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generic_types.tex

Timestamp:

Apr 19, 2017, 10:15:45 AM (9 years ago)

Author:

Thierry Delisle <tdelisle@…>

Branches:

ADT, aaron-thesis, arm-eh, ast-experimental, cleanup-dtors, deferred_resn, demangler, enum, forall-pointer-decay, jacob/cs343-translation, jenkins-sandbox, master, new-ast, new-ast-unique-expr, new-env, no_list, persistent-indexer, pthread-emulation, qualifiedEnum, resolv-new, with_gc

Children:

cd348e7

Parents:

221c2de7 (diff), de4ce0e (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.

Message:

Merge branch 'master' of plg.uwaterloo.ca:software/cfa/cfa-cc

File:

: 1 edited

doc/generic_types/generic_types.tex (modified) (19 diffs)

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doc/generic_types/generic_types.tex

-              r221c2de7
+              r154fdc8
 % take off review (for line numbers) and anonymous (for anonymization) on submission
 % \documentclass[format=acmlarge, anonymous, review]{acmart}
 \documentclass[format=acmlarge,review]{acmart}
+\documentclass[format=acmlarge,anonymous,review]{acmart}
+% \documentclass[format=acmlarge,review]{acmart}
 \usepackage{xspace,calc,comment}
 \usepackage{upquote}                                                                    % switch curled `'" to straight
 \usepackage{listings}                                                                   % format program code
+\usepackage[usenames]{color}
 \makeatletter
+% Default underscore is too low and wide. Cannot use lstlisting "literate" as replacing underscore
+% removes it as a variable-name character so keyworks in variables are highlighted
+\DeclareTextCommandDefault{\textunderscore}{\leavevmode\makebox[1.2ex][c]{\rule{1ex}{0.1ex}}}
 % parindent is relative, i.e., toggled on/off in environments like itemize, so store the value for
 % use rather than use \parident directly.
 …
 \setlength{\parindentlnth}{\parindent}
 \newlength{\gcolumnposn}                                % temporary hack because lstlisting does handle tabs correctly
+\newlength{\gcolumnposn}                                % temporary hack because lstlisting does not handle tabs correctly
 \newlength{\columnposn}
 \setlength{\gcolumnposn}{2.75in}
 …
 \newcommand{\C}[2][\@empty]{\ifx#1\@empty\else\global\setlength{\columnposn}{#1}\global\columnposn=\columnposn\fi\hfill\makebox[\textwidth-\columnposn][l]{\lst@commentstyle{#2}}}
 \newcommand{\CRT}{\global\columnposn=\gcolumnposn}
+% Latin abbreviation
+\newcommand{\abbrevFont}{\textit}       % set empty for no italics
+\newcommand*{\eg}{%
+        \@ifnextchar{,}{\abbrevFont{e}.\abbrevFont{g}.}%
+                {\@ifnextchar{:}{\abbrevFont{e}.\abbrevFont{g}.}%
+                        {\abbrevFont{e}.\abbrevFont{g}.,\xspace}}%
+}%
+\newcommand*{\ie}{%
+        \@ifnextchar{,}{\abbrevFont{i}.\abbrevFont{e}.}%
+                {\@ifnextchar{:}{\abbrevFont{i}.\abbrevFont{e}.}%
+                        {\abbrevFont{i}.\abbrevFont{e}.,\xspace}}%
+}%
+\newcommand*{\etc}{%
+        \@ifnextchar{.}{\abbrevFont{etc}}%
+        {\abbrevFont{etc}.\xspace}%
+}%
+\newcommand{\etal}{%
+        \@ifnextchar{.}{\abbrevFont{et~al}}%
+                {\abbrevFont{et al}.\xspace}%
+}%
 \makeatother
 …
 \newcommand{\CCseventeen}{\rm C\kern-.1em\hbox{+\kern-.25em+}17\xspace} % C++17 symbolic name
 \newcommand{\CCtwenty}{\rm C\kern-.1em\hbox{+\kern-.25em+}20\xspace} % C++20 symbolic name
+\newcommand{\CS}{C\raisebox{-0.7ex}{\Large$^\sharp$}\xspace}
+\newcommand{\CCV}{\rm C\kern-.1em\hbox{+\kern-.25em+}obj\xspace} % C++ virtual symbolic name
+\newcommand{\Csharp}{C\raisebox{-0.7ex}{\Large$^\sharp$}\xspace} % C# symbolic name
 \newcommand{\Textbf}[1]{{\color{red}\textbf{#1}}}
 \newcommand{\TODO}[1]{\textbf{TODO}: {\itshape #1}} % TODO included
 %\newcommand{\TODO}[1]{} % TODO elided
-\newcommand{\eg}{\textit{e}.\textit{g}.,\xspace}
-\newcommand{\ie}{\textit{i}.\textit{e}.,\xspace}
-\newcommand{\etc}{\textit{etc}.,\xspace}
 % CFA programming language, based on ANSI C (with some gcc additions)
 …
 belowskip=3pt,
 % replace/adjust listing characters that look bad in sanserif
 literate={-}{\raisebox{-0.15ex}{\texttt{-}}}1 {^}{\raisebox{0.6ex}{$\scriptscriptstyle\land\,$}}1
         {~}{\raisebox{0.3ex}{$\scriptstyle\sim\,$}}1 {_}{\makebox[1.2ex][c]{\rule{1ex}{0.1ex}}}1 % {`}{\ttfamily\upshape\hspace*{-0.1ex}`}1
         {<-}{$\leftarrow$}2 {=>}{$\Rightarrow$}2,
+literate={-}{\makebox[1.4ex][c]{\raisebox{0.5ex}{\rule{1.2ex}{0.1ex}}}}1 {^}{\raisebox{0.6ex}{$\scriptscriptstyle\land\,$}}1
+        {~}{\raisebox{0.3ex}{$\scriptstyle\sim\,$}}1 % {`}{\ttfamily\upshape\hspace*{-0.1ex}`}1
+        {<-}{$\leftarrow$}2 {=>}{$\Rightarrow$}2 {->}{\makebox[1.4ex][c]{\raisebox{0.5ex}{\rule{1.2ex}{0.1ex}}}\kern-0.3ex\textgreater}2,
 moredelim=**[is][\color{red}]{`}{`},
 }% lstset
 % inline code @...@
 \lstMakeShortInline@
+\lstMakeShortInline@%
 % ACM Information
 …
 \begin{abstract}
+The C programming language is a foundational technology for modern computing with millions of lines of code implementing everything from commercial operating-systems to hobby projects. This installation base and the programmers producing it represent a massive software-engineering investment spanning decades and likely to continue for decades more. Nonetheless, C, first standardized over thirty years ago, lacks many features that make programming in more modern languages safer and more productive. The goal of the \CFA project is to create an extension of C that provides modern safety and productivity features while still ensuring strong backwards compatibility with C and its programmers. Prior projects have attempted similar goals but failed to honour C programming-style; for instance, adding object-oriented or functional programming with garbage collection is a non-starter for many C developers. Specifically, \CFA is designed to have an orthogonal feature-set based closely on the C programming paradigm, so that \CFA features can be added \emph{incrementally} to existing C code-bases, and C programmers can learn \CFA extensions on an as-needed basis, preserving investment in existing code and engineers. This paper describes two \CFA extensions, generic and tuple types, details how their design avoids shortcomings of similar features in C and other C-like languages, and presents experimental results validating the design.
+The C programming language is a foundational technology for modern computing with millions of lines of code implementing everything from commercial operating-systems to hobby projects.
+This installation base and the programmers producing it represent a massive software-engineering investment spanning decades and likely to continue for decades more.
+Nonetheless, C, first standardized over thirty years ago, lacks many features that make programming in more modern languages safer and more productive.
+The goal of the \CFA project is to create an extension of C that provides modern safety and productivity features while still ensuring strong backwards compatibility with C and its programmers.
+Prior projects have attempted similar goals but failed to honour C programming-style; for instance, adding object-oriented or functional programming with garbage collection is a non-starter for many C developers.
+Specifically, \CFA is designed to have an orthogonal feature-set based closely on the C programming paradigm, so that \CFA features can be added \emph{incrementally} to existing C code-bases, and C programmers can learn \CFA extensions on an as-needed basis, preserving investment in existing code and engineers.
+This paper describes two \CFA extensions, generic and tuple types, details how their design avoids shortcomings of similar features in C and other C-like languages, and presents experimental results validating the design.
 \end{abstract}
 …
 \section{Introduction and Background}
+The C programming language is a foundational technology for modern computing with millions of lines of code implementing everything from commercial operating-systems to hobby projects. This installation base and the programmers producing it represent a massive software-engineering investment spanning decades and likely to continue for decades more.
+The \citet{TIOBE} ranks the top 5 most popular programming languages as: Java 16\%, \Textbf{C 7\%}, \Textbf{\CC 5\%}, \CS 4\%, Python 4\% = 36\%, where the next 50 languages are less than 3\% each with a long tail. The top 3 rankings over the past 30 years are:
+\lstDeleteShortInline@
+The C programming language is a foundational technology for modern computing with millions of lines of code implementing everything from commercial operating-systems to hobby projects.
+This installation base and the programmers producing it represent a massive software-engineering investment spanning decades and likely to continue for decades more.
+The \citet{TIOBE} ranks the top 5 most popular programming languages as: Java 16\%, \Textbf{C 7\%}, \Textbf{\CC 5\%}, \Csharp 4\%, Python 4\% = 36\%, where the next 50 languages are less than 3\% each with a long tail.
+The top 3 rankings over the past 30 years are:
+\lstDeleteShortInline@%
 \begin{center}
 \setlength{\tabcolsep}{10pt}
+\begin{tabular}{@{}r|c|c|c|c|c|c|c@{}}
+                & 2017  & 2012  & 2007  & 2002  & 1997  & 1992  & 1987          \\
+\hline
+Java    & 1             & 1             & 1             & 3             & 13    & -             & -                     \\
+\hline
+\Textbf{C}      & \Textbf{2}& \Textbf{2}& \Textbf{2}& \Textbf{1}& \Textbf{1}& \Textbf{1}& \Textbf{1}    \\
+\hline
+\begin{tabular}{@{}rccccccc@{}}
+                & 2017  & 2012  & 2007  & 2002  & 1997  & 1992  & 1987          \\ \hline
+Java    & 1             & 1             & 1             & 1             & 12    & -             & -                     \\
+\Textbf{C}      & \Textbf{2}& \Textbf{2}& \Textbf{2}& \Textbf{2}& \Textbf{1}& \Textbf{1}& \Textbf{1}    \\
 \CC             & 3             & 3             & 3             & 3             & 2             & 2             & 4                     \\
 \end{tabular}
 \end{center}
 \lstMakeShortInline@
 Love it or hate it, C is extremely popular, highly used, and one of the few system's languages.
+\lstMakeShortInline@%
+Love it or hate it, C is extremely popular, highly used, and one of the few systems languages.
 In many cases, \CC is often used solely as a better C.
 Nonetheless, C, first standardized over thirty years ago, lacks many features that make programming in more modern languages safer and more productive.
+\CFA (pronounced ``C-for-all'', and written \CFA or Cforall) is an evolutionary extension of the C programming language that aims to add modern language features to C while maintaining both source compatibility with C and a familiar programming model for programmers. The four key design goals for \CFA~\citep{Bilson03} are:
+\CFA (pronounced ``C-for-all'', and written \CFA or Cforall) is an evolutionary extension of the C programming language that aims to add modern language features to C while maintaining both source compatibility with C and a familiar programming model for programmers.
+The four key design goals for \CFA~\citep{Bilson03} are:
 (1) The behaviour of standard C code must remain the same when translated by a \CFA compiler as when translated by a C compiler;
 (2) Standard C code must be as fast and as small when translated by a \CFA compiler as when translated by a C compiler;
 (3) \CFA code must be at least as portable as standard C code;
 (4) Extensions introduced by \CFA must be translated in the most efficient way possible.
+These goals ensure existing C code-bases can be converted to \CFA incrementally with minimal effort, and C programmers can productively generate \CFA code without training beyond the features being used. In its current implementation, \CFA is compiled by translating it to the GCC-dialect of C~\citep{GCCExtensions}, allowing it to leverage the portability and code optimizations provided by GCC, meeting goals (1)-(3). Ultimately, a compiler is necessary for advanced features and optimal performance.
+This paper identifies shortcomings in existing approaches to generic and variadic data types in C-like languages and presents a design for generic and variadic types avoiding those shortcomings. Specifically, the solution is both reusable and type-checked, as well as conforming to the design goals of \CFA with ergonomic use of existing C abstractions. The new constructs are empirically compared with both standard C and \CC; the results show the new design is comparable in performance.
+These goals ensure existing C code-bases can be converted to \CFA incrementally with minimal effort, and C programmers can productively generate \CFA code without training beyond the features being used.
+\CC is used similarly, but has the disadvantages of multiple legacy design-choices that cannot be updated and active divergence of the language model from C, requiring significant effort and training to incrementally add \CC to a C-based project.
+\CFA is currently implemented as a source-to-source translator from \CFA to the GCC-dialect of C~\citep{GCCExtensions}, allowing it to leverage the portability and code optimizations provided by GCC, meeting goals (1)--(3).
+Ultimately, a compiler is necessary for advanced features and optimal performance.
+This paper identifies shortcomings in existing approaches to generic and variadic data types in C-like languages and presents a design for generic and variadic types avoiding those shortcomings.
+Specifically, the solution is both reusable and type-checked, as well as conforming to the design goals of \CFA with ergonomic use of existing C abstractions.
+The new constructs are empirically compared with both standard C and \CC; the results show the new design is comparable in performance.
 …
 \label{sec:poly-fns}
+\CFA's polymorphism was originally formalized by \citet{Ditchfield92}, and first implemented by \citet{Bilson03}. The signature feature of \CFA is parametric-polymorphic functions where functions are generalized using a @forall@ clause (giving the language its name):
+\CFA's polymorphism was originally formalized by \citet{Ditchfield92}, and first implemented by \citet{Bilson03}.
+The signature feature of \CFA is parametric-polymorphic functions~\citep{forceone:impl,Cormack90,Duggan96} with functions generalized using a @forall@ clause (giving the language its name):
 \begin{lstlisting}
 `forall( otype T )` T identity( T val ) { return val; }
 int forty_two = identity( 42 );                         $\C{// T is bound to int, forty\_two == 42}$
 \end{lstlisting}
+The @identity@ function above can be applied to any complete \emph{object type} (or @otype@). The type variable @T@ is transformed into a set of additional implicit parameters encoding sufficient information about @T@ to create and return a variable of that type. The \CFA implementation passes the size and alignment of the type represented by an @otype@ parameter, as well as an assignment operator, constructor, copy constructor and destructor. If this extra information is not needed, \eg for a pointer, the type parameter can be declared as a \emph{data type} (or @dtype@).
+Here, the runtime cost of polymorphism is spread over each polymorphic call, due to passing more arguments to polymorphic functions; preliminary experiments have shown this overhead is similar to \CC virtual function calls. An advantage of this design is that, unlike \CC template functions, \CFA polymorphic functions are compatible with C \emph{separate} compilation, preventing code bloat.
+Since bare polymorphic-types provide only a narrow set of available operations, \CFA provides a \emph{type assertion} mechanism to provide further type information, where type assertions may be variable or function declarations that depend on a polymorphic type-variable. For example, the function @twice@ can be defined using the \CFA syntax for operator overloading:
+The @identity@ function above can be applied to any complete \emph{object type} (or @otype@).
+The type variable @T@ is transformed into a set of additional implicit parameters encoding sufficient information about @T@ to create and return a variable of that type.
+The \CFA implementation passes the size and alignment of the type represented by an @otype@ parameter, as well as an assignment operator, constructor, copy constructor and destructor.
+If this extra information is not needed, \eg for a pointer, the type parameter can be declared as a \emph{data type} (or @dtype@).
+In \CFA, the polymorphism runtime-cost is spread over each polymorphic call, due to passing more arguments to polymorphic functions;
+the experiments in Section~\ref{sec:eval} show this overhead is similar to \CC virtual-function calls.
+A design advantage is that, unlike \CC template-functions, \CFA polymorphic-functions are compatible with C \emph{separate compilation}, preventing compilation and code bloat.
+Since bare polymorphic-types provide a restricted set of available operations, \CFA provides a \emph{type assertion}~\cite[pp.~37-44]{Alphard} mechanism to provide further type information, where type assertions may be variable or function declarations that depend on a polymorphic type-variable.
+For example, the function @twice@ can be defined using the \CFA syntax for operator overloading:
 \begin{lstlisting}
 forall( otype T `| { T ?+?(T, T); }` ) T twice( T x ) { return x + x; } $\C{// ? denotes operands}$
 int val = twice( twice( 3.7 ) );
 \end{lstlisting}
+which works for any type @T@ with a matching addition operator. The polymorphism is achieved by creating a wrapper function for calling @+@ with @T@ bound to @double@, then passing this function to the first call of @twice@. There is now the option of using the same @twice@ and converting the result to @int@ on assignment, or creating another @twice@ with type parameter @T@ bound to @int@ because \CFA uses the return type~\cite{Ada} in its type analysis. The first approach has a late conversion from @int@ to @double@ on the final assignment, while the second has an eager conversion to @int@. \CFA minimizes the number of conversions and their potential to lose information, so it selects the first approach, which corresponds with C-programmer intuition.
+which works for any type @T@ with a matching addition operator.
+The polymorphism is achieved by creating a wrapper function for calling @+@ with @T@ bound to @double@, then passing this function to the first call of @twice@.
