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@@ -232,10 +232,10 @@
expressiveness that affirms the existence of encodings between expressiveness that affirms the existence of encodings between
handlers, but it provides no information about the computational handlers, but it provides no information about the computational
content of the encodings. Second, using the semantic notion of content of the encodings. Second, using the semantic notion of
\emph{type-respecting expressiveness} I show that for a class of expressiveness I show that for a class of programs a programming
programs a programming language with first-class control language with first-class control (e.g. effect handlers) admits
(e.g. effect handlers) admits asymptotically faster implementations asymptotically faster implementations than possible in a language
than possible in a language without first-class control. without first-class control.
% %
} }
@@ -1937,8 +1937,12 @@ deep handlers, which provide a form of lexical scoping for effect
operations, thus statically binding them to their handlers. operations, thus statically binding them to their handlers.
% %
\citeauthor{Geron19}'s PhD dissertation develops the mathematical \citeauthor{Geron19}'s PhD dissertation develops the mathematical
theory of scoped effect operations, whilst \citet{WuSH14} and theory of scoped effect operations, whilst \citet{BiernackiPPS20}
\citet{BiernackiPPS20} study them from a programming perspective. study them in conjunction with ordinary handlers from a programming
perspective.
% \citet{WuSH14} study scoped effects, which are effects whose payloads
% are effectful computations
% Effect handlers were conceived in the realm of category theory to give % Effect handlers were conceived in the realm of category theory to give
% an algebraic treatment of exception handling~\cite{PlotkinP09}. They % an algebraic treatment of exception handling~\cite{PlotkinP09}. They
@@ -1950,13 +1954,13 @@ either added language-level support for handlers~
or embedded them in or embedded them in
libraries~\cite{KiselyovSS13,KiselyovI15,KiselyovS16,KammarLO13,BrachthauserS17,Brady13,XieL20}. Thus libraries~\cite{KiselyovSS13,KiselyovI15,KiselyovS16,KammarLO13,BrachthauserS17,Brady13,XieL20}. Thus
functional perspectives on effect handlers are plentiful in the functional perspectives on effect handlers are plentiful in the
literature. There are only a a few takes on effect handlers outside literature. Some notable examples of perspectives on effect handlers
functional programming: \citeauthor{Brachthauser20}'s PhD dissertation outside functional programming are: \citeauthor{Brachthauser20}'s PhD
contains an object-oriented perspective on effect handlers in dissertation, which contains an object-oriented perspective on effect
Java~\cite{Brachthauser20}; \citeauthor{Saleh19}'s PhD dissertation handlers in Java~\cite{Brachthauser20}; \citeauthor{Saleh19}'s PhD
offers a logic programming perspective via an effect handlers dissertation offers a logic programming perspective via an effect
extension to Prolog; and \citet{Leijen17b} has an imperative take on handlers extension to Prolog; and \citet{Leijen17b} has an imperative
effect handlers in C. take on effect handlers in C.
\section{Contributions} \section{Contributions}
The key contributions of this dissertation are scattered across the The key contributions of this dissertation are scattered across the
@@ -2092,15 +2096,15 @@ recommend \citeauthor{Harper16}'s book \emph{Practical Foundations for
Programming Languages}~\cite{Harper16} (do not let the ``practical'' Programming Languages}~\cite{Harper16} (do not let the ``practical''
qualifier deceive you) --- the two books complement each other nicely. qualifier deceive you) --- the two books complement each other nicely.
\section{Relations and functions} \section{Relations}
\label{sec:functions} \label{sec:relations}
Relations and functions feature prominently in the design and Relations feature prominently in the design and understanding of the
understanding of the static and dynamic properties of programming static and dynamic properties of programming languages. The interested
languages. The interested reader is likely to already be familiar with reader is likely to already be familiar with the basic concepts of
the basic concepts of relations and functions, although this section relations, although this section briefly introduces the concepts, its
briefly introduces the concepts, its purpose is to introduce the real purpose is to introduce the notation that I am using pervasively
notation that I am using pervasively throughout this dissertation. throughout this dissertation.
% %
I assume familiarity with basic set theory. I assume familiarity with basic set theory.
@@ -2123,7 +2127,7 @@ this raises the question in which order the product operator
taken to be right associative, meaning taken to be right associative, meaning
$A \times B \times C = A \times (B \times C)$. $A \times B \times C = A \times (B \times C)$.
