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• doc/theses/aaron_moss_PhD/phd/background.tex (modified) (14 diffs)

doc/theses/aaron_moss_PhD/phd/background.tex

@@ -7 +7 @@
 The core design of \CFA{} is laid out in Glen Ditchfield's 1992 PhD thesis, \emph{Contextual Polymorphism} \cite{Ditchfield92}; in that thesis, Ditchfield presents the theoretical underpinnings of the \CFA{} polymorphism model. 
 Building on Ditchfield's design for contextual polymorphism as well as KW-C \cite{Buhr94a}, an earlier set of (largely syntactic) extensions to C, Richard Bilson \cite{Bilson03} built the first version of the \CFA{} compiler, \CFACC{}, in the early 2000's. 
-This early \CFACC{} provided basic functionality, but incorporated a number of poor algorithmic choices due to a rushed implementation time frame, and as such lacked the runtime performance required for practical use; this thesis is substantially concerned with rectifying those deficits. 
-
-The \CFA{} project was revived in 2015 with the intention of building a production-ready language and compiler; at the time of this writing, both \CFA{} and \CFACC{} have been under active development continuously since. 
-As this development has been proceeding concurrently with the work described in this thesis, the state of \CFA{} has been somewhat of a moving target; however, Moss~\etal \cite{Moss18} provides a reasonable summary of the current design. 
+This early \CFACC{} provided basic functionality, but incorporated a number of algorithmic choices that have failed to scale as \CFA{} has developed, lacking the runtime performance for practical use; this thesis is substantially concerned with rectifying those deficits. 
+
+The \CFA{} project was revived in 2015 with the intention of building a production-ready language and compiler; at the time of this writing, both \CFA{} and \CFACC{} remain under active development. 
+As this development has been proceeding concurrently with the work described in this thesis, the state of \CFA{} has been somewhat of a moving target; however, Moss~\etal~\cite{Moss18} provides a reasonable summary of the current design. 
 Notable features added during this period include generic types (Chapter~\ref{generic-chap}), constructors and destructors \cite{Schluntz17}, improved support for tuples \cite{Schluntz17}, reference types \cite{Moss18}, first-class concurrent and parallel programming support \cite{Delisle18}, as well as numerous pieces of syntactic sugar and the start of an idiomatic standard library \cite{Moss18}.
 
-\section{\CFA{} Features}
-
 The selection of features presented in this chapter are chosen to elucidate the design constraints of the work presented in this thesis. 
-In some cases the interactions of multiple features make this design a significantly more complex problem than any individual feature would; in other cases a feature that does not by itself add any complexity to expression resolution triggers previously rare edge cases more frequently.
-
-\subsection{Procedural Paradigm}
+In some cases the interactions of multiple features make this design a significantly more complex problem than any individual feature; in other cases a feature that does not by itself add any complexity to expression resolution triggers previously rare edge cases more frequently.
+
+\section{Procedural Paradigm}
 
 It is important to note that \CFA{} is not an object-oriented language. 
-This is a deliberate choice intended to maintain the applicability of the mental model and language idioms already possessed by C programmers. 
-This choice is in marked contrast to \CC{}, which, though it has backward-compatibility with C on the source code level, is a much larger and more complex language, and requires extensive developer re-training to write idiomatic, efficient code in \CC{}'s object-oriented paradigm.
-
-\CFA{} does have a system of implicit type conversions derived from C's ``usual arithmetic conversions'' \cite[\S{}6.3.1.8]{C11}; while these conversions may be thought of as something like an inheritance hierarchy, the underlying semantics are significantly different and such an analogy is loose at best. 
+This is a deliberate choice intended to maintain the applicability of the programming model and language idioms already possessed by C programmers. 
+This choice is in marked contrast to \CC{}, which is a much larger and more complex language, and requires extensive developer re-training to write idiomatic, efficient code in \CC{}'s object-oriented paradigm.
+
 Particularly, \CFA{} has no concept of \emph{subclass}, and thus no need to integrate an inheritance-based form of polymorphism with its parametric and overloading-based polymorphism. 
+While \CFA{} does have a system of implicit type conversions derived from C's ``usual arithmetic conversions'' \cite[\S{}6.3.1.8]{C11} and these conversions may be thought of as something like an inheritance hierarchy, the underlying semantics are significantly different and such an analogy is loose at best. 
 The graph structure of the \CFA{} type conversions (discussed in Section~\ref{conv-cost-sec}) is also markedly different than an inheritance hierarchy; it has neither a top nor a bottom type, and does not satisfy the lattice properties typical of inheritance hierarchies. 
 