+There is now the option of using the same @twice@ and converting the result to @int@ on assignment, or creating another @twice@ with type parameter @T@ bound to @int@ because \CFA uses the return type~\cite{Cormack81,Baker82,Ada}, in its type analysis.
+The first approach has a late conversion from @double@ to @int@ on the final assignment, while the second has an eager conversion to @int@.
+\CFA minimizes the number of conversions and their potential to lose information, so it selects the first approach, which corresponds with C-programmer intuition.
 Crucial to the design of a new programming language are the libraries to access thousands of external software features.
 \CFA inherits a massive compatible library-base, where other programming languages must rewrite or provide fragile inter-language communication with C.
+Like \CC, \CFA inherits a massive compatible library-base, where other programming languages must rewrite or provide fragile inter-language communication with C.
 A simple example is leveraging the existing type-unsafe (@void *@) C @bsearch@ to binary search a sorted floating-point array:
 \begin{lstlisting}
 void * bsearch( const void * key, const void * base, size_t nmemb, size_t size,
                                 int (* compar)(const void *, const void *));
+                                int (* compar)( const void *, const void * ));
 int comp( const void * t1, const void * t2 ) { return *(double *)t1 < *(double *)t2 ? -1 :
                                 *(double *)t2 < *(double *)t1 ? 1 : 0; }
+double vals[10] = { /* 10 floating-point values */ };
+double key = 5.0;
+double key = 5.0, vals[10] = { /* 10 floating-point values */ };
 double * val = (double *)bsearch( &key, vals, 10, sizeof(vals[0]), comp );      $\C{// search sorted array}$
 \end{lstlisting}
 …
         return (T *)bsearch( &key, arr, size, sizeof(T), comp ); }
 forall( otype T | { int ?<?( T, T ); } ) unsigned int bsearch( T key, const T * arr, size_t size ) {
         T *result = bsearch( key, arr, size );  $\C{// call first version}$
+        T * result = bsearch( key, arr, size ); $\C{// call first version}$
         return result ? result - arr : size; }  $\C{// pointer subtraction includes sizeof(T)}$
 double * val = bsearch( 5.0, vals, 10 );        $\C{// selection based on return type}$
 int posn = bsearch( 5.0, vals, 10 );
 \end{lstlisting}
+The nested routine @comp@ provides the hidden interface from typed \CFA to untyped (@void *@) C, plus the cast of the result.
+The nested function @comp@ provides the hidden interface from typed \CFA to untyped (@void *@) C, plus the cast of the result.
+Providing a hidden @comp@ function in \CC is awkward as lambdas do not use C calling-conventions and template declarations cannot appear at block scope.
 As well, an alternate kind of return is made available: position versus pointer to found element.
 \CC's type-system cannot disambiguate between the two versions of @bsearch@ because it does not use the return type in overload resolution, nor can \CC separately compile a templated @bsearch@.
 …
 For example, it is possible to write a type-safe \CFA wrapper @malloc@ based on the C @malloc@:
 \begin{lstlisting}
 forall( dtype T | sized(T) ) T * malloc( void ) { return (T *)(void *)malloc( (size_t)sizeof(T) ); }
+forall( dtype T | sized(T) ) T * malloc( void ) { return (T *)malloc( sizeof(T) ); }
 int * ip = malloc();                                            $\C{// select type and size from left-hand side}$
 double * dp = malloc();
 …
 where the return type supplies the type/size of the allocation, which is impossible in most type systems.
+Call-site inferencing and nested functions provide a localized form of inheritance. For example, the \CFA @qsort@ only sorts in ascending order using @<@. However, it is trivial to locally change this behaviour:
+Call-site inferencing and nested functions provide a localized form of inheritance.
+For example, the \CFA @qsort@ only sorts in ascending order using @<@.
+However, it is trivial to locally change this behaviour:
 \begin{lstlisting}
 forall( otype T | { int ?<?( T, T ); } ) void qsort( const T * arr, size_t size ) { /* use C qsort */ }
 …
 \end{lstlisting}
 Within the block, the nested version of @<@ performs @>@ and this local version overrides the built-in @<@ so it is passed to @qsort@.
 Hence, programmers can easily form a local environments, adding and modifying appropriate functions, to maximize reuse of other existing functions and types.
+Hence, programmers can easily form local environments, adding and modifying appropriate functions, to maximize reuse of other existing functions and types.
 Finally, \CFA allows variable overloading:
+\lstDeleteShortInline@
+\par\smallskip
+\begin{tabular}{@{}l@{\hspace{\parindent}}|@{\hspace{\parindent}}l@{}}
+\begin{lstlisting}
+short int MAX = ...;
+int MAX = ...;
+double MAX = ...;
+\end{lstlisting}
+&
+\begin{lstlisting}
+short int s = MAX;  // select correct MAX
+int i = MAX;
+double d = MAX;
+\end{lstlisting}
+\end{tabular}
+\lstMakeShortInline@
+\smallskip\par\noindent
+Hence, the single name @MAX@ replaces all the C type-specific names: @SHRT_MAX@, @INT_MAX@, @DBL_MAX@.
+\begin{lstlisting}
+short int MAX = ...;   int MAX = ...;  double MAX = ...;
+short int s = MAX;    int i = MAX;    double d = MAX;   $\C{// select correct MAX}$
+\end{lstlisting}
+Here, the single name @MAX@ replaces all the C type-specific names: @SHRT_MAX@, @INT_MAX@, @DBL_MAX@.
 As well, restricted constant overloading is allowed for the values @0@ and @1@, which have special status in C, \eg the value @0@ is both an integer and a pointer literal, so its meaning depends on context.
+In addition, several operations are defined in terms values @0@ and @1@.
+For example,
+In addition, several operations are defined in terms values @0@ and @1@, \eg:
 \begin{lstlisting}
 int x;
+if (x)        // if (x != 0)
+        x++;    //   x += 1;
+\end{lstlisting}
+Every if statement in C compares the condition with @0@, and every increment and decrement operator is semantically equivalent to adding or subtracting the value @1@ and storing the result.
+if (x) x++                                                                      $\C{// if (x != 0) x += 1;}$
+\end{lstlisting}
+Every if and iteration statement in C compares the condition with @0@, and every increment and decrement operator is semantically equivalent to adding or subtracting the value @1@ and storing the result.
 Due to these rewrite rules, the values @0@ and @1@ have the types @zero_t@ and @one_t@ in \CFA, which allows overloading various operations for new types that seamlessly connect to all special @0@ and @1@ contexts.
 The types @zero_t@ and @one_t@ have special built in implicit conversions to the various integral types, and a conversion to pointer types for @0@, which allows standard C code involving @0@ and @1@ to work as normal.
 …
 forall( otype T `| summable( T )` ) T sum( T a[$\,$], size_t size ) {  // use trait
         `T` total = { `0` };                                    $\C{// instantiate T from 0 by calling its constructor}$
+        for ( unsigned int i = 0; i < size; i += 1 )
+                total `+=` a[i];                                        $\C{// select appropriate +}$
+        for ( unsigned int i = 0; i < size; i += 1 ) total `+=` a[i]; $\C{// select appropriate +}$
         return total; }
 \end{lstlisting}
 In fact, the set of operators is incomplete, \eg no assignment, but @otype@ is syntactic sugar for the following implicit trait:
+In fact, the set of @summable@ trait operators is incomplete, as it is missing assignment for type @T@, but @otype@ is syntactic sugar for the following implicit trait:
 \begin{lstlisting}
 trait otype( dtype T | sized(T) ) {  // sized is a pseudo-trait for types with known size and alignment
 …
 \end{lstlisting}
 Given the information provided for an @otype@, variables of polymorphic type can be treated as if they were a complete type: stack-allocatable, default or copy-initialized, assigned, and deleted.
+% As an example, the @sum@ function produces generated code something like the following (simplified for clarity and brevity)\TODO{fix example, maybe elide, it's likely too long with the more complicated function}:
+In summation, the \CFA type-system uses \emph{nominal typing} for concrete types, matching with the C type-system, and \emph{structural typing} for polymorphic types.
+Hence, trait names play no part in type equivalence;
+the names are simply macros for a list of polymorphic assertions, which are expanded at usage sites.
+Nevertheless, trait names form a logical subtype-hierarchy with @dtype@ at the top, where traits often contain overlapping assertions, \eg operator @+@.
+Traits are used like interfaces in Java or abstract base-classes in \CC, but without the nominal inheritance-relationships.
+Instead, each polymorphic function (or generic type) defines the structural type needed for its execution (polymorphic type-key), and this key is fulfilled at each call site from the lexical environment, which is similar to Go~\citep{Go} interfaces.
+Hence, new lexical scopes and nested functions are used extensively to create local subtypes, as in the @qsort@ example, without having to manage a nominal-inheritance hierarchy.
+(Nominal inheritance can be approximated with traits using marker variables or functions, as is done in Go.)
+% Nominal inheritance can be simulated with traits using marker variables or functions:
 % \begin{lstlisting}
+% void abs( size_t _sizeof_M, size_t _alignof_M,
+%               void (*_ctor_M)(void*), void (*_copy_M)(void*, void*),
+%               void (*_assign_M)(void*, void*), void (*_dtor_M)(void*),
+%               _Bool (*_lt_M)(void*, void*), void (*_neg_M)(void*, void*),
+%       void (*_ctor_M_zero)(void*, int),
+%               void* m, void* _rtn ) {                         $\C{// polymorphic parameter and return passed as void*}$
+%                                                                                       $\C{// M zero = { 0 };}$
+%       void* zero = alloca(_sizeof_M);                 $\C{// stack allocate zero temporary}$
+%       _ctor_M_zero(zero, 0);                                  $\C{// initialize using zero\_t constructor}$
+%                                                                                       $\C{// return m < zero ? -m : m;}$
+%       void *_tmp = alloca(_sizeof_M);
+%       _copy_M( _rtn,                                                  $\C{// copy-initialize return value}$
+%               _lt_M( m, zero ) ?                                      $\C{// check condition}$
+%                (_neg_M(m, _tmp), _tmp) :                      $\C{// negate m}$
+%                m);
+%       _dtor_M(_tmp); _dtor_M(zero);                   $\C{// destroy temporaries}$
+% trait nominal(otype T) {
+%     T is_nominal;
+% };
+% int is_nominal;                                                               $\C{// int now satisfies the nominal trait}$
+% \end{lstlisting}
+%
+% Traits, however, are significantly more powerful than nominal-inheritance interfaces; most notably, traits may be used to declare a relationship \emph{among} multiple types, a property that may be difficult or impossible to represent in nominal-inheritance type systems:
+% \begin{lstlisting}
+% trait pointer_like(otype Ptr, otype El) {
+%     lvalue El *?(Ptr);                                                $\C{// Ptr can be dereferenced into a modifiable value of type El}$
 % }
+% struct list {
+%     int value;
+%     list * next;                                                              $\C{// may omit "struct" on type names as in \CC}$
+% };
+% typedef list * list_iterator;
+%
+% lvalue int *?( list_iterator it ) { return it->value; }
 % \end{lstlisting}
+Traits may be used for many of the same purposes as interfaces in Java or abstract base classes in \CC. Unlike Java interfaces or \CC base classes, \CFA types do not explicitly state any inheritance relationship to traits they satisfy, which is a form of structural inheritance, similar to the implementation of an interface in Go~\citep{Go}, as opposed to the nominal inheritance model of Java and \CC.
+Nominal inheritance can be simulated with traits using marker variables or functions:
+\begin{lstlisting}
+trait nominal(otype T) {
+    T is_nominal;
+% In the example above, @(list_iterator, int)@ satisfies @pointer_like@ by the user-defined dereference function, and @(list_iterator, list)@ also satisfies @pointer_like@ by the built-in dereference operator for pointers. Given a declaration @list_iterator it@, @*it@ can be either an @int@ or a @list@, with the meaning disambiguated by context (\eg @int x = *it;@ interprets @*it@ as an @int@, while @(*it).value = 42;@ interprets @*it@ as a @list@).
+% While a nominal-inheritance system with associated types could model one of those two relationships by making @El@ an associated type of @Ptr@ in the @pointer_like@ implementation, few such systems could model both relationships simultaneously.
+\section{Generic Types}
+One of the known shortcomings of standard C is that it does not provide reusable type-safe abstractions for generic data structures and algorithms.
+Broadly speaking, there are three approaches to implement abstract data-structures in C.
+One approach is to write bespoke data structures for each context in which they are needed.
+While this approach is flexible and supports integration with the C type-checker and tooling, it is also tedious and error-prone, especially for more complex data structures.
+A second approach is to use @void *@--based polymorphism, \eg the C standard-library functions @bsearch@ and @qsort@; an approach which does allow reuse of code for common functionality.
+However, basing all polymorphism on @void *@ eliminates the type-checker's ability to ensure that argument types are properly matched, often requiring a number of extra function parameters, pointer indirection, and dynamic allocation that would not otherwise be needed.
+A third approach to generic code is to use preprocessor macros, which does allow the generated code to be both generic and type-checked, but errors may be difficult to interpret.
+Furthermore, writing and using preprocessor macros can be unnatural and inflexible.
+\CC, Java, and other languages use \emph{generic types} to produce type-safe abstract data-types.
+\CFA also implements generic types that integrate efficiently and naturally with the existing polymorphic functions, while retaining backwards compatibility with C and providing separate compilation.
+However, for known concrete parameters, the generic-type definition can be inlined, like \CC templates.
+A generic type can be declared by placing a @forall@ specifier on a @struct@ or @union@ declaration, and instantiated using a parenthesized list of types after the type name:
+\begin{lstlisting}
+forall( otype R, otype S ) struct pair {
+        R first;
+        S second;
 };
+int is_nominal;                                                         $\C{// int now satisfies the nominal trait}$
+\end{lstlisting}
+Traits, however, are significantly more powerful than nominal-inheritance interfaces; most notably, traits may be used to declare a relationship \emph{among} multiple types, a property that may be difficult or impossible to represent in nominal-inheritance type systems:
+\begin{lstlisting}
+trait pointer_like(otype Ptr, otype El) {
+    lvalue El *?(Ptr);                                          $\C{// Ptr can be dereferenced into a modifiable value of type El}$
+}
+struct list {
+    int value;
+    list *next;                                                         $\C{// may omit "struct" on type names as in \CC}$
+};
+typedef list *list_iterator;
+lvalue int *?( list_iterator it ) { return it->value; }
+\end{lstlisting}
+In the example above, @(list_iterator, int)@ satisfies @pointer_like@ by the user-defined dereference function, and @(list_iterator, list)@ also satisfies @pointer_like@ by the built-in dereference operator for pointers. Given a declaration @list_iterator it@, @*it@ can be either an @int@ or a @list@, with the meaning disambiguated by context (\eg @int x = *it;@ interprets @*it@ as an @int@, while @(*it).value = 42;@ interprets @*it@ as a @list@).
+While a nominal-inheritance system with associated types could model one of those two relationships by making @El@ an associated type of @Ptr@ in the @pointer_like@ implementation, few such systems could model both relationships simultaneously.
+\section{Generic Types}
+One of the known shortcomings of standard C is that it does not provide reusable type-safe abstractions for generic data structures and algorithms. Broadly speaking, there are three approaches to create data structures in C. One approach is to write bespoke data structures for each context in which they are needed. While this approach is flexible and supports integration with the C type-checker and tooling, it is also tedious and error-prone, especially for more complex data structures. A second approach is to use @void*@-based polymorphism. This approach is taken by the C standard library functions @qsort@ and @bsearch@, and does allow the use of common code for common functionality. However, basing all polymorphism on @void*@ eliminates the type-checker's ability to ensure that argument types are properly matched, often requires a number of extra function parameters, and also adds pointer indirection and dynamic allocation to algorithms and data structures that would not otherwise require them. A third approach to generic code is to use pre-processor macros to generate it -- this approach does allow the generated code to be both generic and type-checked, though any errors produced may be difficult to interpret. Furthermore, writing and invoking C code as preprocessor macros is unnatural and somewhat inflexible.
+Other C-like languages such as \CC and Java use \emph{generic types} to produce type-safe abstract data types. \CFA implements generic types with some care taken that the generic types design for \CFA integrates efficiently and naturally with the existing polymorphic functions in \CFA while retaining backwards compatibility with C; maintaining separate compilation is a particularly important constraint on the design. However, where the concrete parameters of the generic type are known, there is no extra overhead for the use of a generic type, as for \CC templates.
+A generic type can be declared by placing a @forall@ specifier on a @struct@ or @union@ declaration, and instantiated using a parenthesized list of types after the type name:
+\begin{lstlisting}
+forall(otype R, otype S) struct pair {
+    R first;
+    S second;
+};
+forall(otype T)
+T value( pair(const char*, T) p ) { return p.second; }
+forall(dtype F, otype T)
+T value_p( pair(F*, T*) p ) { return *p.second; }
+pair(const char*, int) p = { "magic", 42 };
+forall( otype T ) T value( pair( const char *, T ) p ) { return p.second; }
+forall( dtype F, otype T ) T value_p( pair( F *, T * ) p ) { return * p.second; }
+pair( const char *, int ) p = { "magic", 42 };
 int magic = value( p );
+pair(void*, int*) q = { 0, &p.second };
+pair( void *, int * ) q = { 0, &p.second };
 magic = value_p( q );
 double d = 1.0;
 pair(double*, double*) r = { &d, &d };
+pair( double *, double * ) r = { &d, &d };
 d = value_p( r );
 \end{lstlisting}
+\CFA classifies generic types as either \emph{concrete} or \emph{dynamic}. Concrete generic types have a fixed memory layout regardless of type parameters, while dynamic generic types vary in their in-memory layout depending on their type parameters. A type may have polymorphic parameters but still be concrete; in \CFA such types are called \emph{dtype-static}. Polymorphic pointers are an example of dtype-static types -- @forall(dtype T) T*@ is a polymorphic type, but for any @T@ chosen, @T*@ has exactly the same in-memory representation as a @void*@, and can therefore be represented by a @void*@ in code generation.
+\CFA generic types may also specify constraints on their argument type to be checked by the compiler. For example, consider the following declaration of a sorted set-type, which ensures that the set key supports equality and relational comparison:
+\begin{lstlisting}
+forall(otype Key | { _Bool ?==?(Key, Key); _Bool ?<?(Key, Key); })
+  struct sorted_set;
+\end{lstlisting}
+\subsection{Concrete Generic Types}
+The \CFA translator instantiates concrete generic types by template-expanding them to fresh struct types; concrete generic types can therefore be used with zero runtime overhead. To enable inter-operation among equivalent instantiations of a generic type, the translator saves the set of instantiations currently in scope and reuses the generated struct declarations where appropriate. For example, a function declaration that accepts or returns a concrete generic type produces a declaration for the instantiated struct in the same scope, which all callers that can see that declaration may reuse. As an example of the expansion, the concrete instantiation for @pair(const char*, int)@ looks like this:
+\CFA classifies generic types as either \emph{concrete} or \emph{dynamic}.
+Concrete types have a fixed memory layout regardless of type parameters, while dynamic types vary in memory layout depending on their type parameters.
+A type may have polymorphic parameters but still be concrete, called \emph{dtype-static}.
+Polymorphic pointers are an example of dtype-static types, \eg @forall(dtype T) T *@ is a polymorphic type, but for any @T@, @T *@  is a fixed-sized pointer, and therefore, can be represented by a @void *@ in code generation.
+\CFA generic types also allow checked argument-constraints.
+For example, the following declaration of a sorted set-type ensures the set key supports equality and relational comparison:
+\begin{lstlisting}
+forall( otype Key | { _Bool ?==?(Key, Key); _Bool ?<?(Key, Key); } ) struct sorted_set;
+\end{lstlisting}
+\subsection{Concrete Generic-Types}
+The \CFA translator template-expands concrete generic-types into new structure types, affording maximal inlining.
+To enable inter-operation among equivalent instantiations of a generic type, the translator saves the set of instantiations currently in scope and reuses the generated structure declarations where appropriate.
+For example, a function declaration that accepts or returns a concrete generic-type produces a declaration for the instantiated struct in the same scope, which all callers may reuse.
+For example, the concrete instantiation for @pair( const char *, int )@ is:
 \begin{lstlisting}
 struct _pair_conc1 {
         const char* first;
+        const char * first;
         int second;
 };
 \end{lstlisting}
+A concrete generic type with dtype-static parameters is also expanded to a struct type, but this struct type is used for all matching instantiations. In the example above, the @pair(F*, T*)@ parameter to @value_p@ is such a type; its expansion looks something like this, and is used as the type of the variables @q@ and @r@ as well, with casts for member access where appropriate:
+A concrete generic-type with dtype-static parameters is also expanded to a structure type, but this type is used for all matching instantiations.
+In the above example, the @pair( F *, T * )@ parameter to @value_p@ is such a type; its expansion is below and it is used as the type of the variables @q@ and @r@ as well, with casts for member access where appropriate:
 \begin{lstlisting}
 struct _pair_conc0 {
         void* first;
         void* second;
+        void * first;
+        void * second;
 };
 \end{lstlisting}
+\subsection{Dynamic Generic Types}
+Though \CFA implements concrete generic types efficiently, it also has a fully general system for computing with dynamic generic types. As mentioned in Section~\ref{sec:poly-fns}, @otype@ function parameters (in fact all @sized@ polymorphic parameters) come with implicit size and alignment parameters provided by the caller. Dynamic generic structs also have implicit size and alignment parameters, and also an \emph{offset array} which contains the offsets of each member of the struct\footnote{Dynamic generic unions need no such offset array, as all members are at offset 0; the size and alignment parameters are still provided for dynamic unions, however.}. Access to members\footnote{The \lstinline@offsetof@ macro is implemented similarly.} of a dynamic generic struct is provided by adding the corresponding member of the offset array to the struct pointer at runtime, essentially moving a compile-time offset calculation to runtime where necessary.
+These offset arrays are statically generated where possible. If a dynamic generic type is declared to be passed or returned by value from a polymorphic function, the translator can safely assume that the generic type is complete (that is, has a known layout) at any call-site, and the offset array is passed from the caller; if the generic type is concrete at the call site the elements of this offset array can even be statically generated using the C @offsetof@ macro. As an example, @p.second@ in the @value@ function above is implemented as @*(p + _offsetof_pair[1])@, where @p@ is a @void*@, and @_offsetof_pair@ is the offset array passed in to @value@ for @pair(const char*, T)@. The offset array @_offsetof_pair@ is generated at the call site as @size_t _offsetof_pair[] = { offsetof(_pair_conc1, first), offsetof(_pair_conc1, second) };@.
+In some cases the offset arrays cannot be statically generated. For instance, modularity is generally provided in C by including an opaque forward-declaration of a struct and associated accessor and mutator routines in a header file, with the actual implementations in a separately-compiled \texttt{.c} file. \CFA supports this pattern for generic types, and in this instance the caller does not know the actual layout or size of the dynamic generic type, and only holds it by pointer. The \CFA translator automatically generates \emph{layout functions} for cases where the size, alignment, and offset array of a generic struct cannot be passed in to a function from that function's caller. These layout functions take as arguments pointers to size and alignment variables and a caller-allocated array of member offsets, as well as the size and alignment of all @sized@ parameters to the generic struct (un-@sized@ parameters are forbidden from the language from being used in a context that affects layout). Results of these layout functions are cached so that they are only computed once per type per function.%, as in the example below for @pair@.
+% \begin{lstlisting}
+% static inline void _layoutof_pair(size_t* _szeof_pair, size_t* _alignof_pair, size_t* _offsetof_pair,
+%               size_t _szeof_R, size_t _alignof_R, size_t _szeof_S, size_t _alignof_S) {
+%     *_szeof_pair = 0; // default values
+%     *_alignof_pair = 1;
+%       // add offset, size, and alignment of first field
+%     _offsetof_pair[0] = *_szeof_pair;
+%     *_szeof_pair += _szeof_R;
+%     if ( *_alignof_pair < _alignof_R ) *_alignof_pair = _alignof_R;
+%       // padding, offset, size, and alignment of second field
+%     if ( *_szeof_pair & (_alignof_S - 1) )
+%               *_szeof_pair += (_alignof_S - ( *_szeof_pair & (_alignof_S - 1) ) );
+%     _offsetof_pair[1] = *_szeof_pair;
+%     *_szeof_pair += _szeof_S;
+%     if ( *_alignof_pair < _alignof_S ) *_alignof_pair = _alignof_S;
+%       // pad to struct alignment
+%     if ( *_szeof_pair & (*_alignof_pair - 1) )
+%               *_szeof_pair += ( *_alignof_pair - ( *_szeof_pair & (*_alignof_pair - 1) ) );
+% }
+% \end{lstlisting}
+Layout functions also allow generic types to be used in a function definition without reflecting them in the function signature. For instance, a function that strips duplicate values from an unsorted @vector(T)@ would likely have a pointer to the vector as its only explicit parameter, but use some sort of @set(T)@ internally to test for duplicate values. This function could acquire the layout for @set(T)@ by calling its layout function with the layout of @T@ implicitly passed into the function.
+Whether a type is concrete, dtype-static, or dynamic is decided based solely on the type parameters and @forall@ clause on the struct declaration. This design allows opaque forward declarations of generic types like @forall(otype T) struct Box;@ -- like in C, all uses of @Box(T)@ can be in a separately compiled translation unit, and callers from other translation units know the proper calling conventions to use. If the definition of a struct type was included in the decision of whether a generic type is dynamic or concrete, some further types may be recognized as dtype-static (\eg @forall(otype T) struct unique_ptr { T* p };@ does not depend on @T@ for its layout, but the existence of an @otype@ parameter means that it \emph{could}.), but preserving separate compilation (and the associated C compatibility) in the existing design is judged to be an appropriate trade-off.
+\subsection{Dynamic Generic-Types}
+Though \CFA implements concrete generic-types efficiently, it also has a fully general system for dynamic generic types.
+As mentioned in Section~\ref{sec:poly-fns}, @otype@ function parameters (in fact all @sized@ polymorphic parameters) come with implicit size and alignment parameters provided by the caller.
+Dynamic generic-types also have an \emph{offset array} containing structure-member offsets.
+A dynamic generic-union needs no such offset array, as all members are at offset 0, but size and alignment are still necessary.
+Access to members of a dynamic structure is provided at runtime via base-displacement addressing with the structure pointer and the member offset (similar to the @offsetof@ macro), moving a compile-time offset calculation to runtime.
+The offset arrays are statically generated where possible.
+If a dynamic generic-type is declared to be passed or returned by value from a polymorphic function, the translator can safely assume the generic type is complete (\ie has a known layout) at any call-site, and the offset array is passed from the caller;
+if the generic type is concrete at the call site, the elements of this offset array can even be statically generated using the C @offsetof@ macro.
+As an example, @p.second@ in the @value@ function above is implemented as @*(p + _offsetof_pair[1])@, where @p@ is a @void *@, and @_offsetof_pair@ is the offset array passed into @value@ for @pair( const char *, T )@.
+The offset array @_offsetof_pair@ is generated at the call site as @size_t _offsetof_pair[] = { offsetof(_pair_conc1, first), offsetof(_pair_conc1, second) }@.
+In some cases the offset arrays cannot be statically generated.
+For instance, modularity is generally provided in C by including an opaque forward-declaration of a structure and associated accessor and mutator functions in a header file, with the actual implementations in a separately-compiled @.c@ file.
+\CFA supports this pattern for generic types, but the caller does not know the actual layout or size of the dynamic generic-type, and only holds it by a pointer.
+The \CFA translator automatically generates \emph{layout functions} for cases where the size, alignment, and offset array of a generic struct cannot be passed into a function from that function's caller.
+These layout functions take as arguments pointers to size and alignment variables and a caller-allocated array of member offsets, as well as the size and alignment of all @sized@ parameters to the generic structure (un@sized@ parameters are forbidden from being used in a context that affects layout).
+Results of these layout functions are cached so that they are only computed once per type per function. %, as in the example below for @pair@.
+Layout functions also allow generic types to be used in a function definition without reflecting them in the function signature.
+For instance, a function that strips duplicate values from an unsorted @vector(T)@ would likely have a pointer to the vector as its only explicit parameter, but use some sort of @set(T)@ internally to test for duplicate values.
+This function could acquire the layout for @set(T)@ by calling its layout function with the layout of @T@ implicitly passed into the function.
+Whether a type is concrete, dtype-static, or dynamic is decided solely on the type parameters and @forall@ clause on a declaration.
+This design allows opaque forward declarations of generic types, \eg @forall(otype T) struct Box@ -- like in C, all uses of @Box(T)@ can be separately compiled, and callers from other translation units know the proper calling conventions to use.
+If the definition of a structure type is included in deciding whether a generic type is dynamic or concrete, some further types may be recognized as dtype-static (\eg @forall(otype T) struct unique_ptr { T * p }@ does not depend on @T@ for its layout, but the existence of an @otype@ parameter means that it \emph{could}.), but preserving separate compilation (and the associated C compatibility) in the existing design is judged to be an appropriate trade-off.
 \subsection{Applications}
 \label{sec:generic-apps}
+The reuse of dtype-static struct instantiations enables some useful programming patterns at zero runtime cost. The most important such pattern is using @forall(dtype T) T*@ as a type-checked replacement for @void*@, as in this example, which takes a @qsort@ or @bsearch@-compatible comparison routine and creates a similar lexicographic comparison for pairs of pointers:
+\begin{lstlisting}
+forall(dtype T)
+int lexcmp( pair(T*, T*)* a, pair(T*, T*)* b, int (*cmp)(T*, T*) ) {
+        int c = cmp(a->first, b->first);
+        if ( c == 0 ) c = cmp(a->second, b->second);
+        return c;
+}
+\end{lstlisting}
+Since @pair(T*, T*)@ is a concrete type, there are no added implicit parameters to @lexcmp@, so the code generated by \CFA is effectively identical to a version of this function written in standard C using @void*@, yet the \CFA version is type-checked to ensure that the fields of both pairs and the arguments to the comparison function match in type.
+Another useful pattern enabled by reused dtype-static type instantiations is zero-cost ``tag'' structs. Sometimes a particular bit of information is only useful for type-checking, and can be omitted at runtime. Tag structs can be used to provide this information to the compiler without further runtime overhead, as in the following example:
+The reuse of dtype-static structure instantiations enables useful programming patterns at zero runtime cost.
+The most important such pattern is using @forall(dtype T) T *@ as a type-checked replacement for @void *@, \eg creating a lexicographic comparison for pairs of pointers used by @bsearch@ or @qsort@:
+\begin{lstlisting}
+forall(dtype T) int lexcmp( pair( T *, T * ) * a, pair( T *, T * ) * b, int (* cmp)( T *, T * ) ) {
+        return cmp( a->first, b->first ) ? : cmp( a->second, b->second );
+}
+\end{lstlisting}
+Since @pair(T *, T * )@ is a concrete type, there are no implicit parameters passed to @lexcmp@, so the generated code is identical to a function written in standard C using @void *@, yet the \CFA version is type-checked to ensure the fields of both pairs and the arguments to the comparison function match in type.
+Another useful pattern enabled by reused dtype-static type instantiations is zero-cost \emph{tag-structures}.
+Sometimes information is only used for type-checking and can be omitted at runtime, \eg:
 \begin{lstlisting}
 forall(dtype Unit) struct scalar { unsigned long value; };
 struct metres {};
 struct litres {};
+forall(dtype U)
+scalar(U) ?+?(scalar(U) a, scalar(U) b) {
+forall(dtype U) scalar(U) ?+?( scalar(U) a, scalar(U) b ) {
         return (scalar(U)){ a.value + b.value };
+}
 scalar(metres) half_marathon = { 21093 };
 scalar(litres) swimming_pool = { 2500000 };
 scalar(metres) marathon = half_marathon + half_marathon;
 scalar(litres) two_pools = swimming_pool + swimming_pool;
+marathon + swimming_pool; // ERROR -- caught by compiler
+\end{lstlisting}
+@scalar@ is a dtype-static type, so all uses of it use a single struct definition, containing only a single @unsigned long@, and can share the same implementations of common routines like @?+?@ -- these implementations may even be separately compiled, unlike \CC template functions. However, the \CFA type-checker ensures that matching types are used by all calls to @?+?@, preventing nonsensical computations like adding the length of a marathon to the volume of an olympic pool.
+marathon + swimming_pool;                                       $\C{// compilation ERROR}$
+\end{lstlisting}
+@scalar@ is a dtype-static type, so all uses have a single structure definition, containing @unsigned long@, and can share the same implementations of common functions like @?+?@.
+These implementations may even be separately compiled, unlike \CC template functions.
+However, the \CFA type-checker ensures matching types are used by all calls to @?+?@, preventing nonsensical computations like adding a length to a volume.