% %
\dhil{Define tuples (and ordered pairs)?} %\dhil{Define tuples (and ordered pairs)?}
% %
% To make the notation more compact for the special case of $n$-fold % To make the notation more compact for the special case of $n$-fold
% product of some set $A$ with itself we write % product of some set $A$ with itself we write
@@ -2138,7 +2142,7 @@ $A \times B \times C = A \times (B \times C)$.
An element $a \in A$ is related to an element $b \in B$ if An element $a \in A$ is related to an element $b \in B$ if
$(a, b) \in R$, sometimes written using infix notation $a\,R\,b$. $(a, b) \in R$, sometimes written using infix notation $a\,R\,b$.
% %
If $A = B$ then $R$ is said to be a homogeneous relation. If $A = B$ then $R$ is said to be a \emph{homogeneous} relation.
\end{definition} \end{definition}
% %
@@ -2218,7 +2222,10 @@ related to one $b \in B$. Note this does not mean that every $a$
\emph{is} related to some $b$. The serial property guarantees that \emph{is} related to some $b$. The serial property guarantees that
every $a \in A$ is related to one or more elements in $B$. every $a \in A$ is related to one or more elements in $B$.
% %
We use these properties to define partial and total functions.
\section{Functions}
\label{sec:functions}
We define partial and total functions in terms of relations.
% %
\begin{definition} \begin{definition}
A partial function $f : A \pto B$ is a functional relation A partial function $f : A \pto B$ is a functional relation
@@ -2259,7 +2266,6 @@ written $\dec{Im}(f)$, is the set of values that it can return, i.e.
\dec{Im}(f) \defas \{\, f(a) \mid a \in \dom(f) \}. \dec{Im}(f) \defas \{\, f(a) \mid a \in \dom(f) \}.
\] \]
\begin{definition} \begin{definition}
A function $f : A \to B$ is injective and surjective if it satisfies A function $f : A \to B$ is injective and surjective if it satisfies
the following criteria, respectively. the following criteria, respectively.
@@ -2330,242 +2336,242 @@ of growth of $f$ is constant.
\] \]
\end{definition} \end{definition}
\section{Typed programming languages} % \section{Typed programming languages}
\label{sec:pls} % \label{sec:pls}
We will be working mostly with statically typed programming % We will be working mostly with statically typed programming
languages. The following definition informally describes the core % languages. The following definition informally describes the core
components used to construct a statically typed programming language. % components used to construct a statically typed programming language.
% % %
The objective here is not to be mathematical % The objective here is not to be mathematical
rigorous.% but rather to give % rigorous.% but rather to give
% an idea of what constitutes a programming language. % % an idea of what constitutes a programming language.
% % %
\begin{definition}
A statically typed programming language $\LLL$ consists of a syntax
$S$, static semantics $T$, and dynamic semantics $E$ where
\begin{itemize}
\item $S$ is a collection of, possibly mutually, inductively defined
syntactic categories (e.g. terms, types, and kinds). Each
syntactic category contains a collection of syntax constructors
with nonnegative arities $\{(\SC_i,\ar_i)\}$ which construct
abstract syntax trees. We insist that $S$ contains at least two
syntactic categories for constructing terms and types of
$\LLL$. We write $\Tm(S)$ for the terms category and $\Ty(S)$ for
the types category;
\item $T : \Tm(S) \times \Ty(S) \to \B$ is \emph{typechecking}
function, which decides whether a given term is well-typed; and
\item $E : P \pto R$ is an \emph{evaluation} function, which maps
well-typed programs
\[
P \defas \{ M \in \Tm(S) \mid A \in \Ty(S), T(M, A)~\text{is true} \},
\]
to some unspecified set of answers $R$.
\end{itemize}
\end{definition}
%
We will always present the syntax of a programming language on
Backus-Naur form.
%
The static semantics will always be given in form a typing judgement
(and a kinding judgement for languages with polymorphism), and the
dynamic semantics will be given as either a reduction relation or an
abstract machine.
We will take the view that an untyped programming language is a
special instance of a statically typed programming language, where the
typechecking function $T$ is the constant function that always returns
true.
Often we will build programming languages incrementally by starting
from a base language and extend it with new facilities. A
\emph{conservative extension} is a particularly well-behaved extension
in the sense that it preserves the all of the behaviour of the
original language.
