Changeset ef42e764 for doc/aaron_comp_II

Timestamp:

Aug 10, 2016, 2:36:17 PM (9 years ago)

Author:

Thierry Delisle <tdelisle@…>

Branches:

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

Children:

5ae36ed

Parents:

e109716 (diff), 3078643 (diff)
Note: this is a merge changeset, the changes displayed below correspond to the merge itself.
Use the (diff) links above to see all the changes relative to each parent.

Message:

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

File:

• doc/aaron_comp_II/comp_II.tex (modified) (10 diffs)

doc/aaron_comp_II/comp_II.tex

@@ -62 +62 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+\newcommand{\bigO}[1]{O\left( #1 \right)}
+
 \begin{document}
 \pagestyle{headings}
@@ -116 +118 @@
 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 and destructor.
+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 and destructor. 
+Here, the runtime cost of polymorphism is spread over each polymorphic call, due to passing more arguments to polymorphic functions; preliminary experiments have shown this overhead to be similar to \CC virtual function calls. 
+Determining if packaging all polymorphic arguments to a function into a virtual function table would reduce the runtime overhead of polymorphic calls is an open research question. 
 
 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:
@@ -129 +133 @@
 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©.
+These type assertions may be either variable or function declarations that 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 is 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 defined as below, using the \CFA syntax for operator overloading:
-\begin{lstlisting}
-forall(otype S | { S ?+?(S, S); })
+For instance, ©twice© could have been defined using the \CFA syntax for operator overloading as:
+\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©. 
+This version of ©twice© works 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 also have had a type parameter named ©T©; \CFA specifies renaming of the type parameters, which would avoid the name conflict with the type variable ©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 CFA compiler 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 which cannot be resolved by CFA. 
-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 more precicely determine when further type assertion satisfaction recursion will not produce a well-typed expression.
+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 is examined as a candidate for its own type assertion unboundedly repeatedly. 
+To avoid infinite loops, the current CFA compiler 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 CFA. 
+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 more precicely determine when further type assertion satisfaction recursion does not produce a well-typed expression.
 
 \subsubsection{Traits}
 \CFA provides \emph{traits} as a means to name a group of type assertions, as in the example below:
 \begin{lstlisting}
-trait has_magnitude(otype T) {
+®trait has_magnitude(otype T)® {
     bool ?<?(T, T);        // comparison operator for T
     T -?(T);               // negation operator for T
@@ -168 +172 @@
 \end{lstlisting}
 
-Semantically, a trait is merely a named list of type assertions, but they can be used in many of the same situations where an interface in Java or an abstract base class in \CC would be used.
+Semantically, traits are simply a named lists 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 of structural inheritance, similar to interface implementation in Go, as opposed to the nominal inheritance model of Java and \CC. 
-% TODO talk about modelling of nominal inheritance with structural inheritance, possibility of investigating some resolver algorithms that require nominal
+Nominal inheritance can be simulated with traits using marker variables or functions:
+\begin{lstlisting}
+trait nominal(otype T) {
+    ®T is_nominal;®
+};
+
+int is_nominal;  // int now satisfies the nominal trait
+{
+    char is_nominal; // char satisfies the nominal trait
+}
+// char no longer satisfies the nominal trait here  
+\end{lstlisting}
+
+Traits, however, are significantly more powerful than nominal-inheritance interfaces; firstly, due to the scoping rules of the declarations which satisfy a trait's type assertions, a type may not satisfy a trait everywhere that the type is declared, as with ©char© and the ©nominal© trait above. 
+Secondly, traits may be used to declare a relationship between multiple types, a property which may be difficult or impossible to represent in nominal-inheritance type systems:
+\begin{lstlisting}
+trait pointer_like(®otype Ptr, otype El®) {
+    lvalue El *?(Ptr); // Ptr can be dereferenced into a modifiable value of type El
+}
+
+struct list {
+    int value;
+    list *next;  // may omit "struct" on type names
+};
+
+typedef list* list_iterator;
+
+lvalue int *?( list_iterator it ) {
+    return it->value;
+}
+\end{lstlisting}
+
+In the example above, ©(list_iterator, int)© satisfies ©pointer_like© by the given function, and ©(list_iterator, list)© also satisfies ©pointer_like© by the built-in pointer dereference operator. 
+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.
+
+The flexibility of \CFA's implicit trait satisfaction mechanism provides user programmers with a great deal of power, but also blocks some optimization approaches for expression resolution. 
+The ability of types to begin to or cease to satisfy traits when declarations go into or out of scope makes caching of trait satisfaction judgements difficult, and the ability of traits to take multiple type parameters could lead to a combinatorial explosion of work in any attempt to pre-compute trait satisfaction relationships. 
+On the other hand, the addition of a nominal inheritance mechanism to \CFA's type system or replacement of \CFA's trait satisfaction system with a more object-oriented inheritance model and investigation of possible expression resolution optimizations for such a system may be an interesting avenue of further research.
 