 \section{Tuples}
 \label{sec:tuples}
+The @pair(R, S)@ generic type used as an example in the previous section can be considered a special case of a more general \emph{tuple} data structure. The authors have implemented tuples in \CFA, with a design particularly motivated by two use cases: \emph{multiple-return-value functions} and \emph{variadic functions}.
+In standard C, functions can return at most one value. This restriction results in code that emulates functions with multiple return values by \emph{aggregation} or by \emph{aliasing}. In the former situation, the function designer creates a record type that combines all of the return values into a single type. Unfortunately, the designer must come up with a name for the return type and for each of its fields. Unnecessary naming is a common programming language issue, introducing verbosity and a complication of the user's mental model. As such, this technique is effective when used sparingly, but can quickly get out of hand if many functions need to return different combinations of types. In the latter approach, the designer simulates multiple return values by passing the additional return values as pointer parameters. The pointer parameters are assigned inside of the routine body to emulate a return. Using this approach, the caller is directly responsible for allocating storage for the additional temporary return values. This responsibility complicates the call site with a sequence of variable declarations leading up to the call. Also, while a disciplined use of @const@ can give clues about whether a pointer parameter is going to be used as an out parameter, it is not immediately obvious from only the routine signature whether the callee expects such a parameter to be initialized before the call. Furthermore, while many C routines that accept pointers are designed so that it is safe to pass @NULL@ as a parameter, there are many C routines that are not null-safe. On a related note, C does not provide a standard mechanism to state that a parameter is going to be used as an additional return value, which makes the job of ensuring that a value is returned more difficult for the compiler.
+C does provide a mechanism for variadic functions through manipulation of @va_list@ objects, but it is notoriously type-unsafe. A variadic function is one that contains at least one parameter, followed by @...@ as the last token in the parameter list. In particular, some form of \emph{argument descriptor} is needed to inform the function of the number of arguments and their types, commonly a format string or counter parameter. It is important to note that both of these mechanisms are inherently redundant, because they require the user to specify information that the compiler knows explicitly. This required repetition is error prone, because it is easy for the user to add or remove arguments without updating the argument descriptor. In addition, C requires the programmer to hard code all of the possible expected types. As a result, it is cumbersome to write a variadic function that is open to extension. For example, consider a simple function that sums $N$ @int@s:
+\begin{lstlisting}
+int sum(int N, ...) {
+  va_list args;
+  va_start(args, N);  // must manually specify last non-variadic argument
+  int ret = 0;
+  while(N) {
+    ret += va_arg(args, int);  // must specify type
+    N--;
+  }
+  va_end(args);
+  return ret;
+}
+sum(3, 10, 20, 30);  // must keep initial counter argument in sync
+\end{lstlisting}
+The @va_list@ type is a special C data type that abstracts variadic argument manipulation. The @va_start@ macro initializes a @va_list@, given the last named parameter. Each use of the @va_arg@ macro allows access to the next variadic argument, given a type. Since the function signature does not provide any information on what types can be passed to a variadic function, the compiler does not perform any error checks on a variadic call. As such, it is possible to pass any value to the @sum@ function, including pointers, floating-point numbers, and structures. In the case where the provided type is not compatible with the argument's actual type after default argument promotions, or if too many arguments are accessed, the behaviour is undefined~\citep{C11}. Furthermore, there is no way to perform the necessary error checks in the @sum@ function at run-time, since type information is not carried into the function body. Since they rely on programmer convention rather than compile-time checks, variadic functions are inherently unsafe.
+In practice, compilers can provide warnings to help mitigate some of the problems. For example, GCC provides the @format@ attribute to specify that a function uses a format string, which allows the compiler to perform some checks related to the standard format specifiers. Unfortunately, this attribute does not permit extensions to the format string syntax, so a programmer cannot extend it to warn for mismatches with custom types.
+In many languages, functions can return at most one value;
+however, many operations have multiple outcomes, some exceptional.
+Consider C's @div@ and @remquo@ functions, which return the quotient and remainder for a division of integer and floating-point values, respectively.
+\begin{lstlisting}
+typedef struct { int quo, rem; } div_t;         $\C{// from include stdlib.h}$
+div_t div( int num, int den );
+double remquo( double num, double den, int * quo );
+div_t qr = div( 13, 5 );                                        $\C{// return quotient/remainder aggregate}$
+int q;
+double r = remquo( 13.5, 5.2, &q );                     $\C{// return remainder, alias quotient}$
+\end{lstlisting}
+@div@ aggregates the quotient/remainder in a structure, while @remquo@ aliases a parameter to an argument.
+Both approaches are awkward.
+Alternatively, a programming language can directly support returning multiple values, \eg in \CFA:
+\begin{lstlisting}
+[ int, int ] div( int num, int den );           $\C{// return two integers}$
+[ double, double ] div( double num, double den ); $\C{// return two doubles}$
+int q, r;                                                                       $\C{// overloaded variable names}$
+double q, r;
+[ q, r ] = div( 13, 5 );                                        $\C{// select appropriate div and q, r}$
+[ q, r ] = div( 13.5, 5.2 );                            $\C{// assign into tuple}$
+\end{lstlisting}
+Clearly, this approach is straightforward to understand and use;
+therefore, why do few programming languages support this obvious feature or provide it awkwardly?
+The answer is that there are complex consequences that cascade through multiple aspects of the language, especially the type-system.
+This section show these consequences and how \CFA handles them.
 \subsection{Tuple Expressions}
+The tuple extensions in \CFA can express multiple return values and variadic function parameters in an efficient and type-safe manner. \CFA introduces \emph{tuple expressions} and \emph{tuple types}. A tuple expression is an expression producing a fixed-size, ordered list of values of heterogeneous types. The type of a tuple expression is the tuple of the subexpression types, or a \emph{tuple type}. In \CFA, a tuple expression is denoted by a comma-separated list of expressions enclosed in square brackets. For example, the expression @[5, 'x', 10.5]@ has type @[int, char, double]@. The previous expression has three \emph{components}. Each component in a tuple expression can be any \CFA expression, including another tuple expression. The order of evaluation of the components in a tuple expression is unspecified, to allow a compiler the greatest flexibility for program optimization. It is, however, guaranteed that each component of a tuple expression is evaluated for side-effects, even if the result is not used. Multiple-return-value functions can equivalently be called \emph{tuple-returning functions}.
+\CFA allows declaration of \emph{tuple variables}, variables of tuple type. For example:
+\begin{lstlisting}
+[int, char] most_frequent(const char*);
+const char* str = "hello, world!";
+[int, char] freq = most_frequent(str);
+printf("%s -- %d %c\n", str, freq);
+\end{lstlisting}
+In this example, the type of the @freq@ and the return type of @most_frequent@ are both tuple types. Also of note is how the tuple expression @freq@ is implicitly flattened into separate @int@ and @char@ arguments to @printf@; this code snippet could have been shortened by replacing the last two lines with @printf("%s -- %d %c\n", str, most_frequent(str));@ using exactly the same mechanism.
+In addition to variables of tuple type, it is also possible to have pointers to tuples, and arrays of tuples. Tuple types can be composed of any types, except for array types, since arrays are not of fixed size, which makes tuple assignment difficult when a tuple contains an array.
+\begin{lstlisting}
+[double, int] di;
+[double, int] * pdi
+[double, int] adi[10];
+\end{lstlisting}
+This example declares a variable of type @[double, int]@, a variable of type pointer to @[double, int]@, and an array of ten @[double, int]@.
+The addition of multiple-return-value functions (MRVF) are useless without a syntax for accepting multiple values at the call-site.
+The simplest mechanism for capturing the return values is variable assignment, allowing the values to be retrieved directly.
+As such, \CFA allows assigning multiple values from a function into multiple variables, using a square-bracketed list of lvalue expressions (as above), called a \emph{tuple}.
+However, functions also use \emph{composition} (nested calls), with the direct consequence that MRVFs must also support composition to be orthogonal with single-returning-value functions (SRVF), \eg:
+\begin{lstlisting}
+printf( "%d %d\n", div( 13, 5 ) );                      $\C{// return values seperated into arguments}$
+\end{lstlisting}
+Here, the values returned by @div@ are composed with the call to @printf@ by flattening the tuple into separate arguments.
+However, the \CFA type-system must support significantly more complex composition:
+\begin{lstlisting}
+[ int, int ] foo$\(_1\)$( int );
+[ double ] foo$\(_2\)$( int );
+void bar( int, double, double );
+bar( foo( 3 ), foo( 3 ) );
+\end{lstlisting}
+The type-resolver only has the tuple return-types to resolve the call to @bar@ as the @foo@ parameters are identical, which involves unifying the possible @foo@ functions with @bar@'s parameter list.
+No combination of @foo@s are an exact match with @bar@'s parameters, so the resolver applies C conversions.
+The minimal cost is @bar( foo@$_1$@( 3 ), foo@$_2$@( 3 ) )@, giving (@int@, {\color{ForestGreen}@int@}, @double@) to (@int@, {\color{ForestGreen}@double@}, @double@) with one {\color{ForestGreen}safe} (widening) conversion from @int@ to @double@ versus ({\color{red}@double@}, {\color{ForestGreen}@int@}, {\color{ForestGreen}@int@}) to ({\color{red}@int@}, {\color{ForestGreen}@double@}, {\color{ForestGreen}@double@}) with one {\color{red}unsafe} (narrowing) conversion from @double@ to @int@ and two safe conversions.
+\subsection{Tuple Variables}
+An important observation from function composition is that new variable names are not required to initialize parameters from an MRVF.
+\CFA also allows declaration of tuple variables that can be initialized from an MRVF, since it can be awkward to declare multiple variables of different types, \eg:
+\begin{lstlisting}
+[ int, int ] qr = div( 13, 5 );                         $\C{// tuple-variable declaration and initialization}$
+[ double, double ] qr = div( 13.5, 5.2 );
+\end{lstlisting}
+where the tuple variable-name serves the same purpose as the parameter name(s).
+Tuple variables can be composed of any types, except for array types, since array sizes are generally unknown.
+One way to access the tuple-variable components is with assignment or composition:
+\begin{lstlisting}
+[ q, r ] = qr;                                                          $\C{// access tuple-variable components}$
+printf( "%d %d\n", qr );
+\end{lstlisting}
+\CFA also supports \emph{tuple indexing} to access single components of a tuple expression:
+\begin{lstlisting}
+[int, int] * p = &qr;                                           $\C{// tuple pointer}$
+int rem = qr.1;                                                         $\C{// access remainder}$
+int quo = div( 13, 5 ).0;                                       $\C{// access quotient}$
+p->0 = 5;                                                                       $\C{// change quotient}$
+bar( qr.1, qr );                                                        $\C{// pass remainder and quotient/remainder}$
+rem = [42, div( 13, 5 )].0.1;                           $\C{// access 2nd component of 1st component of tuple expression}$
+\end{lstlisting}
 \subsection{Flattening and Restructuring}
+In function call contexts, tuples support implicit flattening and restructuring conversions. Tuple flattening recursively expands a tuple into the list of its basic components. Tuple structuring packages a list of expressions into a value of tuple type.
+\begin{lstlisting}
+int f(int, int);
+int g([int, int]);
+int h(int, [int, int]);
+In function call contexts, tuples support implicit flattening and restructuring conversions.
+Tuple flattening recursively expands a tuple into the list of its basic components.
+Tuple structuring packages a list of expressions into a value of tuple type, \eg:
+%\lstDeleteShortInline@%
+%\par\smallskip
+%\begin{tabular}{@{}l@{\hspace{1.5\parindent}}||@{\hspace{1.5\parindent}}l@{}}
+\begin{lstlisting}
+int f( int, int );
+int g( [int, int] );
+int h( int, [int, int] );
 [int, int] x;
 int y;
+f(x);      // flatten
+g(y, 10);  // structure
+h(x, y);   // flatten & structure
+\end{lstlisting}
+In \CFA, each of these calls is valid. In the call to @f@, @x@ is implicitly flattened so that the components of @x@ are passed as the two arguments to @f@. For the call to @g@, the values @y@ and @10@ are structured into a single argument of type @[int, int]@ to match the type of the parameter of @g@. Finally, in the call to @h@, @y@ is flattened to yield an argument list of length 3, of which the first component of @x@ is passed as the first parameter of @h@, and the second component of @x@ and @y@ are structured into the second argument of type @[int, int]@. The flexible structure of tuples permits a simple and expressive function call syntax to work seamlessly with both single- and multiple-return-value functions, and with any number of arguments of arbitrarily complex structure.
+% In {K-W C} \citep{Buhr94a,Till89}, a precursor to \CFA, there were 4 tuple coercions: opening, closing, flattening, and structuring. Opening coerces a tuple value into a tuple of values, while closing converts a tuple of values into a single tuple value. Flattening coerces a nested tuple into a flat tuple, \ie it takes a tuple with tuple components and expands it into a tuple with only non-tuple components. Structuring moves in the opposite direction, \ie it takes a flat tuple value and provides structure by introducing nested tuple components.
+In \CFA, the design has been simplified to require only the two conversions previously described, which trigger only in function call and return situations. Specifically, the expression resolution algorithm examines all of the possible alternatives for an expression to determine the best match. In resolving a function call expression, each combination of function value and list of argument alternatives is examined. Given a particular argument list and function value, the list of argument alternatives is flattened to produce a list of non-tuple valued expressions. Then the flattened list of expressions is compared with each value in the function's parameter list. If the parameter's type is not a tuple type, then the current argument value is unified with the parameter type, and on success the next argument and parameter are examined. If the parameter's type is a tuple type, then the structuring conversion takes effect, recursively applying the parameter matching algorithm using the tuple's component types as the parameter list types. Assuming a successful unification, eventually the algorithm gets to the end of the tuple type, which causes all of the matching expressions to be consumed and structured into a tuple expression. For example, in
+\begin{lstlisting}
+int f(int, [double, int]);
+f([5, 10.2], 4);
+\end{lstlisting}
+There is only a single definition of @f@, and 3 arguments with only single interpretations. First, the argument alternative list @[5, 10.2], 4@ is flattened to produce the argument list @5, 10.2, 4@. Next, the parameter matching algorithm begins, with $P =~$@int@ and $A =~$@int@, which unifies exactly. Moving to the next parameter and argument, $P =~$@[double, int]@ and $A =~$@double@. This time, the parameter is a tuple type, so the algorithm applies recursively with $P' =~$@double@ and $A =~$@double@, which unifies exactly. Then $P' =~$@int@ and $A =~$@double@, which again unifies exactly. At this point, the end of $P'$ has been reached, so the arguments @10.2, 4@ are structured into the tuple expression @[10.2, 4]@. Finally, the end of the parameter list $P$ has also been reached, so the final expression is @f(5, [10.2, 4])@.
+f( x );                 $\C{// flatten}$
+g( y, 10 );             $\C{// structure}$
+h( x, y );              $\C{// flatten and structure}$
+\end{lstlisting}
+%\end{lstlisting}
+%&
+%\begin{lstlisting}
+%\end{tabular}
+%\smallskip\par\noindent
+%\lstMakeShortInline@%
+In the call to @f@, @x@ is implicitly flattened so the components of @x@ are passed as the two arguments.
+In the call to @g@, the values @y@ and @10@ are structured into a single argument of type @[int, int]@ to match the parameter type of @g@.
+Finally, in the call to @h@, @x@ is flattened to yield an argument list of length 3, of which the first component of @x@ is passed as the first parameter of @h@, and the second component of @x@ and @y@ are structured into the second argument of type @[int, int]@.
+The flexible structure of tuples permits a simple and expressive function call syntax to work seamlessly with both SRVF and MRVF, and with any number of arguments of arbitrarily complex structure.
+\subsection{Tuple Assignment}
+An assignment where the left side is a tuple type is called \emph{tuple assignment}.
+There are two kinds of tuple assignment depending on whether the right side of the assignment operator has a tuple type or a non-tuple type, called \emph{multiple} and \emph{mass assignment}, respectively.