 Another aspect of \CFA{}'s procedural paradigm is that it retains C's translation-unit-based encapsulation model, rather than class-based encapsulation such as in \CC{}. 
-This choice implies that that separate compilation must be maintained to allow headers to act as an encapsulation boundary, rather than the header-only libraries used by \CC{} templates.
-
-\subsection{Name Overloading} \label{overloading-sec}
+This design choice implies that separate compilation must be maintained to allow headers to act as an encapsulation boundary, rather than the header-only libraries used by \CC{} templates.
+
+\section{Name Overloading} \label{overloading-sec}
 
 In C, no more than one variable or function in the same scope may share the same name\footnote{Technically, C has multiple separated namespaces, one holding \lstinline{struct}, \lstinline{union}, and \lstinline{enum} tags, one holding labels, one holding \lstinline{typedef} names, variable, function, and enumerator identifiers, and one for each \lstinline{struct} and \lstinline{union} type holding the field names \cite[\S{}6.2.3]{C11}.}, and variable or function declarations in inner scopes with the same name as a declaration in an outer scope hide the outer declaration. 
-This restriction makes finding the proper declaration to match to a variable expression or function application a simple matter of symbol-table lookup, which can be easily and efficiently implemented. 
+This restriction makes finding the proper declaration to match to a variable expression or function application a simple matter of lexically-scoped name lookup, which can be easily and efficiently implemented. 
 \CFA{}, on the other hand, allows overloading of variable and function names so long as the overloaded declarations do not have the same type, avoiding the multiplication of variable and function names for different types common in the C standard library, as in the following example: 
 
@@ -51 +49 @@
 \end{cfa}
 
-While the wisdom of giving both the maximum value of a type and the function to take the maximum of two values the same name is debatable, \eg{} some developers may prefer !MAX! for the former, the guiding philosophy of \CFA{} is ``describe, don't prescribe'' --- we prefer to trust programmers with powerful tools, and there is no technical reason to restrict overloading between variables and functions. 
+The final expression in the preceding example includes a feature of \CFA{} name overloading not shared by \CC{}, the ability to disambiguate expressions based on their return type. This provides greater flexibility and power than the parameter-based overload selection of \CC{}, though at the cost of greater complexity in the resolution algorithm.
+
+While the wisdom of giving both the maximum value of a type and the function to take the maximum of two values the same name in the example above is debatable, \eg{} some developers may prefer !MAX! for the former, the guiding philosophy of \CFA{} is ``describe, don't prescribe'' --- we prefer to trust programmers with powerful tools, and there is no technical reason to restrict overloading between variables and functions. 
 However, the expressivity of \CFA{}'s name overloading does have the consequence that simple table lookup is insufficient to match identifiers to declarations, and a type-matching algorithm must be part of expression resolution.
 
-\subsubsection{Operator Overloading}
+\subsection{Operator Overloading}
 
 C does allow name overloading in one context: operator overloading. 
 From the perspective of the type system, there is nothing special about operators as opposed to other functions, nor is it desirable to restrict the clear and readable syntax of operators to only the built-in types. 
-For these reasons, \CFA{} also allows overloading of operators by writing specially-named functions where !?! stands in for the operands\footnote{This example uses \CFA{}'s reference types, described in Section~\ref{type-features-sec}}:
+For these reasons, \CFA{}, like \CC{} and many other programming languages, allows overloading of operators by writing specially-named functions where !?! stands in for the operands. 
+This syntax is more natural than the operator overloading syntax of \CC{}, which requires ``dummy'' parameters to disambiguate overloads of similarly-named pre- and postfix operators\footnote{This example uses \CFA{}'s reference types, described in Section~\ref{type-features-sec}}:
 
 \begin{cfa}
@@ -74 +75 @@
 Together, \CFA{}'s backward-compatibility with C and the inclusion of this operator overloading feature imply that \CFA{} must select among function overloads using a method compatible with C's ``usual arithmetic conversions'' \cite[\S{}6.3.1.8]{C11}, so as to present user programmers with only a single set of overloading rules.
 
