {A} {B})
+    ∘-resp-≈  : ∀ {A B C} {f h : B ⇒ C} {g i : A ⇒ B} → f ≈ h → g ≈ i → f ∘ g ≈ h ∘ i
+  \end{minted}
+\end{frame}