% inline code ©...© (copyright symbol) emacs: C-q M-)
% red highlighting ®...® (registered trademark symbol) emacs: C-q M-.
% blue highlighting ß...ß (sharp s symbol) emacs: C-q M-_
% green highlighting ¢...¢ (cent symbol) emacs: C-q M-"
% LaTex escape §...§ (section symbol) emacs: C-q M-'
% keyword escape ¶...¶ (pilcrow symbol) emacs: C-q M-^
% math escape $...$ (dollar symbol)

\documentclass[twoside,11pt]{article}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Latex packages used in the document (copied from CFA user manual).
\usepackage[T1]{fontenc}                                % allow Latin1 (extended ASCII) characters
\usepackage{textcomp}
\usepackage[latin1]{inputenc}
\usepackage{fullpage,times,comment}
\usepackage{epic,eepic}
\usepackage{upquote}									% switch curled `'" to straight
\usepackage{calc}
\usepackage{xspace}
\usepackage{graphicx}
\usepackage{varioref}									% extended references
\usepackage{listings}									% format program code
\usepackage[flushmargin]{footmisc}						% support label/reference in footnote
\usepackage{latexsym}                                   % \Box glyph
\usepackage{mathptmx}                                   % better math font with "times"
\usepackage[usenames]{color}
\usepackage[pagewise]{lineno}
\renewcommand{\linenumberfont}{\scriptsize\sffamily}
\input{common}                                          % bespoke macros used in the document
\usepackage[dvips,plainpages=false,pdfpagelabels,pdfpagemode=UseNone,colorlinks=true,pagebackref=true,linkcolor=blue,citecolor=blue,urlcolor=blue,pagebackref=true,breaklinks=true]{hyperref}
\usepackage{breakurl}
\renewcommand{\UrlFont}{\small\sf}

\setlength{\topmargin}{-0.45in}							% move running title into header
\setlength{\headsep}{0.25in}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newsavebox{\LstBox}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{\Huge
\vspace*{1in}
Efficient Type Resolution in \CFA \\
\vspace*{0.25in}
\huge
PhD Comprehensive II Research Proposal
\vspace*{1in}
}

\author{\huge
\vspace*{0.1in}
Aaron Moss \\
\Large Cheriton School of Computer Science \\
\Large University of Waterloo
}

\date{
\today
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\pagestyle{headings}
% changed after setting pagestyle
\renewcommand{\sectionmark}[1]{\markboth{\thesection\quad #1}{\thesection\quad #1}}
\renewcommand{\subsectionmark}[1]{\markboth{\thesubsection\quad #1}{\thesubsection\quad #1}}
\pagenumbering{roman}
\linenumbers                                            % comment out to turn off line numbering

\maketitle
\thispagestyle{empty}

\clearpage
\thispagestyle{plain}
\pdfbookmark[1]{Contents}{section}
\tableofcontents

\clearpage
\thispagestyle{plain}
\pagenumbering{arabic}

\section{Introduction}

\CFA\footnote{Pronounced ``C-for-all'', and written \CFA, CFA, or \CFL.} is an evolutionary modernization of the C programming language currently being designed and built at the University of Waterloo by a team led by Peter Buhr. 
Features added to C by \CFA include name overloading, user-defined operators, parametric-polymorphic routines, and type constructors and destructors, among others. 
These features make \CFA significantly more powerful and expressive than C, but impose a significant compile-time cost to implement, particularly in the expression resolver, which must evaluate the typing rules of a much more complex type system. 
The primary goal of this proposed research project is to develop a sufficiently performant expression resolution algorithm, experimentally validate its performance, and integrate it into \Index*{CFA-CC}, the \CFA reference compiler.
Secondary goals of this project include the development of various new language features for \CFA; parametric-polymorphic (``generic'') types have already been designed and implemented, and reference types and user-defined conversions are under design consideration. 
The experimental performance-testing architecture for resolution algorithms will also be used to determine the compile-time cost of adding such new features to the \CFA type system.