-\subsubsection{Special Literal Types}
+\subsection{Special Literal Types}
 
 Literal !0! is also used polymorphically in C; it may be either integer zero or the null value of any pointer type. 
-\CFA{} provides a special type for the !0! literal, !zero_t!, so that users can define a zero value for their own types without being forced to create a conversion from an integer or pointer type (though \CFA{} also includes implicit conversions from !zero_t! to the integer and pointer types for backward compatibility). 
+\CFA{} provides a special type for the !0! literal, !zero_t!, so that users can define a zero value for their own types without being forced to create a conversion from an integer or pointer type; \CFA{} also includes implicit conversions from !zero_t! to the !int! and pointer type constructors\footnote{See Section~\ref{type-features-sec}} from !zero_t! for backward compatibility. 
 
 According to the C standard \cite[\S{}6.8.4.1]{C11}, !0! is the only false value; any value that compares equal to zero is false, while any value that does not is true. 
 By this rule, Boolean contexts such as !if ( x )! can always be equivalently rewritten as \lstinline{if ( (x) != 0 )}. 
-\CFACC{} applies this rewriting in all Boolean contexts, so any type !T! can be made ``truthy'' (that is, given a Boolean interpretation) in \CFA{} by defining an operator overload \lstinline{int ?!=?(T, zero_t)}; unlike \CC{} prior to the addition of explicit casts in \CCeleven{}, this design does not add comparability or convertibility to arbitrary integer types. 
+\CFACC{} applies this rewriting in all Boolean contexts, so any type !T! can be made ``truthy'' (that is, given a Boolean interpretation) in \CFA{} by defining an operator overload \lstinline{int ?!=?(T, zero_t)}.
+\CC{} takes a different approach to user-defined truthy types, allowing definition of an implicit conversion operator to !bool!; prior to the introduction of the !explicit! keyword for conversion operators in \CCeleven{} this approach also allowed undesired implicit conversions to all other arithmetic types, a shortcoming not shared by the \CFA{} design.
 
 \CFA{} also includes a special type for !1!, !one_t!; like !zero_t!, !one_t! has built-in implicit conversions to the various integral types so that !1! maintains its expected semantics in legacy code. 
@@ -87 +89 @@
 
 \CFA{} previously allowed !0! and !1! to be the names of polymorphic variables, with separate overloads for !int 0!, !int 1!, and !forall(dtype T) T* 0!. 
-As revealed in the work presented here on generic types (Chapter~\ref{generic-chap}), the parametric polymorphic zero variable was not generalizable to other types; though all null pointers have the same in-memory representation, the same cannot be said of the zero values of arbitrary types. 
+As revealed in the work presented here on generic types (Chapter~\ref{generic-chap}), the parametric polymorphic zero variable is not generalizable to other types; though all null pointers have the same in-memory representation, the same cannot be said of the zero values of arbitrary types. 
 As such, variables that could represent !0! and !1! were phased out in favour of functions that could generate those values for a given type as appropriate.
 
-\subsection{Polymorphic Functions} \label{poly-func-sec}
-
-The most significant feature \CFA{} adds is parametric-polymorphic functions. 
+\section{Polymorphic Functions} \label{poly-func-sec}
+
+The most significant type-system feature \CFA{} adds is parametric-polymorphic functions. 
 Such functions are written using a !forall! clause (which gives the language its name):
 
@@ -103 +105 @@
 The type variable !T! is transformed into a set of additional implicit parameters to !identity!, which encode sufficient information about !T! to create and return a variable of that type. 
 \CFA{} passes the size and alignment of the type represented by an !otype! parameter, as well as a default constructor, copy constructor, assignment operator, and destructor. 
-Types which do not have one or more of these operators visible cannot be bound to !otype! parameters. 
+Types which do not have one or more of these operators visible cannot be bound to !otype! parameters, but may be bound to un-constrained !dtype! (``data type'') type variables. 
 In this design, the runtime cost of polymorphism is spread over each polymorphic call, due to passing more arguments to polymorphic functions; the experiments in Chapter~\ref{generic-chap} indicate that this overhead is comparable to that of \CC{} virtual function calls. 
 % \TODO{rerun experiments, possibly look at vtable variant}
@@ -111 +113 @@
 The fact that there is only one implementation of each polymorphic function also reduces compile times relative to the template-expansion approach taken by \CC{}, as well as reducing binary sizes and runtime pressure on instruction cache by re-using a single version of each function.
 