%
\begin{definition}
A programming language $\LLL = (S, T, E)$ is said to be a
\emph{conservative extension} of a language $\LLL' = (S', T', E')$
if the following conditions are met.
\begin{itemize}
\item $\LLL$ syntactically extends $\LLL'$, i.e. $S'$ is a proper
subset of $S$.
\item $\LLL$ preserves the static semantics and dynamic semantics
of $\LLL'$, that is $T(M, A) = T'(M, A)$ and $E(M)$ holds if and
only if $E'(M)$ holds for all types $A \in \Ty(S)$ and programs
$M \in \Tm(S)$.
\end{itemize}
Conversely, $\LLL'$ is a \emph{conservative restriction} of $\LLL$.
\end{definition}
We will often work with translations on syntax between languages. It
is often the case that a syntactic translation is \emph{homomorphic}
on most syntax constructors, which is a technical way of saying it
does not perform any interesting transformation on those
constructors. Therefore we will omit homomorphic cases in definitions
of translations.
%
\begin{definition}
Let $\LLL = (S, T, E)$ and $\LLL' = (S', T', E')$ be programming
languages. A translation $\sembr{-} : S \to S'$ on the syntax
between $\LLL$ and $\LLL'$ is a homomorphism if it distributes over
the syntax constructors, i.e. for every $\{(\SC_i,\ar_i)\}_i \in S$
\[
\sembr{\SC_i(t_1,\dots,t_{\ar_i})} = \sembr{\SC_i}(\sembr{t_1},\cdots,\sembr{t_{\ar_i}}),
\]
%
where $t_1,\dots,t_{\ar_i}$ are abstract syntax trees. We say that
$\sembr{-}$ is homomorphic on $S_i$.
\end{definition}
We will be using a typed variation of \citeauthor{Felleisen91}'s
macro-expressivity to compare the relative expressiveness of different
languages~\cite{Felleisen90,Felleisen91}. Macro-expressivity is a
syntactic notion based on the idea of macro rewrites as found in the
programming language Scheme~\cite{SperberDFSFM10}.
%
Informally, if $\LLL$ admits a translation into a sublanguage $\LLL'$
in a way which respects not only the behaviour of programs but also
their local syntactic structure, then $\LLL'$ macro-expresses
$\LLL$. % If the translation of some $\LLL$-program into $\LLL'$
% requires a complete global restructuring, then we may say that $\LLL'$
% is in some way less expressive than $\LLL$.
%
\begin{definition}[Typability-preserving macro-expressiveness]
Let both $\LLL = (S, T, E)$ and $\LLL' = (S', T', E')$ be
programming languages such that the syntax constructors
$\SC_1,\dots,\SC_k$ are unique to $\LLL$. If there exists a
translation $\sembr{-} : S \to S'$ on the syntax of $\LLL$ that is
homomorphic on all syntax constructors but $\SC_1,\dots,\SC_k$, such
that for all $A \in \Ty(S)$ and $M \in \Tm(S)$
%
\[
T(M,A) = T'(\sembr{M},\sembr{A})~\text{and}~
E(M)~\text{holds if and only if}~E'(\sembr{M})~\text{holds},
\]
%
then we say that $\LLL'$ can \emph{macro-express} the
$\SC_1,\dots,\SC_k$ facilities of $\LLL$.
\end{definition}
%
In general, it is not the case that $\sembr{-}$ preserves types, it is
only required to ensure that the translated term is typable in the
target language $\LLL'$.
\begin{definition}[Type-respecting expressiveness]
Let $\LLL = (S,T,E)$ and $\LLL' = (S',T',E')$ be programming
languages and $A$ be a type such that $A \in \Ty(S)$ and
$A \in \Ty(S')$.
\end{definition}
% \dhil{Definition of (typed) programming language, conservative extension, macro-expressiveness~\cite{Felleisen90,Felleisen91}}
% \begin{definition} % \begin{definition}
% A signature $\Sigma$ is a collection of abstract syntax constructors % A statically typed programming language $\LLL$ consists of a syntax
% with arities $\{(\SC_i, \ar_i)\}_i$, where $\ST_i$ are syntactic % $S$, static semantics $T$, and dynamic semantics $E$ where
% entities and $\ar_i$ are natural numbers. Syntax constructors with
% arities $0$, $1$, $2$, and $3$ are referred to as nullary, unary,
% binary, and ternary, respectively.