\section{\CFA}

To make the scope of the proposed expression resolution problem more explicit, it is necessary to define the features of both C and \CFA (both current and proposed) which affect this algorithm. 
In some cases the interactions of multiple features make expression resolution a significantly more complex problem than any individual feature would; in others a feature which does not by itself add any complexity to expression resolution will trigger previously rare edge cases much more frequently.

\subsection{Polymorphic Functions}
The most significant feature \CFA adds is parametric-polymorphic functions. 
Such functions are written using a ©forall© clause, the feature that gave the language its name:
\begin{lstlisting}
forall(otype T)
T identity(T x) {
    return x;
}

int forty_two = identity(42); // T is bound to int, forty_two == 42
\end{lstlisting}
The ©identity© function above can be applied to any complete object type (or ``©otype©''). 
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. 
The current \CFA implementation passes the size and alignment of the type represented by an ©otype© parameter, as well as an assignment operator, constructor, copy constructor \& destructor.

Since bare polymorphic types do not provide a great range of available operations, \CFA also provides a \emph{type assertion} mechanism to provide further information about a type:
\begin{lstlisting}
forall(otype T | { T twice(T); })
T four_times(T x) {
    return twice( twice(x) );
}

double twice(double d) { return d * 2.0; } // (1)

double magic = four_times(10.5); // T is bound to double, uses (1) to satisfy type assertion
\end{lstlisting}
These type assertions may be either variable or function declarations which depend on a polymorphic type variable. 
©four_times© can only be called with an argument for which there exists a function named ©twice© that can take that argument and return another value of the same type; a pointer to the appropriate ©twice© function will be passed as an additional implicit parameter to the call to ©four_times©.

Monomorphic specializations of polymorphic functions can themselves be used to satisfy type assertions. 
For instance, ©twice© could have been define as below, using the \CFA syntax for operator overloading:
\begin{lstlisting}
forall(otype S | { S ?+?(S, S); })
S twice(S x) { return x + x; }  // (2)
\end{lstlisting} 
This version of ©twice© will work for any type ©S© that has an addition operator defined for it, and it could have been used to satisfy the type assertion on ©four_times©. 
The compiler accomplishes this by creating a wrapper function calling ©twice // (2)© with ©S© bound to ©double©, then providing this wrapper function to ©four_times©\footnote{©twice // (2)© could have had a type parameter named ©T©; \CFA specifies a renaming the type parameters, which would avoid the name conflict with the parameter ©T© of ©four_times©.}. 

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 function will be examined as a candidate for its own type assertion unboundedly repeatedly. 
To avoid infinite loops, the current \Index*{CFA-CC} compiler imposes a fixed limit on the possible depth of recursion, similar to that employed by most \Index*[C++]{\CC} compilers for template expansion; this restriction means that there are some semantically well-typed expressions which cannot be resolved by {CFA-CC}. 
One area of potential improvement this project proposes to investigate is the possibility of using the compiler's knowledge of the current set of declarations to make a more precise judgement of when further type assertion satisfaction recursion will not produce a well-typed expression.

\subsection{Name Overloading}
In C, no more than one function or variable in the same scope may share the same name, and function or variable declarations in inner scopes with the same name as a declaration in an outer scope hide the outer declaration.  
This makes finding the proper declaration to match to a function application or variable expression a simple matter of symbol table 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 function names for different types common in the C standard library, as in the following example:
\begin{lstlisting}
int three = 3;
double three = 3.0;

int thrice(int i) { return i * three; } // uses int three
double thrice(double d) { return d * three; } // uses double three

// thrice(three); // ERROR: ambiguous
int nine = thrice(three);    // uses int thrice and three, based on return type
double nine = thrice(three); // uses double thrice and three, based on return type
\end{lstlisting}

The presence of name overloading in \CFA means that simple table lookup is not sufficient to match identifiers to declarations, and a type matching algorithm must be part of expression resolution.

