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doc/generic_types/generic_types.tex

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 % inline code @...@
 \lstMakeShortInline@%
+\lstMakeShortInline@
 % ACM Information
 …
 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@%
+\lstDeleteShortInline@
 \begin{center}
 \setlength{\tabcolsep}{10pt}
 …
 \end{tabular}
 \end{center}
 \lstMakeShortInline@%
+\lstMakeShortInline@
 Love it or hate it, C is extremely popular, highly used, and one of the few system's languages.
 In many cases, \CC is often used solely as a better C.
 …
 Finally, \CFA allows variable overloading:
 \lstDeleteShortInline@%
+\lstDeleteShortInline@
 \par\smallskip
 \begin{tabular}{@{}l@{\hspace{\parindent}}|@{\hspace{\parindent}}l@{}}
 …
 \end{lstlisting}
 \end{tabular}
+\lstMakeShortInline@
 \smallskip\par\noindent
-\lstMakeShortInline@%
 Hence, 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.
 …
 \begin{lstlisting}
 int x;
+if (x) x++                                                                      $\C{// if (x != 0) x += 1;}$
+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.
 …
 % 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}
 …
 % };
 % typedef list *list_iterator;
+%
+%
 % lvalue int *?( list_iterator it ) { return it->value; }
 % \end{lstlisting}
 …
 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 );
 …
 scalar(metres) marathon = half_marathon + half_marathon;
 scalar(litres) two_pools = swimming_pool + swimming_pool;
 marathon + swimming_pool;                                       $\C{// compilation ERROR}$
+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 @?+?@.
 …
 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}
+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;
+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;                                                                       $\C{// overload variable names}$
+double q, r;
+[ q, r ] = div( 13, 5 );                                        $\C{// select appropriate div and q, r}$
+[ q, r ] = div( 13.5, 5.2 );
+\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 deals with them.
+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 function 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 function signature whether the callee expects such a parameter to be initialized before the call. Furthermore, while many C functions that accept pointers are designed so that it is safe to pass @NULL@ as a parameter, there are many C functions 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.
 \subsection{Tuple Expressions}
+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@.
+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{green}@int@}, @double@) to (@int@, {\color{green}@double@}, @double@) with one {\color{green}safe} (widening) conversion from @int@ to @double@ versus ({\color{red}@double@}, {\color{green}@int@}, {\color{green}@int@}) to ({\color{red}@int@}, {\color{green}@double@}, {\color{green}@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.
+As a consequence, \CFA allows declaration of \emph{tuple variables} that can be initialized from an MRVF, \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}
+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]@.
 \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, \eg:
+\lstDeleteShortInline@%
+\par\smallskip
+\begin{tabular}{@{}l@{\hspace{\parindent}}|@{\hspace{\parindent}}l@{}}
+\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.
+\begin{lstlisting}
+int f(int, int);
+int g([int, int]);
+int h(int, [int, int]);
 [int, int] x;
-\end{lstlisting}
+&
-\begin{lstlisting}
 int y;
+f( x );                 $\C[1in]{// flatten}$
+g( y, 10 );             $\C{// structure}$
+h( x, y );              $\C{// flatten and structure}\CRT{}$
+\end{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{\parindent}}|@{\hspace{\parindent}}l@{}}
+\begin{lstlisting}
+int x = 10;
+double y = 3.5;
+[int, double] z;
+\end{lstlisting}
+&
+\begin{lstlisting}
+z = [x, y];             $\C[1in]{// multiple assignment}$
+[x, y] = z;             $\C{// multiple assignment}$
+z = 10;                 $\C{// mass assignment}$
+[y, x] = 3.14;  $\C{// mass assignment}\CRT{}$
+\end{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}
+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])@.
 \subsection{Member Access}
+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:
+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,
 \begin{lstlisting}
 struct S { int x; double y; char * z; } s;
+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{\parindent}}|@{\hspace{\parindent}}l@{}}
+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.):
 \begin{lstlisting}
 [int, int, long, double] x;
+void f( double, long );
+\end{lstlisting}
+&
+\begin{lstlisting}
+x.[0, 1] = x.[1, 0];    $\C[1in]{// rearrange: [x.0, x.1] = [x.1, x.0]}$
+f( x.[0, 3] );            $\C{// drop: f(x.0, x.3)}\CRT{}$
+[int, int, int] y = x.[2, 0, 2]; // duplicate: [y.0, y.1, y.2] = [x.2, x.0. x.2]
+\end{lstlisting}
+\end{tabular}
+\smallskip\par\noindent
+\lstMakeShortInline@%
+It is also possible for a member access to contain other member accesses, \eg:
+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:
 \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];                                           $\C{// [v.x, [v.y.i, v.y.j], v.z.k]}$
+\end{lstlisting}
+\begin{comment}
+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.
 \subsection{Casting}
 …
 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();  $\C{// (5)}$
+  [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}
 …
 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 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];
+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];
+}
 [int, int, int] x;
 …
 \end{lstlisting}
+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.
+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.
 \subsection{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:
+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:
 \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 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@:
+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 function which can 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 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, as long as each individual type has a @print@ function.
+The individual print functions can be used to build up more complicated @print@ functions, such as for @S@, which is something that 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}
+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 \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.
+% 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)@.
+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 function 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, as long as each individual type has a @print@ function. The individual print functions can be used to build up more complicated @print@ functions, 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)@.
 \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. \eg:
+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:
 \begin{lstlisting}
 [int, int] f() {
+        [double, double] x;
+        [int, double, int] y;
+}
+\end{lstlisting}
+is transformed into:
+\begin{lstlisting}
+// generated before the first 2-tuple
+forall(dtype T0, dtype T1 | sized(T0) | sized(T1)) struct _tuple2 {
+  [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;
+};
+_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;
+};
+_tuple2(int, int) f() {
+        _tuple2(double, double) x;
+        // generated before the first 3-tuple
+        forall(dtype T0, dtype T1, dtype T2 | sized(T0) | sized(T1) | sized(T2)) struct _tuple3 {
+                T0 field_0;
+                T1 field_1;
+                T2 field_2;
+        };
+        _tuple3(int, double, int) y;
+}
+\end{lstlisting}
+        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:
+Becomes:
 \begin{lstlisting}
 (_tuple3(int, char, double)){ 5, 'x', 1.24 };
 \end{lstlisting}
-\begin{comment}
 Since tuples are essentially structures, tuple indexing expressions are just field accesses:
 \begin{lstlisting}
 …
 [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}
 …
 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.}
 \section{Related Work}

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