% \end{definition}
% \begin{definition}[Terms in contexts]
% A context $\mathcal{X}$ is set of variables $\{x_1,\dots,x_n\}$. For
% a signature $\Sigma = \{(\SC_i, \ar_i)\}_i$ the $\Sigma$-terms in
% some context $\mathcal{X}$ are generated according to the following
% inductive rules:
% \begin{itemize} % \begin{itemize}
% \item each variable $x_i \in \mathcal{X}$ is a $\Sigma$-term, % \item $S$ is a collection of, possibly mutually, inductively defined
% \item if $t_1,\dots,t_{\ar_i}$ are $\Sigma$-terms in context % syntactic categories (e.g. terms, types, and kinds). Each
% $\mathcal{X}$, then $\SC_i(t_1,\dots,t_{\ar_i})$ is % syntactic category contains a collection of syntax constructors
% $\Sigma$-term in context $\mathcal{X}$. % with nonnegative arities $\{(\SC_i,\ar_i)\}$ which construct
% \end{itemize} % abstract syntax trees. We insist that $S$ contains at least two
% We write $\mathcal{X} \vdash t$ to indicate that $t$ is a % syntactic categories for constructing terms and types of
% $\Sigma$-term in context $\mathcal{X}$. A $\Sigma$-term $t$ is % $\LLL$. We write $\Tm(S)$ for the terms category and $\Ty(S)$ for
% said to be closed if $\emptyset \vdash t$, i.e. no variables occur % the types category;
% in $t$. % \item $T : \Tm(S) \times \Ty(S) \to \B$ is \emph{typechecking}
% \end{definition} % function, which decides whether a given term is well-typed; and
% \item $E : P \pto R$ is an \emph{evaluation} function, which maps
% \begin{definition} % well-typed programs
% A $\Sigma$-equation is a pair of terms $t_1,t_2 \in \Sigma$ in
% context $\mathcal{X}$, which we write as
% $\mathcal{X} \vdash t_1 = t_2$.
% \end{definition}
% The notation $\mathcal{X} \vdash t_1 = t_2$ begs to be read as a
% logical statement, though, in universal algebra the notation is simply
% a syntax. In order to give it a meaning we must first construct a
% \emph{model} which interprets the syntax. We shall not delve deeper
% here; for our purposes we may think of a $\Sigma$-equation as a
% logical statement (the attentive reader may already have realised that .
% The following definition is an adaptation of
% \citeauthor{Felleisen90}'s definition of programming language to
% define statically typed programming languages~\cite{Felleisen90}.
% %
% \begin{definition}
% %
% A statically typed programming language $\LLL$ consists of the
% following components.
% %
% \begin{itemize}
% \item A signature $\Sigma_T = \{(\ST_i,\ar_i)\}_i$ of \emph{type syntax constructors}.
% \item A signature $\Sigma_t = \{(\SC_i,\ar_i)\}_i$ of \emph{term syntax constructors}.
% \item A signature $\Sigma_p = \{(\SC_i,
% \item A static semantics, which is a function
% $typecheck : \Sigma_t \to \B$, which decides whether a
% given closed term is well-typed.
% \item An operational semantics, which is a partial function
% $eval : P \pto R$, where $P$ is the set of well-typed terms, i.e.
% %
% \[ % \[
% P = \{ t \in \Sigma_t \mid typecheck~t~\text{is true} \}, % P \defas \{ M \in \Tm(S) \mid A \in \Ty(S), T(M, A)~\text{is true} \},
% \] % \]
% % % to some unspecified set of answers $R$.
% and $R$ is some unspecified set of answers.
% \end{itemize} % \end{itemize}
% \end{definition} % \end{definition}
% % % %
% We will always present the syntax of a programming language on
% Backus-Naur form.
% %
% The static semantics will always be given in form a typing judgement
% (and a kinding judgement for languages with polymorphism), and the
% dynamic semantics will be given as either a reduction relation or an
% abstract machine.
% An untyped programming language is just a special instance of a % We will take the view that an untyped programming language is a
% statically typed programming language, where the signature of type % special instance of a statically typed programming language, where the
% syntax constructors is a singleton containing a nullary type syntax % typechecking function $T$ is the constant function that always returns
% constructor for the \emph{universal type}, and the function % true.
% $typecheck$ is a constant function that always returns true.