+%\lstDeleteShortInline@%
+%\par\smallskip
+%\begin{tabular}{@{}l@{\hspace{1.5\parindent}}||@{\hspace{1.5\parindent}}l@{}}
+\begin{lstlisting}
+int x = 10;
+double y = 3.5;
+[int, double] z;
+z = [x, y];                                                                     $\C{// multiple assignment}$
+[x, y] = z;                                                                     $\C{// multiple assignment}$
+z = 10;                                                                         $\C{// mass assignment}$
+[y, x] = 3.14;                                                          $\C{// mass assignment}$
+\end{lstlisting}
+%\end{lstlisting}
+%&
+%\begin{lstlisting}
+%\end{tabular}
+%\smallskip\par\noindent
+%\lstMakeShortInline@%
+Both kinds of tuple assignment have parallel semantics, so that each value on the left and right side is evaluated before any assignments occur.
+As a result, it is possible to swap the values in two variables without explicitly creating any temporary variables or calling a function, \eg, @[x, y] = [y, x]@.
+This semantics means mass assignment differs from C cascading assignment (\eg @a = b = c@) in that conversions are applied in each individual assignment, which prevents data loss from the chain of conversions that can happen during a cascading assignment.
+For example, @[y, x] = 3.14@ performs the assignments @y = 3.14@ and @x = 3.14@, yielding @y == 3.14@ and @x == 3@;
+whereas C cascading assignment @y = x = 3.14@ performs the assignments @x = 3.14@ and @y = x@, yielding @3@ in @y@ and @x@.
+Finally, tuple assignment is an expression where the result type is the type of the left-hand side of the assignment, just like all other assignment expressions in C.
+This example shows mass, multiple, and cascading assignment used in one expression:
+\begin{lstlisting}
+void f( [int, int] );
+f( [x, y] = z = 1.5 );                                          $\C{// assignments in parameter list}$
+\end{lstlisting}
 \subsection{Member Access}
+At times, it is desirable to access a single component of a tuple-valued expression without creating unnecessary temporary variables to assign to. Given a tuple-valued expression @e@ and a compile-time constant integer $i$ where $0 \leq i < n$, where $n$ is the number of components in @e@, @e.i@ accesses the $i$\textsuperscript{th} component of @e@. For example,
+\begin{lstlisting}
+[int, double] x;
+[char *, int] f();
+void g(double, int);
+[int, double] * p;
+int y = x.0;  // access int component of x
+y = f().1;  // access int component of f
+p->0 = 5;  // access int component of tuple pointed-to by p
+g(x.1, x.0);  // rearrange x to pass to g
+double z = [x, f()].0.1;  // access second component of first component of tuple expression
+\end{lstlisting}
+As seen above, tuple-index expressions can occur on any tuple-typed expression, including tuple-returning functions, square-bracketed tuple expressions, and other tuple-index expressions, provided the retrieved component is also a tuple. This feature was proposed for {K-W C}, but never implemented~\citep[p.~45]{Till89}.
+It is possible to access multiple fields from a single expression using a \emph{member-access tuple expression}. The result is a single tuple expression whose type is the tuple of the types of the members. For example,
+It is also possible to access multiple fields from a single expression using a \emph{member-access}.
+The result is a single tuple-valued expression whose type is the tuple of the types of the members, \eg:
 \begin{lstlisting}
 struct S { int x; double y; char * z; } s;
+s.[x, y, z];
+\end{lstlisting}
+Here, the type of @s.[x, y, z]@ is @[int, double, char *]@. A member tuple expression has the form @a.[x, y, z];@ where @a@ is an expression with type @T@, where @T@ supports member access expressions, and @x, y, z@ are all members of @T@ with types @T$_x$@, @T$_y$@, and @T$_z$@ respectively. Then the type of @a.[x, y, z]@ is @[T$_x$, T$_y$, T$_z$]@.
+Since tuple index expressions are a form of member-access expression, it is possible to use tuple-index expressions in conjunction with member tuple expressions to manually restructure a tuple (\eg rearrange components, drop components, duplicate components, etc.):
+s.[x, y, z] = 0;
+\end{lstlisting}
+Here, the mass assignment sets all members of @s@ to zero.
+Since tuple-index expressions are a form of member-access expression, it is possible to use tuple-index expressions in conjunction with member tuple expressions to manually restructure a tuple (\eg rearrange, drop, and duplicate components).
+%\lstDeleteShortInline@%
+%\par\smallskip
+%\begin{tabular}{@{}l@{\hspace{1.5\parindent}}||@{\hspace{1.5\parindent}}l@{}}
 \begin{lstlisting}
 [int, int, long, double] x;
+void f(double, long);
+f(x.[0, 3]);          // f(x.0, x.3)
+x.[0, 1] = x.[1, 0];  // [x.0, x.1] = [x.1, x.0]
+[long, int, long] y = x.[2, 0, 2];
+\end{lstlisting}
+It is possible for a member tuple expression to contain other member access expressions:
+void f( double, long );
+x.[0, 1] = x.[1, 0];                                            $\C{// rearrange: [x.0, x.1] = [x.1, x.0]}$
+f( x.[0, 3] );                                                          $\C{// drop: f(x.0, x.3)}$
+[int, int, int] y = x.[2, 0, 2];                        $\C{// duplicate: [y.0, y.1, y.2] = [x.2, x.0.x.2]}$
+\end{lstlisting}
+%\end{lstlisting}
+%&
+%\begin{lstlisting}
+%\end{tabular}
+%\smallskip\par\noindent
+%\lstMakeShortInline@%
+It is also possible for a member access to contain other member accesses, \eg:
 \begin{lstlisting}
 struct A { double i; int j; };
 struct B { int * k; short l; };
 struct C { int x; A y; B z; } v;
+v.[x, y.[i, j], z.k];
+\end{lstlisting}
+This expression is equivalent to @[v.x, [v.y.i, v.y.j], v.z.k]@. That is, the aggregate expression is effectively distributed across the tuple, which allows simple and easy access to multiple components in an aggregate, without repetition. It is guaranteed that the aggregate expression to the left of the @.@ in a member tuple expression is evaluated exactly once. As such, it is safe to use member tuple expressions on the result of a side-effecting function.
+\subsection{Tuple Assignment}
+In addition to tuple-index expressions, individual components of tuples can be accessed by a \emph{destructuring assignment} which has a tuple expression with lvalue components on its left-hand side. More generally, an assignment where the left-hand side of the assignment operator has a tuple type is called \emph{tuple assignment}. There are two kinds of tuple assignment depending on whether the right-hand side of the assignment operator has a tuple type or a non-tuple type, called \emph{multiple assignment} and \emph{mass assignment}, respectively.
+\begin{lstlisting}
+int x;
+double y;
+[int, double] z;
+[y, x] = 3.14;  // mass assignment
+[x, y] = z;     // multiple assignment
+z = 10;         // mass assignment
+z = [x, y];     // multiple assignment
+\end{lstlisting}
+Let $L_i$ for $i$ in $[0, n)$ represent each component of the flattened left-hand side, $R_i$ represent each component of the flattened right-hand side of a multiple assignment, and $R$ represent the right-hand side of a mass assignment.
+For a multiple assignment to be valid, both tuples must have the same number of elements when flattened. Multiple assignment assigns $R_i$ to $L_i$ for each $i$.
+That is, @?=?(&$L_i$, $R_i$)@ must be a well-typed expression. In the previous example, @[x, y] = z@, @z@ is flattened into @z.0, z.1@, and the assignments @x = z.0@ and @y = z.1@ are executed.
+A mass assignment assigns the value $R$ to each $L_i$. For a mass assignment to be valid, @?=?(&$L_i$, $R$)@ must be a well-typed expression. This rule differs from C cascading assignment (\eg @a=b=c@) in that conversions are applied to $R$ in each individual assignment, which prevents data loss from the chain of conversions that can happen during a cascading assignment. For example, @[y, x] = 3.14@ performs the assignments @y = 3.14@ and @x = 3.14@, which results in the value @3.14@ in @y@ and the value @3@ in @x@. On the other hand, the C cascading assignment @y = x = 3.14@ performs the assignments @x = 3.14@ and @y = x@, which results in the value @3@ in @x@, and as a result the value @3@ in @y@ as well.
+Both kinds of tuple assignment have parallel semantics, such that each value on the left side and right side is evaluated \emph{before} any assignments occur. As a result, it is possible to swap the values in two variables without explicitly creating any temporary variables or calling a function:
+\begin{lstlisting}
+int x = 10, y = 20;
+[x, y] = [y, x];
+\end{lstlisting}
+After executing this code, @x@ has the value @20@ and @y@ has the value @10@.
+Tuple assignment is an expression where the result type is the type of the left-hand side of the assignment, just like all other assignment expressions in C. This definition allows cascading tuple assignment and use of tuple assignment in other expression contexts, an occasionally useful idiom to keep code succinct and reduce repetition.
+% In \CFA, tuple assignment is an expression where the result type is the type of the left-hand side of the assignment, as in normal assignment. That is, a tuple assignment produces the value of the left-hand side after assignment. These semantics allow cascading tuple assignment to work out naturally in any context where a tuple is permitted. These semantics are a change from the original tuple design in {K-W C}~\citep{Till89}, wherein tuple assignment was a statement that allows cascading assignments as a special case. This decision was made in an attempt to fix what was seen as a problem with assignment, wherein it can be used in many different locations, such as in function-call argument position. While permitting assignment as an expression does introduce the potential for subtle complexities, it is impossible to remove assignment expressions from \CFA without affecting backwards compatibility with C. Furthermore, there are situations where permitting assignment as an expression improves readability by keeping code succinct and reducing repetition, and complicating the definition of tuple assignment puts a greater cognitive burden on the user. In another language, tuple assignment as a statement could be reasonable, but it would be inconsistent for tuple assignment to be the only kind of assignment in \CFA that is not an expression.
+v.[x, y.[i, j], z.k];                                           $\C{// [v.x, [v.y.i, v.y.j], v.z.k]}$
+\end{lstlisting}
+\begin{comment}
 \subsection{Casting}
+In C, the cast operator is used to explicitly convert between types. In \CFA, the cast operator has a secondary use as type ascription. That is, a cast can be used to select the type of an expression when it is ambiguous, as in the call to an overloaded function:
+In C, the cast operator is used to explicitly convert between types.
+In \CFA, the cast operator has a secondary use as type ascription.
+That is, a cast can be used to select the type of an expression when it is ambiguous, as in the call to an overloaded function:
 \begin{lstlisting}
 int f();     // (1)
 …
 \end{lstlisting}
+Since casting is a fundamental operation in \CFA, casts should be given a meaningful interpretation in the context of tuples. Taking a look at standard C provides some guidance with respect to the way casts should work with tuples:
+Since casting is a fundamental operation in \CFA, casts should be given a meaningful interpretation in the context of tuples.
+Taking a look at standard C provides some guidance with respect to the way casts should work with tuples:
 \begin{lstlisting}
 int f();
 …
 (int)g();  // (2)
 \end{lstlisting}
+In C, (1) is a valid cast, which calls @f@ and discards its result. On the other hand, (2) is invalid, because @g@ does not produce a result, so requesting an @int@ to materialize from nothing is nonsensical. Generalizing these principles, any cast wherein the number of components increases as a result of the cast is invalid, while casts that have the same or fewer number of components may be valid.
+Formally, a cast to tuple type is valid when $T_n \leq S_m$, where $T_n$ is the number of components in the target type and $S_m$ is the number of components in the source type, and for each $i$ in $[0, n)$, $S_i$ can be cast to $T_i$. Excess elements ($S_j$ for all $j$ in $[n, m)$) are evaluated, but their values are discarded so that they are not included in the result expression. This approach follows naturally from the way that a cast to @void@ works in C.
+In C, (1) is a valid cast, which calls @f@ and discards its result.
+On the other hand, (2) is invalid, because @g@ does not produce a result, so requesting an @int@ to materialize from nothing is nonsensical.
+Generalizing these principles, any cast wherein the number of components increases as a result of the cast is invalid, while casts that have the same or fewer number of components may be valid.
+Formally, a cast to tuple type is valid when $T_n \leq S_m$, where $T_n$ is the number of components in the target type and $S_m$ is the number of components in the source type, and for each $i$ in $[0, n)$, $S_i$ can be cast to $T_i$.
+Excess elements ($S_j$ for all $j$ in $[n, m)$) are evaluated, but their values are discarded so that they are not included in the result expression.
+This approach follows naturally from the way that a cast to @void@ works in C.
 For example, in
 \begin{lstlisting}
+  [int, int, int] f();
+  [int, [int, int], int] g();
+  ([int, double])f();           $\C{// (1)}$
+  ([int, int, int])g();         $\C{// (2)}$
+  ([void, [int, int]])g();      $\C{// (3)}$
+  ([int, int, int, int])g();    $\C{// (4)}$
+  ([int, [int, int, int]])g();  $\C{// (5)}$
+\end{lstlisting}
+(1) discards the last element of the return value and converts the second element to @double@. Since @int@ is effectively a 1-element tuple, (2) discards the second component of the second element of the return value of @g@. If @g@ is free of side effects, this expression is equivalent to @[(int)(g().0), (int)(g().1.0), (int)(g().2)]@.
+[int, int, int] f();
+[int, [int, int], int] g();
+([int, double])f();           $\C{// (1)}$
+([int, int, int])g();         $\C{// (2)}$
+([void, [int, int]])g();      $\C{// (3)}$
+([int, int, int, int])g();    $\C{// (4)}$
+([int, [int, int, int]])g();  $\C{// (5)}$
+\end{lstlisting}
+(1) discards the last element of the return value and converts the second element to @double@.
+Since @int@ is effectively a 1-element tuple, (2) discards the second component of the second element of the return value of @g@.
+If @g@ is free of side effects, this expression is equivalent to @[(int)(g().0), (int)(g().1.0), (int)(g().2)]@.
 Since @void@ is effectively a 0-element tuple, (3) discards the first and third return values, which is effectively equivalent to @[(int)(g().1.0), (int)(g().1.1)]@).
+Note that a cast is not a function call in \CFA, so flattening and structuring conversions do not occur for cast expressions\footnote{User-defined conversions have been considered, but for compatibility with C and the existing use of casts as type ascription, any future design for such conversions would require more precise matching of types than allowed for function arguments and parameters.}. As such, (4) is invalid because the cast target type contains 4 components, while the source type contains only 3. Similarly, (5) is invalid because the cast @([int, int, int])(g().1)@ is invalid. That is, it is invalid to cast @[int, int]@ to @[int, int, int]@.
+Note that a cast is not a function call in \CFA, so flattening and structuring conversions do not occur for cast expressions\footnote{User-defined conversions have been considered, but for compatibility with C and the existing use of casts as type ascription, any future design for such conversions would require more precise matching of types than allowed for function arguments and parameters.}.
+As such, (4) is invalid because the cast target type contains 4 components, while the source type contains only 3.
+Similarly, (5) is invalid because the cast @([int, int, int])(g().1)@ is invalid.
+That is, it is invalid to cast @[int, int]@ to @[int, int, int]@.
+\end{comment}
 \subsection{Polymorphism}
+Tuples also integrate with \CFA polymorphism as a special sort of generic type. Due to the implicit flattening and structuring conversions involved in argument passing, @otype@ and @dtype@ parameters are restricted to matching only with non-tuple types.
+\begin{lstlisting}
+forall(otype T, dtype U)
+void f(T x, U * y);
+f([5, "hello"]);
+\end{lstlisting}
+In this example, @[5, "hello"]@ is flattened, so that the argument list appears as @5, "hello"@. The argument matching algorithm binds @T@ to @int@ and @U@ to @const char*@, and calls the function as normal.
+Tuples, however, may contain polymorphic components. For example, a plus operator can be written to add two triples of a type together.
+\begin{lstlisting}
+forall(otype T | { T ?+?(T, T); })
+[T, T, T] ?+?([T, T, T] x, [T, T, T] y) {
+  return [x.0+y.0, x.1+y.1, x.2+y.2];
+Tuples also integrate with \CFA polymorphism as a kind of generic type.
+Due to the implicit flattening and structuring conversions involved in argument passing, @otype@ and @dtype@ parameters are restricted to matching only with non-tuple types, \eg:
+\begin{lstlisting}
+forall(otype T, dtype U) void f( T x, U * y );
+f( [5, "hello"] );
+\end{lstlisting}
+where @[5, "hello"]@ is flattened, giving argument list @5, "hello"@, and @T@ binds to @int@ and @U@ binds to @const char@.
+Tuples, however, may contain polymorphic components.
+For example, a plus operator can be written to add two triples together.
+\begin{lstlisting}
+forall(otype T | { T ?+?( T, T ); }) [T, T, T] ?+?( [T, T, T] x, [T, T, T] y ) {
+        return [x.0 + y.0, x.1 + y.1, x.2 + y.2];
+}
 [int, int, int] x;
 …
 \end{lstlisting}
+Flattening and restructuring conversions are also applied to tuple types in polymorphic type assertions. Previously in \CFA, it has been assumed that assertion arguments must match the parameter type exactly, modulo polymorphic specialization (\ie no implicit conversions are applied to assertion arguments). In the example below:
+\begin{lstlisting}
+int f([int, double], double);
+forall(otype T, otype U | { T f(T, U, U); })
+void g(T, U);
+g(5, 10.21);
+\end{lstlisting}