-\subsubsection{Type Assertions}
-
-Since bare polymorphic types do not provide a great range of available operations, \CFA{} provides a \emph{type assertion} mechanism to provide further information about a type:
+\subsection{Type Assertions}
+
+Since bare polymorphic types do not provide a great range of available operations, \CFA{} provides a \emph{type assertion} mechanism to provide further information about a type\footnote{Subscript not included in source code}:
 
 \begin{cfa}
 forall(otype T `| { T twice(T); }`)
 T four_times(T x) { return twice( twice(x) ); }
-double twice(double d) { return d * 2.0; }  $\C[2.75in]{// (1)}$
+double twice$\(_1\)$(double d) { return d * 2.0; }
 
 double ans = four_times( 10.5 );  $\C[2.75in]{// T bound to double, ans == 42.0}$
@@ -127 +129 @@
 
 Monomorphic specializations of polymorphic functions can themselves be used to satisfy type assertions. 
-For instance, !twice! could have been defined like this:
+For instance, !twice! could have been defined like this\footnotemark[5]:
 
 \begin{cfa}
 forall(otype S | { S ?+?(S, S); })
-S twice(S x) { return x + x; }  $\C[2.75in]{// (2)}
-\end{cfa}
-
-This version of !twice! works for any type !S! that has an addition operator defined for it, and it could be used to satisfy the type assertion on !four_times!. 
-\CFACC{} accomplishes this by creating a wrapper function calling !twice//(2)! with !S! bound to !double!, then providing this wrapper function to !four_times!\footnote{\lstinline{twice // (2)} could also have had a type parameter named \lstinline{T}; \CFA{} specifies renaming of the type parameters, which would avoid the name conflict with the type variable \lstinline{T} of \lstinline{four_times}}.
+S twice$\(_2\)$(S x) { return x + x; }
+\end{cfa}
+
+Specializing this polymorphic function with !S = double! produces a monomorphic function which can  be used to satisfy the type assertion on !four_times!. 
+\CFACC{} accomplishes this by creating a wrapper function calling !twice!$_2$ with !S! bound to !double!, then providing this wrapper function to !four_times!\footnote{\lstinline{twice}$_2$ could also have had a type parameter named \lstinline{T}; \CFA{} specifies renaming of the type parameters, which would avoid the name conflict with the type variable \lstinline{T} of \lstinline{four_times}}.
+However, !twice!$_2$ also works for any type !S! that has an addition operator defined for it. 
 
 Finding appropriate functions to satisfy type assertions is essentially a recursive case of expression resolution, as it takes a name (that of the type assertion) and attempts to match it to a suitable declaration in the current scope. 
 If a polymorphic function can be used to satisfy one of its own type assertions, this recursion may not terminate, as it is possible that that function is examined as a candidate for its own assertion unboundedly repeatedly. 
-To avoid such infinite loops, \CFACC{} imposes a fixed limit on the possible depth of recursion, similar to that employed by most \CC{} compilers for template expansion; this restriction means that there are some semantically well-typed expressions that cannot be resolved by \CFACC{}.
-
-% \subsubsection{Deleted Declarations}
+To avoid such infinite loops, \CFACC{} imposes a fixed limit on the possible depth of recursion, similar to that employed by most \CC{} compilers for template expansion; this restriction means that there are some otherwise semantically well-typed expressions that cannot be resolved by \CFACC{}.
+
+% \subsection{Deleted Declarations}
 
 % Particular type combinations can be exempted from matching a given polymorphic function through use of a \emph{deleted function declaration}:
@@ -153 +156 @@
 % If this deleted declaration is selected as the unique minimal-cost interpretation of an expression than an error is produced. 
 