% A statically polymorphic typed programming language $\LLL$ also
% includes a set of $\LLL$-kinds of kind syntax constructors as well as
% a function $kindcheck_\LLL : \LLL\text{-types} \to \B$ which checks
% whether a given type is well-kinded.
% Often we will build programming languages incrementally by starting
% from a base language and extend it with new facilities. A
% \emph{conservative extension} is a particularly well-behaved extension
% in the sense that it preserves the all of the behaviour of the
% original language.
% %
% \begin{definition} % \begin{definition}
% A context free grammar is a quadruple $(N, \Sigma, R, S)$, where % A programming language $\LLL = (S, T, E)$ is said to be a
% \begin{enumerate} % \emph{conservative extension} of a language $\LLL' = (S', T', E')$
% \item $N$ is a finite set called the nonterminals. % if the following conditions are met.
% \item $\Sigma$ is a finite set, such that % \begin{itemize}
% $\Sigma \cap V = \emptyset$, called the alphabet (or terminals). % \item $\LLL$ syntactically extends $\LLL'$, i.e. $S'$ is a proper
% \item $R$ is a finite set of rules, each rule is on the form % subset of $S$.
% \[ % \item $\LLL$ preserves the static semantics and dynamic semantics
% P ::= (N \cup \Sigma)^\ast, \quad\text{where}~P \in N. % of $\LLL'$, that is $T(M, A) = T'(M, A)$ and $E(M)$ holds if and
% \] % only if $E'(M)$ holds for all types $A \in \Ty(S)$ and programs
% \item $S$ is the initial nonterminal. % $M \in \Tm(S)$.
% \end{enumerate} % \end{itemize}
% Conversely, $\LLL'$ is a \emph{conservative restriction} of $\LLL$.
% \end{definition} % \end{definition}
% We will often work with translations on syntax between languages. It
% is often the case that a syntactic translation is \emph{homomorphic}
% on most syntax constructors, which is a technical way of saying it
% does not perform any interesting transformation on those
% constructors. Therefore we will omit homomorphic cases in definitions
% of translations.
% %
% \begin{definition}
% Let $\LLL = (S, T, E)$ and $\LLL' = (S', T', E')$ be programming
% languages. A translation $\sembr{-} : S \to S'$ on the syntax
% between $\LLL$ and $\LLL'$ is a homomorphism if it distributes over
% the syntax constructors, i.e. for every $\{(\SC_i,\ar_i)\}_i \in S$
% \[
% \sembr{\SC_i(t_1,\dots,t_{\ar_i})} = \sembr{\SC_i}(\sembr{t_1},\cdots,\sembr{t_{\ar_i}}),
% \]
% %
% where $t_1,\dots,t_{\ar_i}$ are abstract syntax trees. We say that
% $\sembr{-}$ is homomorphic on $S_i$.
% \end{definition}
% \chapter{State of effectful programming} % We will be using a typed variation of \citeauthor{Felleisen91}'s
% \label{ch:related-work} % macro-expressivity to compare the relative expressiveness of different
% languages~\cite{Felleisen90,Felleisen91}. Macro-expressivity is a
% syntactic notion based on the idea of macro rewrites as found in the
% programming language Scheme~\cite{SperberDFSFM10}.
% %
% Informally, if $\LLL$ admits a translation into a sublanguage $\LLL'$
% in a way which respects not only the behaviour of programs but also
% their local syntactic structure, then $\LLL'$ macro-expresses
% $\LLL$. % If the translation of some $\LLL$-program into $\LLL'$
% % requires a complete global restructuring, then we may say that $\LLL'$
% % is in some way less expressive than $\LLL$.
% %
% \begin{definition}[Typeability-preserving macro-expressiveness]
% Let both $\LLL = (S, T, E)$ and $\LLL' = (S', T', E')$ be
% programming languages such that the syntax constructors
% $\SC_1,\dots,\SC_k$ are unique to $\LLL$. If there exists a
% translation $\sembr{-} : S \to S'$ on the syntax of $\LLL$ that is
% homomorphic on all syntax constructors but $\SC_1,\dots,\SC_k$, such
% that for all $A \in \Ty(S)$ and $M \in \Tm(S)$
% %
% \[
% T(M,A) = T'(\sembr{M},\sembr{A})~\text{and}~
% E(M)~\text{holds if and only if}~E'(\sembr{M})~\text{holds},
% \]
% %
% then we say that $\LLL'$ can \emph{macro-express} the
% $\SC_1,\dots,\SC_k$ facilities of $\LLL$.