\subsection{Implicit Conversions}
In addition to the multiple interpretations of an expression produced by name overloading, \CFA also supports all of the implicit conversions present in C, producing further candidate interpretations for expressions. 
C does not have a traditionally-defined inheritance hierarchy of types, but the C standard's rules for the ``usual arithmetic conversions'' define which of the built-in types are implicitly convertable to which other types, and the relative cost of any pair of such conversions from a single source type. 
\CFA adds to the usual arithmetic conversions rules for determining 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©. 
The expression resolution problem, then, 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. 
Note that which subexpression interpretation is minimal-cost may require contextual information to disambiguate.

\subsubsection{User-generated Implicit Conversions}
One possible additional feature to \CFA included in this research proposal is \emph{user-generated implicit conversions}. 
Such a conversion system should be simple for user programmers to utilize, and fit naturally with the existing design of implicit conversions in C; ideally it would also be sufficiently powerful to encode C's usual arithmetic conversions itself, so that \CFA only has one set of rules for conversions. 

Glen Ditchfield \textbf{TODO CITE} has laid out a framework for using polymorphic conversion constructor functions to create a directed acyclic graph (DAG) of conversions. 
A monomorphic variant of these functions can be used to mark a conversion arc in the DAG as only usable as the final step in a conversion. 
With these two types of conversion arcs, separate DAGs can be created for the safe and the unsafe conversions, and conversion cost can be represented as path length through the DAG. 
Open research questions on this topic include whether a conversion graph can be generated that represents each allowable conversion in C with a unique minimal-length path, such that the path lengths accurately represent the relative costs of the conversions, whether such a graph representation can be usefully augmented to include user-defined types as well as built-in types, and whether the graph can be efficiently represented and included in the expression resolver.

\subsection{Constructors \& Destructors}
Rob Shluntz, a current member of the \CFA research team, has added constructors and destructors to \CFA. 
Each type has an overridable default-generated zero-argument constructor, copy constructor, assignment operator, and destructor; for struct types these functions each call their equivalents on each field of the struct. 
This affects expression resolution because an ©otype© type variable ©T© implicitly adds four type assertions, one for each of these four functions, so assertion resolution is pervasive in \CFA polymorphic functions, even those without any explicit type assertions. 

\subsection{Generic Types}
The author has added a generic type capability to \CFA, designed to efficiently and naturally integrate with \CFA's existing polymorphic functions. 
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; }

pair(const char*, int) p = { "magic", 42 };
int magic = value( &p );
\end{lstlisting}
For \emph{concrete} generic types, that is, those where none of the type parameters depend on polymorphic type variables (like ©pair(const char*, int)© above), the struct is essentially template expanded to a new struct type; for \emph{polymorphic} generic types (such as ©pair(const char*, T)© above), member access is handled by a runtime calculation of the field offset, based on the size and alignment information of the polymorphic parameter type. 
The default-generated constructors, destructor \& assignment operator for a generic type are polymorphic functions with the same list of type parameters as the generic type definition.

Aside from giving users the ability to create more parameterized types than just the built-in pointer, array \& function types, the combination of generic types with polymorphic functions and implicit conversions makes the edge case where a polymorphic function can match its own assertions much more common, as follows:
\begin{itemize} 
\item Given an expression in an untyped context, such as a top-level function call with no assignment of return values, apply a polymorphic implicit conversion to the expression that can produce multiple types (the built-in conversion from ©void*© to any other pointer type is one, but not the only).
\item When attempting to use a generic type with ©otype© parameters (such as ©box© above) for the result type of the expression, the resolver will also need to decide what type to use for the ©otype© parameters on the constructors and related functions, and will have no constraints on what they may be.
\item Attempting to match some yet-to-be-determined specialization of the generic type to this ©otype© parameter will create a recursive case of the default constructor, \etc matching their own type assertions, creating an unboundedly deep nesting of the generic type inside itself.
\end{itemize}
As discussed above, any \CFA expression resolver must handle this possible infinite recursion somehow, but the combination of generic types with other language features makes this particular edge case occur somewhat frequently in user code.