-\subsubsection{Traits}
+\subsection{Traits}
 
 \CFA{} provides \emph{traits} as a means to name a group of type assertions, as in the example below\footnote{This example uses \CFA{}'s reference types and constructors, described in Section~\ref{type-features-sec}.}:
@@ -172 +175 @@
 
 Semantically, traits are simply a named list of type assertions, but they may be used for many of the same purposes that interfaces in Java or abstract base classes in \CC{} are used for. 
-Unlike Java interfaces or \CC{} base classes, \CFA{} types do not explicitly state any inheritance relationship to traits they satisfy; this can be considered a form a structural inheritance, similar to interface implementation in Go, as opposed to the nominal inheritance model of Java and \CC{}. 
+Unlike Java interfaces or \CC{} base classes, \CFA{} types do not explicitly state any inheritance relationship to traits they satisfy; this can be considered a form of structural inheritance, similar to interface implementation in Go, as opposed to the nominal inheritance model of Java and \CC{}. 
 Nominal inheritance can be simulated in \CFA{} using marker variables in traits:
 
@@ -187 +190 @@
 
 Traits, however, are significantly more powerful than nominal-inheritance interfaces; firstly, due to the scoping rules of the declarations that satisfy a trait's type assertions, a type may not satisfy a trait everywhere that that type is declared, as with !char! and the !nominal! trait above. 
-Secondly, because \CFA{} is not object-oriented and types do not have a closed set of methods, existing C library types can be extended to implement a trait simply by writing the requisite functions. 
+Secondly, because \CFA{} is not object-oriented and types do not have a closed set of methods, existing C library types can be extended to implement a trait simply by writing the requisite functions\footnote{\CC{} only allows partial extension of C types, because constructors, destructors, conversions, and the assignment, indexing, and function-call operators may only be defined in a class; \CFA{} does not have any of these restrictions.}. 
 Finally, traits may be used to declare a relationship among multiple types, a property that may be difficult or impossible to represent in nominal-inheritance type systems\footnote{This example uses \CFA{}'s reference types, described in Section~\ref{type-features-sec}.}: 
 
@@ -207 +210 @@
 \end{cfa}
 
-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. 
+In this 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 !int!, while !(*it).value = 42;! interprets !*it! as !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. 
@@ -214 +217 @@
 The ability of types to begin or cease to satisfy traits when declarations go into or out of scope makes caching of trait satisfaction judgments difficult, and the ability of traits to take multiple type parameters can lead to a combinatorial explosion of work in any attempt to pre-compute trait satisfaction relationships. 
 
-\subsection{Implicit Conversions} \label{implicit-conv-sec}
-
-In addition to the multiple interpretations of an expression produced by name overloading and polymorphic functions, for backward compatibility \CFA{} must support all of the implicit conversions present in C, producing further candidate interpretations for expressions. 
-As mentioned above, C does not have an inheritance hierarchy of types, but the C standard's rules for the ``usual arithmetic conversions' \cite[\S{}6.3.1.8]{C11} define which of the built-in types are implicitly convertible to which other types, as well as which implicit conversions to apply for mixed arguments to binary operators. 
+\section{Implicit Conversions} \label{implicit-conv-sec}
+
+In addition to the multiple interpretations of an expression produced by name overloading and polymorphic functions, \CFA{} must support all of the implicit conversions present in C for backward compatibility, producing further candidate interpretations for expressions. 
+As mentioned above, C does not have an inheritance hierarchy of types, but the C standard's rules for the ``usual arithmetic conversions'' \cite[\S{}6.3.1.8]{C11} define which of the built-in types are implicitly convertible to which other types, as well as which implicit conversions to apply for mixed arguments to binary operators. 
 \CFA{} adds rules to the usual arithmetic conversions defining the cost of binding a polymorphic type variable in a function call; such bindings are cheaper than any \emph{unsafe} (narrowing) conversion, \eg{} !int! to !char!, but more expensive than any \emph{safe} (widening) conversion, \eg{} !int! to !double!. 
 One contribution of this thesis, discussed in Section~\ref{conv-cost-sec}, is a number of refinements to this cost model to more efficiently resolve polymorphic function calls. 
 