% \end{definition}
% %
% In general, it is not the case that $\sembr{-}$ preserves types, it is
% only required to ensure that the translated term is typeable in the
% target language $\LLL'$.
% % \section{Type and effect systems} % \begin{definition}[Type-respecting expressiveness]
% % \section{Monadic programming} % Let $\LLL = (S,T,E)$ and $\LLL' = (S',T',E')$ be programming
% \section{Golden age of impurity} % languages and $A$ be a type such that $A \in \Ty(S)$ and
% \section{Monadic enlightenment} % $A \in \Ty(S')$.
% \dhil{Moggi's seminal work applies the notion of monads to effectful % \end{definition}
% programming by modelling effects as monads. More importantly,
% Moggi's work gives a precise characterisation of what's \emph{not}
% an effect}
% \section{Direct-style revolution}
% \subsection{Monadic reflection: best of both worlds} % % \dhil{Definition of (typed) programming language, conservative extension, macro-expressiveness~\cite{Felleisen90,Felleisen91}}
% % \begin{definition}
% % A signature $\Sigma$ is a collection of abstract syntax constructors
% % with arities $\{(\SC_i, \ar_i)\}_i$, where $\ST_i$ are syntactic
% % entities and $\ar_i$ are natural numbers. Syntax constructors with
% % arities $0$, $1$, $2$, and $3$ are referred to as nullary, unary,
% % binary, and ternary, respectively.
% % \end{definition}
% % \begin{definition}[Terms in contexts]
% % A context $\mathcal{X}$ is set of variables $\{x_1,\dots,x_n\}$. For
% % a signature $\Sigma = \{(\SC_i, \ar_i)\}_i$ the $\Sigma$-terms in
% % some context $\mathcal{X}$ are generated according to the following
% % inductive rules:
% % \begin{itemize}
% % \item each variable $x_i \in \mathcal{X}$ is a $\Sigma$-term,
% % \item if $t_1,\dots,t_{\ar_i}$ are $\Sigma$-terms in context
% % $\mathcal{X}$, then $\SC_i(t_1,\dots,t_{\ar_i})$ is
% % $\Sigma$-term in context $\mathcal{X}$.
% % \end{itemize}
% % We write $\mathcal{X} \vdash t$ to indicate that $t$ is a
% % $\Sigma$-term in context $\mathcal{X}$. A $\Sigma$-term $t$ is
% % said to be closed if $\emptyset \vdash t$, i.e. no variables occur
% % in $t$.
% % \end{definition}
% % \begin{definition}
% % A $\Sigma$-equation is a pair of terms $t_1,t_2 \in \Sigma$ in
% % context $\mathcal{X}$, which we write as
% % $\mathcal{X} \vdash t_1 = t_2$.
% % \end{definition}
% % The notation $\mathcal{X} \vdash t_1 = t_2$ begs to be read as a
% % logical statement, though, in universal algebra the notation is simply
% % a syntax. In order to give it a meaning we must first construct a
% % \emph{model} which interprets the syntax. We shall not delve deeper
% % here; for our purposes we may think of a $\Sigma$-equation as a
% % logical statement (the attentive reader may already have realised that .
% % The following definition is an adaptation of
% % \citeauthor{Felleisen90}'s definition of programming language to
% % define statically typed programming languages~\cite{Felleisen90}.
% % %
% % \begin{definition}
% % %
% % A statically typed programming language $\LLL$ consists of the
% % following components.
% % %
% % \begin{itemize}
% % \item A signature $\Sigma_T = \{(\ST_i,\ar_i)\}_i$ of \emph{type syntax constructors}.
% % \item A signature $\Sigma_t = \{(\SC_i,\ar_i)\}_i$ of \emph{term syntax constructors}.
% % \item A signature $\Sigma_p = \{(\SC_i,
% % \item A static semantics, which is a function
% % $typecheck : \Sigma_t \to \B$, which decides whether a
% % given closed term is well-typed.
% % \item An operational semantics, which is a partial function
% % $eval : P \pto R$, where $P$ is the set of well-typed terms, i.e.
% % %
% % \[
% % P = \{ t \in \Sigma_t \mid typecheck~t~\text{is true} \},
% % \]
% % %
% % and $R$ is some unspecified set of answers.