\subsection{Tuple Types}
\CFA adds \emph{tuple types} to C, a facility for referring to multiple values with a single identifier. 
A variable may name a tuple, and a function may return one. 
Particularly relevantly for resolution, a tuple may be automatically \emph{destructured} into a list of values, as in the ©swap© function below:
\begin{lstlisting}
[char, char] x = [ '!', '?' ];
int x = 42;

forall(otype T) [T, T] swap( T a, T b ) { return [b, a]; }

x = swap( x ); // destructure [char, char] x into two elements of parameter list
// ^ can't use int x for parameter, not enough arguments to swap
\end{lstlisting}
Tuple destructuring means that the mapping from the position of a subexpression in the argument list to the position of a paramter in the function declaration is not straightforward, as some arguments may be expandable to different numbers of parameters, like ©x© above.

\subsection{Reference Types}
The author, in collaboration with the rest of the \CFA research team, has been designing \emph{reference types} for \CFA. 
Given some type ©T©, a ©T&© (``reference to ©T©'') is essentially an automatically dereferenced pointer; with these semantics most of the C standard's discussions of lvalues can be expressed in terms of references instead, with the benefit of being able to express the difference between the reference and non-reference version of a type in user code. 
References preserve C's existing qualifier-dropping lvalue-to-rvalue conversion (\eg a ©const volatile int&© can be implicitly converted to a bare ©int©); the reference proposal also adds a rvalue-to-lvalue conversion to \CFA, implemented by storing the value in a new compiler-generated temporary and passing a reference to the temporary. 
These two conversions can chain, producing a qualifier-dropping conversion for references, for instance converting a reference to a ©const int© into a reference to a non-©const int© by copying the originally refered to value into a fresh temporary and taking a reference to this temporary. 
These reference conversions may also chain with the other implicit type conversions. 
The main implication of this for expression resolution is the multiplication of available implicit conversions, though in a restricted context that may be able to be treated efficiently as a special case.

\subsection{Literal Types}
Another proposal currently under consideration for the \CFA type system is assigning special types to the literal values ©0© and ©1©.%, say ©zero_t© and ©one_t©. 
Implicit conversions from these types would allow ©0© and ©1© to be considered as values of many different types, depending on context, allowing expression desugarings like ©if ( x ) {}© $\Rightarrow$ ©if ( x != 0 ) {}© to be implemented efficiently and precicely. 
This is a generalization of C's existing behaviour of treating ©0© as either an integer zero or a null pointer constant, and treating either of those values as boolean false. 
The main implication for expression resolution is that the frequently encountered expressions ©0© and ©1© may have a significant number of valid interpretations.

\subsection{Deleted Function Declarations}
One final proposal for \CFA with an impact on the expression resolver is \emph{deleted function declarations}; in \CCeleven, a function declaration can be deleted as below:
\begin{lstlisting}
int somefn(char) = delete;
\end{lstlisting}
To add a similar feature to \CFA would involve including the deleted function declarations in expression resolution along with the normal declarations, but producing a compiler error if the deleted function was the best resolution. 
How conflicts should be handled between resolution of an expression to both a deleted and a non-deleted function is a small but open research question.

\section{Expression Resolution}
% TODO cite Baker, Cormack, etc.

\section{Completion Timeline}
The following is a preliminary estimate of the time necessary to complete the major components of this research project:
\begin{center}
\begin{tabular}{ | r @{--} l | p{4in} | }
\hline       May 2015 & April 2016   & Project familiarization and generic types design \& implementation. \\
\hline       May 2016 & April 2017   & Design \& implement prototype resolver and run performance experiments. \\
\hline       May 2017 & August 2017  & Integrate new language features and best-performing resolver prototype into {CFA-CC}. \\
\hline September 2017 & January 2018 & Thesis writing \& defense. \\
\hline
\end{tabular}
\end{center}

\section{Conclusion}

\newpage

\bibliographystyle{plain}
\bibliography{cfa}


\addcontentsline{toc}{section}{\indexname} % add index name to table of contents
\begin{theindex}
Italic page numbers give the location of the main entry for the referenced term.
Plain page numbers denote uses of the indexed term.
Entries for grammar non-terminals are italicized.
A typewriter font is used for grammar terminals and program identifiers.
\indexspace
\input{comp_II.ind}
\end{theindex}

\end{document}