-The expression resolution problem which is the focus of Chapter~\ref{resolution-chap} is to find the unique minimal-cost interpretation of each expression in the program, where all identifiers must be matched to a declaration, and implicit conversions or polymorphic bindings of the result of an expression may increase the cost of the expression. 
-While semantically valid \CFA{} code always has such a unique minimal-cost interpretation, \CFACC{} must also be able to detect and report as errors expressions which have either no interpretation or multiple ambiguous minimal-cost interpretations. 
+The expression resolution problem, which is the focus of Chapter~\ref{resolution-chap}, is to find the unique minimal-cost interpretation of each expression in the program, where all identifiers must be matched to a declaration, and implicit conversions or polymorphic bindings of the result of an expression may increase the cost of the expression. 
+While semantically valid \CFA{} code always has such a unique minimal-cost interpretation, \CFACC{} must also be able to detect and report as errors expressions that have either no interpretation or multiple ambiguous minimal-cost interpretations. 
 Note that which subexpression interpretation is minimal-cost may require contextual information to disambiguate. 
 For instance, in the example in Section~\ref{overloading-sec}, !max(max, -max)! cannot be unambiguously resolved, but !int m = max(max, -max)! has a single minimal-cost resolution. 
-While the interpretation !int m = (int)max((double)max, -(double)max)! is also a valid interpretation, it is not minimal-cost due to the unsafe cast from the !double! result of !max! to !int!\footnote{The two \lstinline{double} casts function as type ascriptions selecting \lstinline{double max} rather than casts from \lstinline{int max} to \lstinline{double}, and as such are zero-cost.}. 
+While the interpretation !int m = (int)max((double)max, -(double)max)! is also a valid interpretation, it is not minimal-cost due to the unsafe cast from the !double! result of !max! to !int!\footnote{The two \lstinline{double} casts function as type ascriptions selecting \lstinline{double max} rather than casts from \lstinline{int max} to \lstinline{double}, and as such are zero-cost. The \lstinline{int} to \lstinline{double} conversion could be forced if desired with two casts: \lstinline{(double)(int)max}}. 
 These contextual effects make the expression resolution problem for \CFA{} both theoretically and practically difficult, but the observation driving the work in Chapter~\ref{resolution-chap} is that of the many top-level expressions in a given program, most are straightforward and idiomatic so that programmers writing and maintaining the code can easily understand them; it follows that effective heuristics for common cases can bring down compiler runtime enough that a small proportion of harder-to-resolve expressions does not inordinately increase overall compiler runtime or memory usage.
 
-\subsection{Type Features} \label{type-features-sec}
-
-The name overloading and polymorphism features of \CFA{} have the greatest effect on language design and compiler runtime, but there are a number of other features in the type system which have a smaller effect but are useful for code examples. 
+\section{Type Features} \label{type-features-sec}
+
+The name overloading and polymorphism features of \CFA{} have the greatest effect on language design and compiler runtime, but there are a number of other features in the type system that have a smaller effect but are useful for code examples. 
 These features are described here. 
 
-\subsubsection{Reference Types}
+\subsection{Reference Types}
 
 One of the key ergonomic improvements in \CFA{} is reference types, designed and implemented by Robert Schluntz \cite{Schluntz17}. 
 Given some type !T!, a !T&! (``reference to !T!'') is essentially an automatically dereferenced pointer. 
-These types allow seamless pass-by-reference for function parameters, without the extraneous dereferencing syntax present in C; they also allow easy easy aliasing of nested values with a similarly convenient syntax. 
-A particular improvement is removing syntactic special cases for operators which take or return mutable values; for example, the use !a += b! of a compound assignment operator now matches its signature, !int& ?+=?(int&, int)!, as opposed to the previous syntactic special cases to automatically take the address of the first argument to !+=! and to mark its return value as mutable.
-
-The C standard makes heavy use of the concept of \emph{lvalue}, an expression with a memory address; its complement, \emph{rvalue} (a non-addressable expression) is not explicitly named. 
+These types allow seamless pass-by-reference for function parameters, without the extraneous dereferencing syntax present in C; they also allow easy aliasing of nested values with a similarly convenient syntax. 
+A particular improvement is removing syntactic special cases for operators that take or return mutable values; for example, the use !a += b! of a compound assignment operator now matches its signature, !int& ?+=?(int&, int)!, as opposed to the syntactic special cases in earlier versions of \CFA{} to automatically take the address of the first argument to !+=! and to mark its return value as mutable.
+
+The C standard makes heavy use of the concept of \emph{lvalue}, an expression with a memory address; its complement, \emph{rvalue} (a non-addressable expression) is not explicitly named in the standard. 
 In \CFA{}, the distinction between lvalue and rvalue can be re-framed in terms of reference and non-reference types, with the benefit of being able to express the difference in user code. 
 \CFA{} references preserve the existing qualifier-dropping implicit lvalue-to-rvalue conversion from C (\eg{} a !const volatile int&! can be implicitly copied to a bare !int!)
-To make reference types more easily usable in legacy pass-by-value code, \CFA{} also adds an implicit rvalue-to-lvalue conversion, implemented by storing the value in a fresh compiler-generated temporary variable and passing a reference to that temporary. 
-To mitigate the ``!const! hell'' problem present in \CC{}, there is also a qualifier-dropping lvalue-to-lvalue conversion, also implemented by copying into a temporary:
+To make reference types more easily usable in legacy pass-by-value code, \CFA{} also adds an implicit rvalue-to-lvalue conversion, implemented by storing the value in a compiler-generated temporary variable and passing a reference to that temporary. 
+To mitigate the ``!const! hell'' problem present in \CC{}, there is also a qualifier-dropping lvalue-to-lvalue conversion implemented by copying into a temporary:
 