% % \end{itemize}
% % \end{definition}
% % %
% % An untyped programming language is just a special instance of a
% % statically typed programming language, where the signature of type
% % syntax constructors is a singleton containing a nullary type syntax
% % constructor for the \emph{universal type}, and the function
% % $typecheck$ is a constant function that always returns true.
% % A statically polymorphic typed programming language $\LLL$ also
% % includes a set of $\LLL$-kinds of kind syntax constructors as well as
% % a function $kindcheck_\LLL : \LLL\text{-types} \to \B$ which checks
% % whether a given type is well-kinded.
% % \begin{definition}
% % A context free grammar is a quadruple $(N, \Sigma, R, S)$, where
% % \begin{enumerate}
% % \item $N$ is a finite set called the nonterminals.
% % \item $\Sigma$ is a finite set, such that
% % $\Sigma \cap V = \emptyset$, called the alphabet (or terminals).
% % \item $R$ is a finite set of rules, each rule is on the form
% % \[
% % P ::= (N \cup \Sigma)^\ast, \quad\text{where}~P \in N.
% % \]
% % \item $S$ is the initial nonterminal.
% % \end{enumerate}
% % \end{definition}
% % \chapter{State of effectful programming}
% % \label{ch:related-work}
% % % \section{Type and effect systems}
% % % \section{Monadic programming}
% % \section{Golden age of impurity}
% % \section{Monadic enlightenment}
% % \dhil{Moggi's seminal work applies the notion of monads to effectful
% % programming by modelling effects as monads. More importantly,
% % Moggi's work gives a precise characterisation of what's \emph{not}
% % an effect}
% % \section{Direct-style revolution}
% % \subsection{Monadic reflection: best of both worlds}
\chapter{Continuations} \chapter{Continuations}
\label{ch:continuations} \label{ch:continuations}
@@ -6065,7 +6071,7 @@ semantics of \BCalc{}. In this section, we state and prove some
customary metatheoretic properties about \BCalc{}. customary metatheoretic properties about \BCalc{}.
% %
We begin by showing that substitutions preserve typability. We begin by showing that substitutions preserve typeability.
% %
\begin{lemma}[Preservation of typing under substitution]\label{lem:base-language-subst} \begin{lemma}[Preservation of typing under substitution]\label{lem:base-language-subst}
Let $\sigma$ be any type substitution and $V \in \ValCat$ be any Let $\sigma$ be any type substitution and $V \in \ValCat$ be any
@@ -14418,17 +14424,18 @@ setting.
% %
We will restrict our attention to the calculi $\HCalc$, $\SCalc$, and We will restrict our attention to the calculi $\HCalc$, $\SCalc$, and
$\HPCalc$ and use the notion of \emph{typability-preserving $\HPCalc$ and use the notion of \emph{typeability-preserving
macro-expressiveness} to determine whether handlers are macro-expressiveness} to determine whether handlers are
interdefinable. In our particular setting, typability-preserving interdefinable~\cite{ForsterKLP19}. In our particular setting,
macro-expressiveness asks whether there exists a \emph{local} typeability-preserving macro-expressiveness asks whether there exists
transformation that can transform one kind of handler into another a \emph{local} transformation that can transform one kind of handler
kind of handler, whilst preserving typability in the image of the into another kind of handler, whilst preserving typeability in the
transformation. By mandating that the transform is local we rule out image of the transformation. By mandating that the transform is local
the possibility of rewriting the entire program in, say, CPS notation we rule out the possibility of rewriting the entire program in, say,
to implement deep and shallow handlers as in Chapter~\ref{ch:cps}. CPS notation to implement deep and shallow handlers as in
Chapter~\ref{ch:cps}.
% %
In this chapter we use the notion of typability-preserving In this chapter we use the notion of typeability-preserving
macro-expressiveness to show that shallow handlers and general macro-expressiveness to show that shallow handlers and general
recursion can simulate deep handlers up to congruence, and that deep recursion can simulate deep handlers up to congruence, and that deep
handlers can simulate shallow handlers up to administrative handlers can simulate shallow handlers up to administrative
@@ -14527,7 +14534,7 @@ The deep semantics are simulated by generating the name $env$ for the
shallow handlers and recursively apply the handler under the modified shallow handlers and recursively apply the handler under the modified
resumption. resumption.