+If assertion arguments must match exactly, then the call to @g@ cannot be resolved, since the expected type of @f@ is flat, while the only @f@ in scope requires a tuple type. Since tuples are fluid, this requirement reduces the usability of tuples in polymorphic code. To ease this pain point, function parameter and return lists are flattened for the purposes of type unification, which allows the previous example to pass expression resolution.
+This relaxation is made possible by extending the existing thunk generation scheme, as described by \citet{Bilson03}. Now, whenever a candidate's parameter structure does not exactly match the formal parameter's structure, a thunk is generated to specialize calls to the actual function:
+\begin{lstlisting}
+int _thunk(int _p0, double _p1, double _p2) {
+  return f([_p0, _p1], _p2);
+}
+\end{lstlisting}
+Essentially, this thunk provides flattening and structuring conversions to inferred functions, improving the compatibility of tuples and polymorphism. These thunks take advantage of GCC C nested functions to produce closures that have the usual function pointer signature.
+Flattening and restructuring conversions are also applied to tuple types in polymorphic type assertions.
+\begin{lstlisting}
+int f( [int, double], double );
+forall(otype T, otype U | { T f( T, U, U ); }) void g( T, U );
+g( 5, 10.21 );
+\end{lstlisting}
+Hence, function parameter and return lists are flattened for the purposes of type unification allowing the example to pass expression resolution.
+This relaxation is possible by extending the thunk scheme described by \citet{Bilson03}.
+Whenever a candidate's parameter structure does not exactly match the formal parameter's structure, a thunk is generated to specialize calls to the actual function:
+\begin{lstlisting}
+int _thunk( int _p0, double _p1, double _p2 ) { return f( [_p0, _p1], _p2 ); }
+\end{lstlisting}
+so the thunk provides flattening and structuring conversions to inferred functions, improving the compatibility of tuples and polymorphism.
+These thunks take advantage of GCC C nested-functions to produce closures that have the usual function pointer signature.
 \subsection{Variadic Tuples}
+To define variadic functions, \CFA adds a new kind of type parameter, @ttype@. Matching against a @ttype@ (``tuple type'') parameter consumes all remaining argument components and packages them into a tuple, binding to the resulting tuple of types. In a given parameter list, there should be at most one @ttype@ parameter that must occur last, otherwise the call can never resolve, given the previous rule. This idea essentially matches normal variadic semantics, with a strong feeling of similarity to \CCeleven variadic templates. As such, @ttype@ variables are also referred to as \emph{argument} or \emph{parameter packs} in this paper.
+Like variadic templates, the main way to manipulate @ttype@ polymorphic functions is through recursion. Since nothing is known about a parameter pack by default, assertion parameters are key to doing anything meaningful. Unlike variadic templates, @ttype@ polymorphic functions can be separately compiled.
+For example, the C @sum@ function at the beginning of Section~\ref{sec:tuples} could be written using @ttype@ as:
+\begin{lstlisting}
+int sum(){ return 0; }        // (0)
+forall(ttype Params | { int sum(Params); })
+int sum(int x, Params rest) { // (1)
+  return x+sum(rest);
+}
+sum(10, 20, 30);
+\end{lstlisting}
+Since (0) does not accept any arguments, it is not a valid candidate function for the call @sum(10, 20, 30)@.
+In order to call (1), @10@ is matched with @x@, and the argument resolution moves on to the argument pack @rest@, which consumes the remainder of the argument list and @Params@ is bound to @[20, 30]@.
+In order to finish the resolution of @sum@, an assertion parameter that matches @int sum(int, int)@ is required.
+Like in the previous iteration, (0) is not a valid candidate, so (1) is examined with @Params@ bound to @[int]@, requiring the assertion @int sum(int)@.
+Next, (0) fails, and to satisfy (1) @Params@ is bound to @[]@, requiring an assertion @int sum()@.
+Finally, (0) matches and (1) fails, which terminates the recursion.
+Effectively, this algorithm traces as @sum(10, 20, 30)@ $\rightarrow$ @10+sum(20, 30)@ $\rightarrow$ @10+(20+sum(30))@ $\rightarrow$ @10+(20+(30+sum()))@ $\rightarrow$ @10+(20+(30+0))@.
+As a point of note, this version does not require any form of argument descriptor, since the \CFA type system keeps track of all of these details. It might be reasonable to take the @sum@ function a step further to enforce a minimum number of arguments:
+\begin{lstlisting}
+int sum(int x, int y){
+  return x+y;
+}
+forall(ttype Params | { int sum(int, Params); })
+int sum(int x, int y, Params rest) {
+  return sum(x+y, rest);
+}
+\end{lstlisting}
+One more iteration permits the summation of any summable type, as long as all arguments are the same type:
+\label{sec:variadic-tuples}
+To define variadic functions, \CFA adds a new kind of type parameter, @ttype@ (tuple type).
+Matching against a @ttype@ parameter consumes all remaining argument components and packages them into a tuple, binding to the resulting tuple of types.
+In a given parameter list, there must be at most one @ttype@ parameter that occurs last, which matches normal variadic semantics, with a strong feeling of similarity to \CCeleven variadic templates.
+As such, @ttype@ variables are also called \emph{argument packs}.
+Like variadic templates, the main way to manipulate @ttype@ polymorphic functions is via recursion.
+Since nothing is known about a parameter pack by default, assertion parameters are key to doing anything meaningful.
+Unlike variadic templates, @ttype@ polymorphic functions can be separately compiled.
+For example, a generalized @sum@ function written using @ttype@:
+\begin{lstlisting}
+int sum$\(_0\)$() { return 0; }
+forall(ttype Params | { int sum( Params ); } ) int sum$\(_1\)$( int x, Params rest ) {
+        return x + sum( rest );
+}
+sum( 10, 20, 30 );
+\end{lstlisting}
+Since @sum@\(_0\) does not accept any arguments, it is not a valid candidate function for the call @sum(10, 20, 30)@.
+In order to call @sum@\(_1\), @10@ is matched with @x@, and the argument resolution moves on to the argument pack @rest@, which consumes the remainder of the argument list and @Params@ is bound to @[20, 30]@.
+The process continues, @Params@ is bound to @[]@, requiring an assertion @int sum()@, which matches @sum@\(_0\) and terminates the recursion.
+Effectively, this algorithm traces as @sum(10, 20, 30)@ $\rightarrow$ @10 + sum(20, 30)@ $\rightarrow$ @10 + (20 + sum(30))@ $\rightarrow$ @10 + (20 + (30 + sum()))@ $\rightarrow$ @10 + (20 + (30 + 0))@.
+It is reasonable to take the @sum@ function a step further to enforce a minimum number of arguments:
+\begin{lstlisting}
+int sum( int x, int y ) { return x + y; }
+forall(ttype Params | { int sum( int, Params ); } ) int sum( int x, int y, Params rest ) {
+        return sum( x + y, rest );
+}
+\end{lstlisting}
+One more step permits the summation of any summable type with all arguments of the same type:
 \begin{lstlisting}
 trait summable(otype T) {
   T ?+?(T, T);
+        T ?+?( T, T );
 };
+forall(otype R | summable(R))
+R sum(R x, R y){
+  return x+y;
+}
+forall(otype R, ttype Params
+  | summable(R)
+  | { R sum(R, Params); })
+R sum(R x, R y, Params rest) {
+  return sum(x+y, rest);
+}
+\end{lstlisting}
+Unlike C, it is not necessary to hard code the expected type. This code is naturally open to extension, in that any user-defined type with a @?+?@ operator is automatically able to be used with the @sum@ function. That is to say, the programmer who writes @sum@ does not need full program knowledge of every possible data type, unlike what is necessary to write an equivalent function using the standard C mechanisms. Summing arbitrary heterogeneous lists is possible with similar code by adding the appropriate type variables and addition operators.
+It is also possible to write a type-safe variadic print routine which can replace @printf@:
+forall(otype R | summable( R ) ) R sum( R x, R y ) {
+        return x + y;
+}
+forall(otype R, ttype Params | summable(R) | { R sum(R, Params); } ) R sum(R x, R y, Params rest) {
+        return sum( x + y, rest );
+}
+\end{lstlisting}
+Unlike C variadic functions, it is unnecessary to hard code the number and expected types.
+Furthermore, this code is extendable so any user-defined type with a @?+?@ operator.
+Summing arbitrary heterogeneous lists is possible with similar code by adding the appropriate type variables and addition operators.
+It is also possible to write a type-safe variadic print function to replace @printf@:
 \begin{lstlisting}
 struct S { int x, y; };
+forall(otype T, ttype Params |
+  { void print(T); void print(Params); })
+void print(T arg, Params rest) {
+  print(arg);
+  print(rest);
+}
+void print(char * x) { printf("%s", x); }
+void print(int x) { printf("%d", x);  }
+void print(S s) { print("{ ", s.x, ",", s.y, " }"); }
+print("s = ", (S){ 1, 2 }, "\n");
+\end{lstlisting}
+This example routine showcases a variadic-template-like decomposition of the provided argument list. The individual @print@ routines allow printing a single element of a type. The polymorphic @print@ allows printing any list of types, as long as each individual type has a @print@ function. The individual print functions can be used to build up more complicated @print@ routines, such as for @S@, which is something that cannot be done with @printf@ in C.
+It is also possible to use @ttype@ polymorphism to provide arbitrary argument forwarding functions. For example, it is possible to write @new@ as a library function:
+\begin{lstlisting}
+struct Pair(otype R, otype S);
+forall(otype R, otype S)
+void ?{}(Pair(R, S) *, R, S);  // (1)
+forall(dtype T, ttype Params | sized(T) | { void ?{}(T *, Params); })
+T * new(Params p) {
+  return ((T*)malloc( sizeof(T) )){ p }; // construct into result of malloc
+}
+Pair(int, char) * x = new(42, '!');
+\end{lstlisting}
+The @new@ function provides the combination of type-safe @malloc@ with a constructor call, so that it becomes impossible to forget to construct dynamically allocated objects. This function provides the type-safety of @new@ in \CC, without the need to specify the allocated type again, thanks to return-type inference.
+In the call to @new@, @Pair(double, char)@ is selected to match @T@, and @Params@ is expanded to match @[double, char]@. The constructor (1) may be specialized to  satisfy the assertion for a constructor with an interface compatible with @void ?{}(Pair(int, char) *, int, char)@.
+\TODO{Check if we actually can use ttype parameters on generic types (if they set the complete flag, it should work, or nearly so).}
+forall(otype T, ttype Params | { void print(T); void print(Params); }) void print(T arg, Params rest) {
+        print(arg);  print(rest);
+}
+void print( char * x ) { printf( "%s", x ); }
+void print( int x ) { printf( "%d", x ); }
+void print( S s ) { print( "{ ", s.x, ",", s.y, " }" ); }
+print( "s = ", (S){ 1, 2 }, "\n" );
+\end{lstlisting}
+This example showcases a variadic-template-like decomposition of the provided argument list.
+The individual @print@ functions allow printing a single element of a type.
+The polymorphic @print@ allows printing any list of types, where as each individual type has a @print@ function.
+The individual print functions can be used to build up more complicated @print@ functions, such as @S@, which cannot be done with @printf@ in C.
+Finally, it is possible to use @ttype@ polymorphism to provide arbitrary argument forwarding functions.
+For example, it is possible to write @new@ as a library function:
+\begin{lstlisting}
+forall( otype R, otype S ) void ?{}( pair(R, S) *, R, S );
+forall( dtype T, ttype Params | sized(T) | { void ?{}( T *, Params ); } ) T * new( Params p ) {
+        return ((T *)malloc()){ p };                    $\C{// construct into result of malloc}$
+}
+pair( int, char ) * x = new( 42, '!' );
+\end{lstlisting}
+The @new@ function provides the combination of type-safe @malloc@ with a \CFA constructor call, making it impossible to forget constructing dynamically allocated objects.
+This function provides the type-safety of @new@ in \CC, without the need to specify the allocated type again, thanks to return-type inference.
 \subsection{Implementation}
+Tuples are implemented in the \CFA translator via a transformation into generic types. For each $N$, the first time an $N$-tuple is seen in a scope a generic type with $N$ type parameters is generated. For example:
+Tuples are implemented in the \CFA translator via a transformation into generic types.
+For each $N$, the first time an $N$-tuple is seen in a scope a generic type with $N$ type parameters is generated, \eg:
 \begin{lstlisting}
 [int, int] f() {
+  [double, double] x;
+  [int, double, int] y;
+}
+\end{lstlisting}
+Is transformed into:
+\begin{lstlisting}
+forall(dtype T0, dtype T1 | sized(T0) | sized(T1))
+struct _tuple2 {  // generated before the first 2-tuple
+  T0 field_0;
+  T1 field_1;
+        [double, double] x;
+        [int, double, int] y;
+}
+\end{lstlisting}
+is transformed into:
+\begin{lstlisting}
+forall(dtype T0, dtype T1 | sized(T0) | sized(T1)) struct _tuple2 {
+        T0 field_0;                                                             $\C{// generated before the first 2-tuple}$
+        T1 field_1;
 };
 _tuple2(int, int) f() {
+  _tuple2(double, double) x;
+  forall(dtype T0, dtype T1, dtype T2 | sized(T0) | sized(T1) | sized(T2))
+  struct _tuple3 {  // generated before the first 3-tuple
+    T0 field_0;
+    T1 field_1;
+    T2 field_2;
+  };
+  _tuple3_(int, double, int) y;
+}
+\end{lstlisting}
+Tuple expressions are then simply converted directly into compound literals:
+\begin{lstlisting}
+[5, 'x', 1.24];
+\end{lstlisting}
+Becomes:
+\begin{lstlisting}
+(_tuple3(int, char, double)){ 5, 'x', 1.24 };
+\end{lstlisting}
+        _tuple2(double, double) x;
+        forall(dtype T0, dtype T1, dtype T2 | sized(T0) | sized(T1) | sized(T2)) struct _tuple3 {
+                T0 field_0;                                                     $\C{// generated before the first 3-tuple}$
+                T1 field_1;
+                T2 field_2;
+        };
+        _tuple3(int, double, int) y;
+}
+\end{lstlisting}
+Tuple expressions are then simply converted directly into compound literals, \eg @[5, 'x', 1.24]@ becomes @(_tuple3(int, char, double)){ 5, 'x', 1.24 }@.
+\begin{comment}
 Since tuples are essentially structures, tuple indexing expressions are just field accesses:
 \begin{lstlisting}
 …
 f(x.field_0, (_tuple2){ x.field_1, 'z' });
 \end{lstlisting}
+Note that due to flattening, @x@ used in the argument position is converted into the list of its fields. In the call to @f@, the second and third argument components are structured into a tuple argument. Similarly, tuple member expressions are recursively expanded into a list of member access expressions.
+Expressions that may contain side effects are made into \emph{unique expressions} before being expanded by the flattening conversion. Each unique expression is assigned an identifier and is guaranteed to be executed exactly once:
+Note that due to flattening, @x@ used in the argument position is converted into the list of its fields.
+In the call to @f@, the second and third argument components are structured into a tuple argument.
+Similarly, tuple member expressions are recursively expanded into a list of member access expressions.
+Expressions that may contain side effects are made into \emph{unique expressions} before being expanded by the flattening conversion.
+Each unique expression is assigned an identifier and is guaranteed to be executed exactly once:
 \begin{lstlisting}
 void g(int, double);
 …
 [int, double] _unq0;
 g(
   (_unq0_finished_ ? _unq0 : (_unq0 = f(), _unq0_finished_ = 1, _unq0)).0,
   (_unq0_finished_ ? _unq0 : (_unq0 = f(), _unq0_finished_ = 1, _unq0)).1,
+        (_unq0_finished_ ? _unq0 : (_unq0 = f(), _unq0_finished_ = 1, _unq0)).0,
+        (_unq0_finished_ ? _unq0 : (_unq0 = f(), _unq0_finished_ = 1, _unq0)).1,
 );
 \end{lstlisting}
+Since argument evaluation order is not specified by the C programming language, this scheme is built to work regardless of evaluation order. The first time a unique expression is executed, the actual expression is evaluated and the accompanying boolean is set to true. Every subsequent evaluation of the unique expression then results in an access to the stored result of the actual expression. Tuple member expressions also take advantage of unique expressions in the case of possible impurity.
+Currently, the \CFA translator has a very broad, imprecise definition of impurity, where any function call is assumed to be impure. This notion could be made more precise for certain intrinsic, auto-generated, and builtin functions, and could analyze function bodies when they are available to recursively detect impurity, to eliminate some unique expressions.
+The various kinds of tuple assignment, constructors, and destructors generate GNU C statement expressions. A variable is generated to store the value produced by a statement expression, since its fields may need to be constructed with a non-trivial constructor and it may need to be referred to multiple time, \eg in a unique expression. The use of statement expressions allows the translator to arbitrarily generate additional temporary variables as needed, but binds the implementation to a non-standard extension of the C language. However, there are other places where the \CFA translator makes use of GNU C extensions, such as its use of nested functions, so this restriction is not new.
+Since argument evaluation order is not specified by the C programming language, this scheme is built to work regardless of evaluation order.
+The first time a unique expression is executed, the actual expression is evaluated and the accompanying boolean is set to true.
+Every subsequent evaluation of the unique expression then results in an access to the stored result of the actual expression.
+Tuple member expressions also take advantage of unique expressions in the case of possible impurity.
+Currently, the \CFA translator has a very broad, imprecise definition of impurity, where any function call is assumed to be impure.
+This notion could be made more precise for certain intrinsic, auto-generated, and builtin functions, and could analyze function bodies when they are available to recursively detect impurity, to eliminate some unique expressions.