 \begin{cfa}
@@ -273 +276 @@
 For better compatibility with C, the \CFA{} team has chosen not to differentiate function overloads based on top-level reference types, and as such their contribution to the difficulty of \CFA{} expression resolution is largely restricted to the implementation details of normalization conversions and adapters. 
 
-\subsubsection{Resource Management}
+\subsection{Resource Management}
 
 \CFA{} also supports the RAII (``Resource Acquisition is Initialization'') idiom originated by \CC{}, thanks to the object lifetime work of Robert Schluntz \cite{Schluntz17}. 
@@ -326 +329 @@
 \end{cfa}
 
-\subsubsection{Tuple Types}
+\subsection{Tuple Types}
 
 \CFA{} adds \emph{tuple types} to C, a syntactic facility for referring to lists of values anonymously or with a single identifier. 
 An identifier may name a tuple, a function may return one, and a tuple may be implicitly \emph{destructured} into its component values. 
 The implementation of tuples in \CFACC{}'s code generation is based on the generic types introduced in Chapter~\ref{generic-chap}, with one compiler-generated generic type for each tuple arity. 
-This allows tuples to take advantage of the same runtime optimizations available to generic types, while reducing code bloat. 
+This reuse allows tuples to take advantage of the same runtime optimizations available to generic types, while reducing code bloat. 
 An extended presentation of the tuple features of \CFA{} can be found in \cite{Moss18}, but the following example demonstrates the basic features:
 
 \begin{cfa}
-[char, char] x = ['!', '?']; $\C{// (1); tuple type and expression syntax}$
-int x = 2; $\C{// (2)}$
+[char, char] x$\(_1\)$ = ['!', '?']; $\C{// tuple type and expression syntax}$
+int x$\(_2\)$ = 2;
 
 forall(otype T)
-[T, T] swap( T a, T b ) { $\C{// (3)}$
+[T, T] swap$\(_1\)$( T a, T b ) {
         return [b, a]; $\C{// one-line swap syntax}$
 }
 
-x = swap( x ); $\C{// destructure [char, char] x into two elements}$
-$\C{// cannot use int x, not enough arguments}$
-
-void swap( int, char, char ); $\C{// (4)}$
-
-swap( x, x ); $\C{// (4) on (2), (1)}$
-$\C{// not (3) on (2), (2) due to polymorphism cost}$
+x = swap( x ); $\C{// destructure x\(_1\) into two elements}$
+$\C{// cannot use x\(_2\), not enough arguments}$
+
+void swap$\(_2\)$( int, char, char );
+
+swap( x, x ); $\C{// swap\(_2\)( x\(_2\), x\(_1\) )}$
+$\C{// not swap\(_1\)( x\(_2\), x\(_2\) ) due to polymorphism cost}$
 \end{cfa}
 
 Tuple destructuring breaks the one-to-one relationship between identifiers and values. 
-This precludes some argument-parameter matching strategies for expression resolution, as well as cheap interpretation filters based on comparing number of parameters and arguments. 
-As an example, in the call to !swap( x, x )! above, the second !x! can be resolved starting at the second or third parameter of !swap!, depending which interpretation of !x! was chosen for the first argument.
+Hence, some argument-parameter matching strategies for expression resolution are precluded, as well as cheap interpretation filters based on comparing number of parameters and arguments. 
+As an example, in the call to !swap( x, x )! above, the second !x! can be resolved starting at the second or third parameter of !swap!$_2$, depending which interpretation of !x! is chosen for the first argument.
 
 \section{Conclusion}