The translation commutes with substitution and preserves typability. The translation commutes with substitution and preserves typeability.
% %
\begin{lemma}\label{lem:dstrans-subst} \begin{lemma}\label{lem:dstrans-subst}
Let $\sigma$ denote a substitution. The translation $\dstrans{-}$ Let $\sigma$ denote a substitution. The translation $\dstrans{-}$
@@ -14721,13 +14728,13 @@ $\BigO(1)$ time in the source, but in the image it takes $\BigO(k)$
time. time.
% %
Thus this translation is more of theoretical significance than Thus this translation is more of theoretical significance than
practical interest. It also demonstrates that typability-preserving practical interest. It also demonstrates that typeability-preserving
macro-expressiveness is rather coarse-grained notion of macro-expressiveness is rather coarse-grained notion of
expressiveness, as it blindly considers whether some construct is expressiveness, as it blindly considers whether some construct is
computable using another construct without considering the computable using another construct without considering the
computational cost. computational cost.
The translation commutes with substitution and preserves typability. The translation commutes with substitution and preserves typeability.
% %
\begin{lemma}\label{lem:sdtrans-subst} \begin{lemma}\label{lem:sdtrans-subst}
Let $\sigma$ denote a substitution. The translation $\sdtrans{-}$ Let $\sigma$ denote a substitution. The translation $\sdtrans{-}$
@@ -14758,7 +14765,7 @@ If $\Delta; \Gamma \vdash M : C$ then $\sdtrans{\Delta};
\newcommand{\approxa}{\gtrsim} \newcommand{\approxa}{\gtrsim}
As with the implementation of deep handlers as shallow handlers, the As with the implementation of deep handlers as shallow handlers, the
implementation is again given by a typability-preserving local implementation is again given by a typeability-preserving local
translation. However, this time the administrative overhead is more translation. However, this time the administrative overhead is more
significant. Reduction up to congruence is insufficient and we require significant. Reduction up to congruence is insufficient and we require
a more semantic notion of administrative reduction. a more semantic notion of administrative reduction.
@@ -15295,7 +15302,7 @@ handler into an ordinary deep handler that makes use of the
parameter-passing idiom to maintain the state of the handled parameter-passing idiom to maintain the state of the handled
computation~\cite{Pretnar15}. computation~\cite{Pretnar15}.
The translation commutes with substitution and preserves typability. The translation commutes with substitution and preserves typeability.
% %
\begin{lemma}\label{lem:pd-subst} \begin{lemma}\label{lem:pd-subst}
Let $\sigma$ denote a substitution. The translation $\PD{-}$ Let $\sigma$ denote a substitution. The translation $\PD{-}$
@@ -15401,7 +15408,7 @@ myself on the inherited efficiency of effect handlers
(c.f. Chapter~\ref{ch:handlers-efficiency}) and \citet{ForsterKLP17}, who (c.f. Chapter~\ref{ch:handlers-efficiency}) and \citet{ForsterKLP17}, who
investigate various relationships between effect handlers, delimited investigate various relationships between effect handlers, delimited
control in the form of shift/reset, and monadic reflection using the control in the form of shift/reset, and monadic reflection using the
notions of typability-preserving macro-expressiveness and untyped notions of typeability-preserving macro-expressiveness and untyped
macro-expressiveness~\cite{ForsterKLP17,ForsterKLP19}. They show that macro-expressiveness~\cite{ForsterKLP17,ForsterKLP19}. They show that
in an untyped setting all three are interdefinable, whereas in a in an untyped setting all three are interdefinable, whereas in a
simply typed setting effect handlers cannot macro-express simply typed setting effect handlers cannot macro-express
@@ -15426,7 +15433,7 @@ the captured context and continuation invocation context to coincide.
% \label{ch:expressiveness} % \label{ch:expressiveness}
% \section{Notions of expressiveness} % \section{Notions of expressiveness}
% Felleisen's macro-expressiveness, Longley's type-respecting % Felleisen's macro-expressiveness, Longley's type-respecting
% expressiveness, Kammar's typability-preserving expressiveness. % expressiveness, Kammar's typeability-preserving expressiveness.
\chapter{Asymptotic speedup with effect handlers} \chapter{Asymptotic speedup with effect handlers}
\label{ch:handlers-efficiency} \label{ch:handlers-efficiency}