Index: doc/theses/fangren_yu_COOP_S20/Report.tex
===================================================================
--- doc/theses/fangren_yu_COOP_S20/Report.tex	(revision 6cc913e03dd85dca12198d3ca298c52197d4c03b)
+++ doc/theses/fangren_yu_COOP_S20/Report.tex	(revision 2ff78aaef345296b17e6dbe4ee5576d18fd0256f)
@@ -112,5 +112,5 @@
 \begin{itemize}
 \item
-type declaration: @struct@, @union@, @typedef@ or type parameter \TODO{(see Appendix A.3)}
+type declaration: @struct@, @union@, @typedef@ or type parameter (see Appendix A.1)
 \item
 variable declaration
@@ -374,8 +374,4 @@
 
 \subsubsection{Source: \lstinline{AST/SymbolTable.hpp}}
-
-\TODO{Add something here}
-
-
 \subsubsection{Source: \lstinline{SymTab/Indexer.h}}
 
@@ -530,5 +526,18 @@
 Each pair of compatible branch expression types produce a possible interpretation, and the cost is defined as the sum of the expression costs plus the sum of conversion costs to the common type.
 
-\TODO{Write a specification for expression costs.}
+\subsection{Conversion and Application Cost}
+There were some unclear parts in the previous documentation of cost system, as described in the Moss thesis \cite{Moss19}, section 4.1.2. Some clarification are presented in this section.
+
+\begin{enumerate}
+\item
+Conversion to a type denoted by parameter may incur additional cost if the match is not exact. For example, if a function is declared to accept @(T, T)@ and receives @(int, long)@, @T@ is deducted @long@ and an additional widening conversion cost is added for @int@ to @T@.
+
+\item
+The specialization level of a function is the sum of the least depth of an appearance of type parameter (counting pointers, references and parameterized types), plus the number of assertions. A higher specialization level is favored if conversion cost of arguments are equal.
+
+\item
+Coercion of pointer types is only allowed in explicit cast expressions; the only allowed implicit pointer casts are adding qualifiers to the base type and cast to @void*@, and those counts as safe conversions. Note that implicit cast from @void*@ to other pointer types is no longer valid, as opposed to standard C. 
+
+\end{enumerate}
 
 
@@ -547,6 +556,5 @@
 At the call site, implicit parameters are automatically inserted by the compiler.
 
-\TODO{Explain how recursive assertion satisfaction and polymorphic recursion work.}
-
+Implementation of implicit parameters is discussed in Appendix A.3.
 
 \section{Tests}
@@ -589,4 +597,108 @@
 It is suggested to run performance tests with optimization (@g++@ flag @-O3@).
 
+\section{Appendix}
+
+\subsection{Kinds of Type Parameters}
+The type parameters in a @forall@ clause has three different kinds:
+\begin{enumerate}
+\item
+@dtype@: any data type (built-in or user defined). There is also a difference between opaque types (incomplete types, those with only a forward declaration) and concrete types. Only concrete types can be directly used as a variable type. \CFA provides the @otype@ shorthand to require a type parameter as concrete, which also implicitly asserts the existence of its constructor and destructor\footnote{\CFA implements the same automatic resource management (RAII) semantics as \CC.}.
+\item
+@ftype@: any function type. Since @ftype@ does not provide any information about parameter types of a function, it is rarely used. The main purpose of introducing @ftype@ is to disallow a function to match a pointer overload, since variables and functions can have the same names.
+\item
+@ttype@: tuple (variadic) type. @ttype@ parameter may only appear as type of the last parameter in a function, and it provides a type-safe way to implement variadic functions. Note however, that it has certain restrictions, as described in the implementation section below.
+
+\end{enumerate}
+
+\subsection{GNU C Nested Functions}
+
+\CFA is designed to be mostly compatible with GNU C, an extension to ISO C99 and C11 standards. The \CFA compiler also implements some language features by GCC extensions, most notably nested functions.
+
+In ISO C, function definitions are not allowed to be nested. GCC allows nested functions with full lexical scoping. The following example is taken from GCC documentation\footnote{\url{https://gcc.gnu.org/onlinedocs/gcc/Nested-Functions.html}}:
+
+\begin{C++}
+bar (int *array, int offset, int size)
+{
+  int access (int *array, int index)
+    { return array[index + offset]; }
+  int i;
+  /* ... */
+  for (i = 0; i < size; i++)
+    /* ... */ access (array, i) /* ... */
+}
+\end{C++}
+
+GCC nested functions behave identically to \CC lambda functions with default by-reference capture (stack-allocated, lifetime ends upon exiting the block declared in), while also possible to be passed as arguments with standard function pointer types.
+
+\subsection{Implementation of Parametric Functions}
+\CFA implements parametric functions using the implicit parameter approach: required assertions are passed to the callee by function pointers; size of a parametric type must also be known if referenced directly (i.e. not as a pointer). 
+
+The implementation is similar to the one from Scala\footnote{\url{https://www.scala-lang.org/files/archive/spec/2.13/07-implicits.html}}, with some notable differences in resolution:
+\begin{enumerate}
+\item
+All types are function declarations are candidates of implicit parameters.
+\item
+The parameter (assertion) name must match the actual declarations.
+\item
+Currently, assertions are all functions. Note that since \CFA has variable overloading, implicit value parameters might also be supported in the future.
+\end{enumerate}
+
+For example, the \CFA function declaration
+
+\begin{cfa}
+forall(otype T | {int foo(T, int);})
+int bar(T);
+\end{cfa}
+
+after implicit parameter expansion, has the actual signature\footnote{\textbf{otype} also requires the type to have constructor and destructor, which are the first two function pointers preceding the one for \textbf{foo}.}
+
+\begin{C++}
+int bar(T, size_t, void (*)(T&), void (*)(T&), int (*)(T, int));
+\end{C++}
+
+The implicit parameter approach has an apparent issue: when the satisfying declaration is also parametric, it may require its own implicit parameters too. That also causes the supplied implicit parameter to have a different \textbf{actual} type than the \textbf{nominal} type, so it cannot be passed directly. Therefore, a wrapper with matching actual type must be created, and here it is where GCC nested function is used internally by the compiler.
+
+Consider the following program:
+\begin{cfa}
+int assertion(int);
+
+forall (otype T | {int assertion(T);})
+void foo(T);
+
+forall (otype T | {void foo(T);})
+void bar(T t) {
+	foo(t);
+}
+\end{cfa}
+
+\CFA compiler translates the program to non-parametric form\footnote{In the final code output, T needs to be replaced by an opaque type, and arguments must be accessed by a frame pointer offset table, due to the unknown sizes. The presented code here is simplified for better understanding.}
+
+\begin{C++}
+// ctor, dtor and size arguments are omitted
+void foo(T, int (*)(T));
+
+void bar(T t, void (*foo)(T)) {
+	foo(t);
+}
+\end{C++}
+
+However, when @bar(1)@ is called, @foo@ cannot be directly provided as an argument:
+
+\begin{C++}
+bar(1, foo); // WRONG: foo has different actual type
+\end{C++}
+
+and an additional step is required:
+
+\begin{C++}
+{
+	void _foo_wrapper(int t) {
+		foo(t, assertion);
+	}
+	bar(1, _foo_wrapper);
+}
+\end{C++}
+
+Nested assertions and implicit parameter creation may continue indefinitely. This is a limitation of implicit parameter implementation. In particular, polymorphic variadic recursion must be structural (i.e. number of arguments decreases in any possible recursive calls), otherwise code generation gets into an infinite loop. \CFA compiler sets a limit on assertion depth and reports an error if assertion resolution does not terminate within the limit.
 
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