+The various kinds of tuple assignment, constructors, and destructors generate GNU C statement expressions.
+A variable is generated to store the value produced by a statement expression, since its fields may need to be constructed with a non-trivial constructor and it may need to be referred to multiple time, \eg in a unique expression.
+The use of statement expressions allows the translator to arbitrarily generate additional temporary variables as needed, but binds the implementation to a non-standard extension of the C language.
+However, there are other places where the \CFA translator makes use of GNU C extensions, such as its use of nested functions, so this restriction is not new.
+\end{comment}
 \section{Evaluation}
+\TODO{Magnus suggests we need some graphs, it's kind of a done thing that the reviewers will be looking for. Also, we've made some unsubstantiated claims about the runtime performance of \CFA, which some micro-benchmarks could help with. I'm thinking a simple stack push and pop, with an idiomatic \lstinline@void*@, \CFA, \CC template and \CC virtual inheritance versions (the void* and virtual inheritance versions likely need to be linked lists, or clumsy in their API -- possibly both versions) to test generics, and variadic print to test tuples. We measure SLOC, runtime performance, executable size (making sure to include benchmarks for multiple types in the executable), and possibly manually count the number of places where the programmer must provide un-type-checked type information. Appendices don't count against our page limit, so we might want to include the source code for the benchmarks (or at least the relevant implementation details) in one.}
+\label{sec:eval}
+Though \CFA provides significant added functionality over C, these features have a low runtime penalty.
+In fact, \CFA's features for generic programming can enable faster runtime execution than idiomatic @void *@-based C code.
+This claim is demonstrated through a set of generic-code-based micro-benchmarks in C, \CFA, and \CC (see stack implementations in Appendix~\ref{sec:BenchmarkStackImplementation}).
+Since all these languages share a subset essentially comprising standard C, maximal-performance benchmarks would show little runtime variance, other than in length and clarity of source code.
+A more illustrative benchmark measures the costs of idiomatic usage of each language's features.
+Figure~\ref{fig:BenchmarkTest} shows the \CFA benchmark tests for a generic stack based on a singly linked-list, a generic pair-data-structure, and a variadic @print@ routine similar to that in Section~\ref{sec:variadic-tuples}.
+The benchmark test is similar for C and \CC.
+The experiment uses element types @int@ and @pair(_Bool, char)@, and pushes $N=40M$ elements on a generic stack, copies the stack, clears one of the stacks, finds the maximum value in the other stack, and prints $N/2$ (to reduce graph height) constants.
+\begin{figure}
+\begin{lstlisting}[xleftmargin=3\parindentlnth,aboveskip=0pt,belowskip=0pt]
+int main( int argc, char * argv[] ) {
+        FILE * out = fopen( "cfa-out.txt", "w" );
+        int maxi = 0, vali = 42;
+        stack(int) si, ti;
+        REPEAT_TIMED( "push_int", N, push( &si, vali ); )
+        TIMED( "copy_int", ti = si; )
+        TIMED( "clear_int", clear( &si ); )
+        REPEAT_TIMED( "pop_int", N,
+                int xi = pop( &ti ); if ( xi > maxi ) { maxi = xi; } )
+        REPEAT_TIMED( "print_int", N/2, print( out, vali, ":", vali, "\n" ); )
+        pair(_Bool, char) maxp = { (_Bool)0, '\0' }, valp = { (_Bool)1, 'a' };
+        stack(pair(_Bool, char)) sp, tp;
+        REPEAT_TIMED( "push_pair", N, push( &sp, valp ); )
+        TIMED( "copy_pair", tp = sp; )
+        TIMED( "clear_pair", clear( &sp ); )
+        REPEAT_TIMED( "pop_pair", N,
+                pair(_Bool, char) xp = pop( &tp ); if ( xp > maxp ) { maxp = xp; } )
+        REPEAT_TIMED( "print_pair", N/2, print( out, valp, ":", valp, "\n" ); )
+        fclose(out);
+}
+\end{lstlisting}
+\caption{\CFA Benchmark Test}
+\label{fig:BenchmarkTest}
+\end{figure}
+The structure of each benchmark implemented is: C with @void *@-based polymorphism, \CFA with the presented features, \CC with templates, and \CC using only class inheritance for polymorphism, called \CCV.
+The \CCV variant illustrates an alternative object-oriented idiom where all objects inherit from a base @object@ class, mimicking a Java-like interface;
+hence runtime checks are necessary to safely down-cast objects.
+The most notable difference among the implementations is in memory layout of generic types: \CFA and \CC inline the stack and pair elements into corresponding list and pair nodes, while C and \CCV lack such a capability and instead must store generic objects via pointers to separately-allocated objects.
+For the print benchmark, idiomatic printing is used: the C and \CFA variants used @stdio.h@, while the \CC and \CCV variants used @iostream@; preliminary tests show this distinction has negligible runtime impact.
+Note, the C benchmark uses unchecked casts as there is no runtime mechanism to perform such checks, while \CFA and \CC provide type-safety statically.
+Figure~\ref{fig:eval} and Table~\ref{tab:eval} show the results of running the benchmark in Figure~\ref{fig:BenchmarkTest} and its C, \CC, and \CCV equivalents.
+The graph plots the median of 5 consecutive runs of each program, with an initial warm-up run omitted.
+All code is compiled at \texttt{-O2} by GCC or G++ 6.2.0, with all \CC code compiled as \CCfourteen.
+The benchmarks are run on an Ubuntu 16.04 workstation with 16 GB of RAM and a 6-core AMD FX-6300 CPU with 3.5 GHz maximum clock frequency.
+\begin{figure}
+\centering
+\input{timing}
+\caption{Benchmark Timing Results (smaller is better)}
+\label{fig:eval}
+\end{figure}
+\begin{table}
+\caption{Properties of benchmark code}
+\label{tab:eval}
+\newcommand{\CT}[1]{\multicolumn{1}{c}{#1}}
+\begin{tabular}{rrrrr}
+                                                                        & \CT{C}        & \CT{\CFA}     & \CT{\CC}      & \CT{\CCV}             \\ \hline
+maximum memory usage (MB)                       & 10001         & 2502          & 2503          & 11253                 \\
+source code size (lines)                        & 247           & 222           & 165           & 339                   \\
+redundant type annotations (lines)      & 39            & 2                     & 2                     & 15                    \\
+binary size (KB)                                        & 14            & 229           & 18            & 38                    \\
+\end{tabular}
+\end{table}
+The C and \CCV variants are generally the slowest with the largest memory footprint, because of their less-efficient memory layout and the pointer-indirection necessary to implement generic types;
+this inefficiency is exacerbated by the second level of generic types in the pair-based benchmarks.
+By contrast, the \CFA and \CC variants run in roughly equivalent time for both the integer and pair of @_Bool@ and @char@ because the storage layout is equivalent, with the inlined libraries (\ie no separate compilation) and greater maturity of the \CC compiler contributing to its lead.
+\CCV is slower than C largely due to the cost of runtime type-checking of down-casts (implemented with @dynamic_cast@);
+There are two outliers in the graph for \CFA: all prints and pop of @pair@.
+Both of these cases result from the complexity of the C-generated polymorphic code, so that the GCC compiler is unable to optimize some dead code and condense nested calls.
+A compiler designed for \CFA could easily perform these optimizations.
+Finally, the binary size for \CFA is larger because of static linking with the \CFA libraries.
+\CFA is also competitive in terms of source code size, measured as a proxy for programmer effort. The line counts in Table~\ref{tab:eval} include implementations of @pair@ and @stack@ types for all four languages for purposes of direct comparison, though it should be noted that \CFA and \CC have pre-written data structures in their standard libraries that programmers would generally use instead. Use of these standard library types has minimal impact on the performance benchmarks, but shrinks the \CFA and \CC benchmarks to 73 and 54 lines, respectively.
+On the other hand, C does not have a generic collections-library in its standard distribution, resulting in frequent reimplementation of such collection types by C programmers.
+\CCV does not use the \CC standard template library by construction, and in fact includes the definition of @object@ and wrapper classes for @bool@, @char@, @int@, and @const char *@ in its line count, which inflates this count somewhat, as an actual object-oriented language would include these in the standard library;
+with their omission the \CCV line count is similar to C.
+We justify the given line count by noting that many object-oriented languages do not allow implementing new interfaces on library types without subclassing or wrapper types, which may be similarly verbose.
+Raw line-count, however, is a fairly rough measure of code complexity;
+another important factor is how much type information the programmer must manually specify, especially where that information is not checked by the compiler.
+Such unchecked type information produces a heavier documentation burden and increased potential for runtime bugs, and is much less common in \CFA than C, with its manually specified function pointers arguments and format codes, or \CCV, with its extensive use of un-type-checked downcasts (\eg @object@ to @integer@ when popping a stack, or @object@ to @printable@ when printing the elements of a @pair@).
+To quantify this, the ``redundant type annotations'' line in Table~\ref{tab:eval} counts the number of lines on which the type of a known variable is re-specified, either as a format specifier, explicit downcast, type-specific function, or by name in a @sizeof@, struct literal, or @new@ expression.
+The \CC benchmark uses two redundant type annotations to create a new stack nodes, while the C and \CCV benchmarks have several such annotations spread throughout their code.
+The two instances in which the \CFA benchmark still uses redundant type specifiers are to cast the result of a polymorphic @malloc@ call (the @sizeof@ argument is inferred by the compiler).
+These uses are similar to the @new@ expressions in \CC, though the \CFA compiler's type resolver should shortly render even these type casts superfluous.
 \section{Related Work}
+\CC is the existing language it is most natural to compare \CFA to, as they are both more modern extensions to C with backwards source compatibility. The most fundamental difference in approach between \CC and \CFA is their approach to this C compatibility. \CC does provide fairly strong source backwards compatibility with C, but is a dramatically more complex language than C, and imposes a steep learning curve to use many of its extension features. For instance, in a break from general C practice, template code is typically written in header files, with a variety of subtle restrictions implied on its use by this choice, while the other polymorphism mechanism made available by \CC, class inheritance, requires programmers to learn an entirely new object-oriented programming paradigm; the interaction between templates and inheritance is also quite complex. \CFA, by contrast, has a single facility for polymorphic code, one which supports separate compilation and the existing procedural paradigm of C code. A major difference between the approaches of \CC and \CFA to polymorphism is that the set of assumed properties for a type is \emph{explicit} in \CFA. One of the major limiting factors of \CC's approach is that templates cannot be separately compiled, and, until concepts~\citep{C++Concepts} are standardized (currently anticipated for \CCtwenty), \CC provides no way to specify the requirements of a generic function in code beyond compilation errors for template expansion failures. By contrast, the explicit nature of assertions in \CFA allows polymorphic functions to be separately compiled, and for their requirements to be checked by the compiler; similarly, \CFA generic types may be opaque, unlike \CC template classes.
+Cyclone also provides capabilities for polymorphic functions and existential types~\citep{Grossman06}, similar in concept to \CFA's @forall@ functions and generic types. Cyclone existential types can include function pointers in a construct similar to a virtual function table, but these pointers must be explicitly initialized at some point in the code, a tedious and potentially error-prone process. Furthermore, Cyclone's polymorphic functions and types are restricted in that they may only abstract over types with the same layout and calling convention as @void*@, in practice only pointer types and @int@ - in \CFA terms, all Cyclone polymorphism must be dtype-static. This design provides the efficiency benefits discussed in Section~\ref{sec:generic-apps} for dtype-static polymorphism, but is more restrictive than \CFA's more general model.
+Apple's Objective-C \citep{obj-c-book} is another industrially successful set of extensions to C. The Objective-C language model is a fairly radical departure from C, adding object-orientation and message-passing. Objective-C implements variadic functions using the C @va_arg@ mechanism, and did not support type-checked generics until recently \citep{xcode7}, historically using less-efficient and more error-prone runtime checking of object types instead. The GObject framework \citep{GObject} also adds object-orientation with runtime type-checking and reference-counting garbage-collection to C; these are much more intrusive feature additions than those provided by \CFA, in addition to the runtime overhead of reference-counting. The Vala programming language \citep{Vala} compiles to GObject-based C, and so adds the burden of learning a separate language syntax to the aforementioned demerits of GObject as a modernization path for existing C code-bases. Java \citep{Java8} has had generic types and variadic functions since Java~5; Java's generic types are type-checked at compilation and type-erased at runtime, similar to \CFA's, though in Java each object carries its own table of method pointers, while \CFA passes the method pointers separately so as to maintain a C-compatible struct layout. Java variadic functions are simply syntactic sugar for an array of a single type, and therefore less useful than \CFA's heterogeneously-typed variadic functions. Java is also a garbage-collected, object-oriented language, with the associated resource usage and C-interoperability burdens.
+D \citep{D}, Go \citep{Go}, and Rust \citep{Rust} are modern, compiled languages with abstraction features similar to \CFA traits, \emph{interfaces} in D and Go and \emph{traits} in Rust. However, each language represents dramatic departures from C in terms of language model, and none has the same level of compatibility with C as \CFA. D and Go are garbage-collected languages, imposing the associated runtime overhead. The necessity of accounting for data transfer between the managed Go runtime and the unmanaged C runtime complicates foreign-function interface between Go and C. Furthermore, while generic types and functions are available in Go, they are limited to a small fixed set provided by the compiler, with no language facility to define more. D restricts garbage collection to its own heap by default, while Rust is not garbage-collected, and thus has a lighter-weight runtime that is more easily interoperable with C. Rust also possesses much more powerful abstraction capabilities for writing generic code than Go. On the other hand, Rust's borrow-checker, while it does provide strong safety guarantees, is complex and difficult to learn, and imposes a distinctly idiomatic programming style on Rust. \CFA, with its more modest safety features, is significantly easier to port C code to, while maintaining the idiomatic style of the original source.
+\section{Conclusion \& Future Work}
+There is ongoing work on a wide range of \CFA feature extensions, including reference types, exceptions, and concurrent programming primitives. In addition to this work, there are some interesting future directions the polymorphism design could take. Notably, \CC template functions trade compile time and code bloat for optimal runtime of individual instantiations of polymorphic functions. \CFA polymorphic functions, by contrast, use an approach that is essentially dynamic virtual dispatch. The runtime overhead of this approach is low, but not as low as \CC template functions, and it may be beneficial to provide a mechanism for particularly performance-sensitive code to close this gap. Further research is needed, but two promising approaches are to allow an annotation on polymorphic function call sites that tells the translator to create a template-specialization of the function (provided the code is visible in the current translation unit) or placing an annotation on polymorphic function definitions that instantiates a version of the polymorphic function specialized to some set of types. These approaches are not mutually exclusive, and would allow these performance optimizations to be applied only where most useful to increase performance, without suffering the code bloat or loss of generality of a template expansion approach where it is unnecessary.
+In conclusion, the authors' design for generic types and tuples, unlike those available in existing work, is both reusable and type-checked, while still supporting a full range of C features, including separately-compiled modules. We have experimentally validated the performance of our design against both \CC and standard C, showing it is \TODO{shiny, cap'n}.
+\subsection{Polymorphism}
+\CC is the most similar language to \CFA;
+both are extensions to C with source and runtime backwards compatibility.
+The fundamental difference is in their engineering approach to C compatibility and programmer expectation.
+While \CC provides good backwards compatibility with C, it has a steep learning curve for many of its extensions.
+For example, polymorphism is provided via three disjoint mechanisms: overloading, inheritance, and templates.
+The overloading is restricted because resolution does not using the return type, inheritance requires learning object-oriented programming and coping with a restricted nominal-inheritance hierarchy, templates cannot be separately compiled resulting in compilation/code bloat and poor error messages, and determining how these mechanisms interact and which to use is confusing.
+In contrast, \CFA has a single facility for polymorphic code supporting type-safe separate-compilation of polymorphic functions and generic (opaque) types, which uniformly leverage the C procedural paradigm.
+The key mechanism to support separate compilation is \CFA's \emph{explicit} use of assumed properties for a type.
+Until \CC~\citet{C++Concepts} are standardized (anticipated for \CCtwenty), \CC provides no way to specify the requirements of a generic function in code beyond compilation errors during template expansion;
+furthermore, \CC concepts are restricted to template polymorphism.
+Cyclone~\citep{Grossman06} also provides capabilities for polymorphic functions and existential types, similar to \CFA's @forall@ functions and generic types.
+Cyclone existential types can include function pointers in a construct similar to a virtual function-table, but these pointers must be explicitly initialized at some point in the code, a tedious and potentially error-prone process.
+Furthermore, Cyclone's polymorphic functions and types are restricted to abstraction over types with the same layout and calling convention as @void *@, \ie only pointer types and @int@.
+In \CFA terms, all Cyclone polymorphism must be dtype-static.
+While the Cyclone design provides the efficiency benefits discussed in Section~\ref{sec:generic-apps} for dtype-static polymorphism, it is more restrictive than \CFA's general model.
+\citet{Smith98} present Polymorphic C, an ML dialect with polymorphic functions and C-like syntax and pointer types; it lacks many of C's features, however, most notably structure types, and so is not a practical C replacement.
+\citet{obj-c-book} is an industrially successful extension to C.
+However, Objective-C is a radical departure from C, using an object-oriented model with message-passing.
+Objective-C did not support type-checked generics until recently \citet{xcode7}, historically using less-efficient runtime checking of object types.
+The~\citet{GObject} framework also adds object-oriented programming with runtime type-checking and reference-counting garbage-collection to C;
+these features are more intrusive additions than those provided by \CFA, in addition to the runtime overhead of reference-counting.
+\citet{Vala} compiles to GObject-based C, adding the burden of learning a separate language syntax to the aforementioned demerits of GObject as a modernization path for existing C code-bases.
+Java~\citep{Java8} included generic types in Java~5, which are type-checked at compilation and type-erased at runtime, similar to \CFA's.
+However, in Java, each object carries its own table of method pointers, while \CFA passes the method pointers separately to maintain a C-compatible layout.
+Java is also a garbage-collected, object-oriented language, with the associated resource usage and C-interoperability burdens.
+D~\citep{D}, Go, and~\citet{Rust} are modern, compiled languages with abstraction features similar to \CFA traits, \emph{interfaces} in D and Go and \emph{traits} in Rust.
+However, each language represents a significant departure from C in terms of language model, and none has the same level of compatibility with C as \CFA.
+D and Go are garbage-collected languages, imposing the associated runtime overhead.
+The necessity of accounting for data transfer between managed runtimes and the unmanaged C runtime complicates foreign-function interfaces to C.
+Furthermore, while generic types and functions are available in Go, they are limited to a small fixed set provided by the compiler, with no language facility to define more.
+D restricts garbage collection to its own heap by default, while Rust is not garbage-collected, and thus has a lighter-weight runtime more interoperable with C.
+Rust also possesses much more powerful abstraction capabilities for writing generic code than Go.
+On the other hand, Rust's borrow-checker provides strong safety guarantees but is complex and difficult to learn and imposes a distinctly idiomatic programming style.
+\CFA, with its more modest safety features, allows direct ports of C code while maintaining the idiomatic style of the original source.
+\subsection{Tuples/Variadics}
+Many programming languages have some form of tuple construct and/or variadic functions, \eg SETL, C, KW-C, \CC, D, Go, Java, ML, and Scala.
+SETL~\cite{SETL} is a high-level mathematical programming language, with tuples being one of the primary data types.
+Tuples in SETL allow subscripting, dynamic expansion, and multiple assignment.
+C provides variadic functions through @va_list@ objects, but the programmer is responsible for managing the number of arguments and their types, so the mechanism is type unsafe.
+KW-C~\cite{Buhr94a}, a predecessor of \CFA, introduced tuples to C as an extension of the C syntax, taking much of its inspiration from SETL.
+The main contributions of that work were adding MRVF, tuple mass and multiple assignment, and record-field access.
+\CCeleven introduced @std::tuple@ as a library variadic template structure.
+Tuples are a generalization of @std::pair@, in that they allow for arbitrary length, fixed-size aggregation of heterogeneous values.
+Operations include @std::get<N>@ to extract vales, @std::tie@ to create a tuple of references used for assignment, and lexicographic comparisons.
+\CCseventeen proposes \emph{structured bindings}~\cite{Sutter15} to eliminate pre-declaring variables and use of @std::tie@ for binding the results.
+This extension requires the use of @auto@ to infer the types of the new variables, so complicated expressions with a non-obvious type must be documented with some other mechanism.
+Furthermore, structured bindings are not a full replacement for @std::tie@, as it always declares new variables.
+Like \CC, D provides tuples through a library variadic-template structure.
+Go does not have tuples but supports MRVF.
+Java's variadic functions appear similar to C's but are type-safe using homogeneous arrays, which are less useful than \CFA's heterogeneously-typed variadic functions.
+Tuples are a fundamental abstraction in most functional programming languages, such as Standard ML~\cite{sml} and~\cite{Scala}, which decompose tuples using pattern matching.
+\section{Conclusion and Future Work}
+The goal of \CFA is to provide an evolutionary pathway for large C development-environments to be more productive and safer, while respecting the talent and skill of C programmers.
+While other programming languages purport to be a better C, they are in fact new and interesting languages in their own right, but not C extensions.
+The purpose of this paper is to introduce \CFA, and showcase two language features that illustrate the \CFA type-system and approaches taken to achieve the goal of evolutionary C extension.
+The contributions are a powerful type-system using parametric polymorphism and overloading, generic types, and tuples, which all have complex interactions.
+The work is a challenging design, engineering, and implementation exercise.
+On the surface, the project may appear as a rehash of similar mechanisms in \CC.
+However, every \CFA feature is different than its \CC counterpart, often with extended functionality, better integration with C and its programmers, and always supporting separate compilation.
+All of these new features are being used by the \CFA development-team to build the \CFA runtime-system.
+Finally, we demonstrate that \CFA performance for some idiomatic cases is better than C and close to \CC, showing the design is practically applicable.
+There is ongoing work on a wide range of \CFA feature extensions, including reference types, exceptions, concurrent primitives and modules.
+(While all examples in the paper compile and run, a public beta-release of \CFA will take another 8--12 months to finalize these additional extensions.)
+In addition, there are interesting future directions for the polymorphism design.
+Notably, \CC template functions trade compile time and code bloat for optimal runtime of individual instantiations of polymorphic functions.
+\CFA polymorphic functions use dynamic virtual-dispatch;
+the runtime overhead of this approach is low, but not as low as inlining, and it may be beneficial to provide a mechanism for performance-sensitive code.
+Two promising approaches are an @inline@ annotation at polymorphic function call sites to create a template-specialization of the function (provided the code is visible) or placing an @inline@ annotation on polymorphic function-definitions to instantiate a specialized version for some set of types.
+These approaches are not mutually exclusive and allow performance optimizations to be applied only when necessary, without suffering global code-bloat.
+In general, we believe separate compilation, producing smaller code, works well with loaded hardware-caches, which may offset the benefit of larger inlined-code.
 \begin{acks}
 The authors would like to thank Magnus Madsen for valuable editorial feedback.
 This work is supported in part by a corporate partnership with \grantsponsor{Huawei}{Huawei Ltd.}{http://www.huawei.com}\ and the first author's \grantsponsor{NSERC-PGS}{NSERC PGS D}{http://www.nserc-crsng.gc.ca/Students-Etudiants/PG-CS/BellandPostgrad-BelletSuperieures_eng.asp} scholarship.
+The authors would like to recognize the design assistance of Glen Ditchfield, Richard Bilson, and Thierry Delisle on the features described in this paper, and thank Magnus Madsen and the three anonymous reviewers for valuable feedback.
+This work is supported in part by a corporate partnership with \grantsponsor{Huawei}{Huawei Ltd.}{http://www.huawei.com}, and Aaron Moss and Peter Buhr are funded by the \grantsponsor{Natural Sciences and Engineering Research Council} of Canada.
+% the first author's \grantsponsor{NSERC-PGS}{NSERC PGS D}{http://www.nserc-crsng.gc.ca/Students-Etudiants/PG-CS/BellandPostgrad-BelletSuperieures_eng.asp} scholarship.
 \end{acks}
 \bibliographystyle{ACM-Reference-Format}
 \bibliography{cfa}
+\appendix
+\section{Benchmark Stack Implementation}
+\label{sec:BenchmarkStackImplementation}
+\lstset{basicstyle=\linespread{0.9}\sf\small}
+Throughout, @/***/@ designates a counted redundant type annotation.
+\medskip\noindent
+\CFA
+\begin{lstlisting}[xleftmargin=2\parindentlnth,aboveskip=0pt,belowskip=0pt]
+forall(otype T) struct stack_node {
+        T value;
+        stack_node(T) * next;
+};
+forall(otype T) void ?{}(stack(T) * s) { (&s->head){ 0 }; }
+forall(otype T) void ?{}(stack(T) * s, stack(T) t) {
+        stack_node(T) ** crnt = &s->head;
+        for ( stack_node(T) * next = t.head; next; next = next->next ) {
+                *crnt = ((stack_node(T) *)malloc()){ next->value }; /***/
+                stack_node(T) * acrnt = *crnt;
+                crnt = &acrnt->next;
+        }
+        *crnt = 0;
+}
+forall(otype T) stack(T) ?=?(stack(T) * s, stack(T) t) {
+        if ( s->head == t.head ) return *s;
+        clear(s);
+        s{ t };
+        return *s;
+}
+forall(otype T) void ^?{}(stack(T) * s) { clear(s); }
+forall(otype T) _Bool empty(const stack(T) * s) { return s->head == 0; }
+forall(otype T) void push(stack(T) * s, T value) {
+        s->head = ((stack_node(T) *)malloc()){ value, s->head }; /***/
+}
+forall(otype T) T pop(stack(T) * s) {
+        stack_node(T) * n = s->head;
+        s->head = n->next;
+        T x = n->value;
+        ^n{};
+        free(n);
+        return x;
+}
+forall(otype T) void clear(stack(T) * s) {
+        for ( stack_node(T) * next = s->head; next; ) {
+                stack_node(T) * crnt = next;
+                next = crnt->next;
+                delete(crnt);
+        }
+        s->head = 0;
+}
+\end{lstlisting}
+\medskip\noindent
+\CC
+\begin{lstlisting}[xleftmargin=2\parindentlnth,aboveskip=0pt,belowskip=0pt]
+template<typename T> class stack {
+        struct node {
+                T value;
+                node * next;
+                node( const T & v, node * n = nullptr ) : value(v), next(n) {}
+        };
+        node * head;
+        void copy(const stack<T>& o) {
+                node ** crnt = &head;
+                for ( node * next = o.head;; next; next = next->next ) {
+                        *crnt = new node{ next->value }; /***/
+                        crnt = &(*crnt)->next;
+                }
+                *crnt = nullptr;
+        }
+  public:
+        stack() : head(nullptr) {}
+        stack(const stack<T>& o) { copy(o); }
+        stack(stack<T> && o) : head(o.head) { o.head = nullptr; }
+        ~stack() { clear(); }
+        stack & operator= (const stack<T>& o) {
+                if ( this == &o ) return *this;
+                clear();
+                copy(o);
+                return *this;
+        }
+        stack & operator= (stack<T> && o) {
+                if ( this == &o ) return *this;
+                head = o.head;
+                o.head = nullptr;
+                return *this;
+        }
+        bool empty() const { return head == nullptr; }
+        void push(const T & value) { head = new node{ value, head };  /***/ }
+        T pop() {
+                node * n = head;
+                head = n->next;
+                T x = std::move(n->value);
+                delete n;
+                return x;
+        }
+        void clear() {
+                for ( node * next = head; next; ) {
+                        node * crnt = next;
+                        next = crnt->next;
+                        delete crnt;
+                }
+                head = nullptr;
+        }
+};
+\end{lstlisting}
+\medskip\noindent
+C
+\begin{lstlisting}[xleftmargin=2\parindentlnth,aboveskip=0pt,belowskip=0pt]
+struct stack_node {
+        void * value;
+        struct stack_node * next;
+};
+struct stack new_stack() { return (struct stack){ NULL }; /***/ }
+void copy_stack(struct stack * s, const struct stack * t, void * (*copy)(const void *)) {
+        struct stack_node ** crnt = &s->head;
+        for ( struct stack_node * next = t->head; next; next = next->next ) {
+                *crnt = malloc(sizeof(struct stack_node)); /***/
+                **crnt = (struct stack_node){ copy(next->value) }; /***/
+                crnt = &(*crnt)->next;
+        }
+        *crnt = 0;
+}
+_Bool stack_empty(const struct stack * s) { return s->head == NULL; }
+void push_stack(struct stack * s, void * value) {
+        struct stack_node * n = malloc(sizeof(struct stack_node)); /***/
+        *n = (struct stack_node){ value, s->head }; /***/
+        s->head = n;
+}
+void * pop_stack(struct stack * s) {
+        struct stack_node * n = s->head;
+        s->head = n->next;
+        void * x = n->value;
+        free(n);
+        return x;
+}
+void clear_stack(struct stack * s, void (*free_el)(void *)) {
+        for ( struct stack_node * next = s->head; next; ) {
+                struct stack_node * crnt = next;
+                next = crnt->next;
+                free_el(crnt->value);
+                free(crnt);
+        }
+        s->head = NULL;
+}
+\end{lstlisting}
+\medskip\noindent
+\CCV
+\begin{lstlisting}[xleftmargin=2\parindentlnth,aboveskip=0pt,belowskip=0pt]
+stack::node::node( const object & v, node * n ) : value( v.new_copy() ), next( n ) {}
+void stack::copy(const stack & o) {
+        node ** crnt = &head;
+        for ( node * next = o.head; next; next = next->next ) {
+                *crnt = new node{ *next->value };
+                crnt = &(*crnt)->next;
+        }
+        *crnt = nullptr;
+}
+stack::stack() : head(nullptr) {}
+stack::stack(const stack & o) { copy(o); }
+stack::stack(stack && o) : head(o.head) { o.head = nullptr; }
+stack::~stack() { clear(); }
+stack & stack::operator= (const stack & o) {
+        if ( this == &o ) return *this;
+        clear();
+        copy(o);
+        return *this;
+}
+stack & stack::operator= (stack && o) {
+        if ( this == &o ) return *this;
+        head = o.head;
+        o.head = nullptr;
+        return *this;
+}
+bool stack::empty() const { return head == nullptr; }
+void stack::push(const object & value) { head = new node{ value, head }; /***/ }
+ptr<object> stack::pop() {
+        node * n = head;
+        head = n->next;
+        ptr<object> x = std::move(n->value);
+        delete n;
+        return x;
+}
+void stack::clear() {
+        for ( node * next = head; next; ) {
+                node * crnt = next;
+                next = crnt->next;
+                delete crnt;
+        }
+        head = nullptr;
+}
+\end{lstlisting}
+\begin{comment}
+\subsubsection{bench.h}
+(\texttt{bench.hpp} is similar.)
+\lstinputlisting{evaluation/bench.h}
+\subsection{C}
+\subsubsection{c-stack.h} ~
+\lstinputlisting{evaluation/c-stack.h}
+\subsubsection{c-stack.c} ~
+\lstinputlisting{evaluation/c-stack.c}
+\subsubsection{c-pair.h} ~
+\lstinputlisting{evaluation/c-pair.h}
+\subsubsection{c-pair.c} ~
+\lstinputlisting{evaluation/c-pair.c}
+\subsubsection{c-print.h} ~
+\lstinputlisting{evaluation/c-print.h}
+\subsubsection{c-print.c} ~
+\lstinputlisting{evaluation/c-print.c}
+\subsubsection{c-bench.c} ~
+\lstinputlisting{evaluation/c-bench.c}
+\subsection{\CFA}
+\subsubsection{cfa-stack.h} ~
+\lstinputlisting{evaluation/cfa-stack.h}
+\subsubsection{cfa-stack.c} ~
+\lstinputlisting{evaluation/cfa-stack.c}
+\subsubsection{cfa-print.h} ~
+\lstinputlisting{evaluation/cfa-print.h}
+\subsubsection{cfa-print.c} ~
+\lstinputlisting{evaluation/cfa-print.c}
+\subsubsection{cfa-bench.c} ~
+\lstinputlisting{evaluation/cfa-bench.c}
+\subsection{\CC}
+\subsubsection{cpp-stack.hpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-stack.hpp}
+\subsubsection{cpp-print.hpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-print.hpp}
+\subsubsection{cpp-bench.cpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-bench.cpp}
+\subsection{\CCV}
+\subsubsection{object.hpp} ~
+\lstinputlisting[language=c++]{evaluation/object.hpp}
+\subsubsection{cpp-vstack.hpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-vstack.hpp}
+\subsubsection{cpp-vstack.cpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-vstack.cpp}
+\subsubsection{cpp-vprint.hpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-vprint.hpp}
+\subsubsection{cpp-vbench.cpp} ~
+\lstinputlisting[language=c++]{evaluation/cpp-vbench.cpp}
+\end{comment}
 \end{document}

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