 \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.
+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 ©struct©, ©union©, and ©enum© tags, one holding labels, one holding typedef names, variable, function, and enumerator identifiers, and one for each ©struct© or ©union© type holding the field names.}, and variable or function 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 variable expression or function application 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 variable and function names for different types common in the C standard library, as in the following example:
+\begin{lstlisting}
+#include <limits.h>
+
+int max(int a, int b) { return a < b ? b : a; }  // (1)
+double max(double a, double b) { return a < b ? b : a; }  // (2)
+
+int max = INT_MAX;     // (3)
+double max = DBL_MAX;  // (4)
+
+max(7, -max);   // uses (1) and (3), by matching int type of 7 
+max(max, 3.14); // uses (2) and (4), by matching double type of 3.14
+
+max(max, -max);  // ERROR: ambiguous
+int m = max(max, -max); // uses (1) and (3) twice, by return type
+\end{lstlisting}
+
+The presence of name overloading in \CFA means that simple table lookup is insufficient to match identifiers to declarations, and a type matching algorithm must be part of expression resolution.
 
 \subsection{Implicit Conversions}
@@ -194 +239 @@
 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.
+Note that which subexpression interpretation is minimal-cost may require contextual information to disambiguate. 
+For instance, in the example in the previous subsection, ©max(max, -max)© cannot be unambiguously resolved, but ©int m = max(max, -max);© has a single minimal-cost resolution. 
+©int m = (int)max((double)max, -(double)max)© is also be a valid interpretation, but is not minimal-cost due to the unsafe cast from the ©double© result of ©max© to ©int© (the two ©double© casts function as type ascriptions selecting ©double max© rather than casts from ©int max© to ©double©, and as such are zero-cost).
 
 \subsubsection{User-generated Implicit Conversions}
@@ -201 +249 @@
 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. 
 
-Ditchfield\cite{Ditchfield:conversions} has laid out a framework for using polymorphic conversion constructor functions to create a directed acyclic graph (DAG) of conversions. 
+Ditchfield~\cite{Ditchfield:conversions} 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.
+\begin{figure}[h]
+\centering
+\includegraphics{conversion_dag}
+\caption{A portion of the implicit conversion DAG for built-in types.}
+\end{figure}
+As can be seen in the example DAG above, there are either safe or unsafe paths between each of the arithmetic types listed; the ``final'' arcs are important both to avoid creating cycles in the signed-unsigned conversions, and to disambiguate potential diamond conversions (\eg, if the ©int© to ©unsigned int© conversion was not marked final there would be two length-two paths from ©int© to ©unsigned long©, and it would be impossible to choose which one; however, since the ©unsigned int© to ©unsigned long© arc can not be traversed after the final ©int© to ©unsigned int© arc, there is a single unambiguous conversion path from ©int© to ©unsigned long©).
+
+Open research questions on this topic include:
+\begin{itemize}
+\item Can a conversion graph 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?
+\item Can such a graph representation can be usefully augmented to include user-defined types as well as built-in types?
+\item Can the graph can be efficiently represented and used in the expression resolver?
+\end{itemize}
 
 \subsection{Constructors and 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. 
+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. 
+The following example shows the implicitly-generated code in green:
+\begin{lstlisting}
+struct kv {
+    int key;
+    char *value;
+};
+
+¢void ?{}(kv *this) {
+    ?{}(&this->key);
+    ?{}(&this->value);
+}
+void ?{}(kv *this, kv that) {
+    ?{}(&this->key, that.key);
+    ?{}(&this->value, that.value);
+}
+kv ?=?(kv *this, kv that) {
+    ?=?(&this->key, that.key);
+    ?=?(&this->value, that.value);
+    return *this;
+}
+void ^?{}(kv *this) {
+    ^?{}(&this->key);
+    ^?{}(&this->value);
+}¢
+
+forall(otype T ¢| { void ?{}(T*); void ?{}(T*, T); T ?=?(T*, T); void ^?{}(T*); }¢)
+void foo(T);
+\end{lstlisting}
 
 \subsection{Generic Types}
 I have already 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:
+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 {
@@ -229 +317 @@
 The default-generated constructors, destructor and 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 and 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:
+Aside from giving users the ability to create more parameterized types than just the built-in pointer, array and function types, the combination of generic types with polymorphic functions and implicit conversions makes the edge case where the resolver may enter an infinite loop much more common, as in the following code example: 
+\begin{lstlisting}
+forall(otype T) struct box { T x; };
+
+void f(void*); // (1)
+
+forall(otype S)
+void f(box(S)* b) { // (2)
+        f(®(void*)0®);
+}
+\end{lstlisting}
+
+The loop in the resolver happens 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.
+\item Since there is an implicit conversion from ©void*© to any pointer type, the highlighted expression can be interpreted as either a ©void*©, matching ©f // (1)©, or a ©box(S)*© for some type ©S©, matching ©f // (2)©.
+\item To determine the cost of the ©box(S)© interpretation, a type must be found for ©S© which satisfies the ©otype© implicit type assertions (assignment operator, default and copy constructors, and destructor); one option is ©box(S2)© for some type ©S2©.
+\item The assignment operator, default and copy constructors, and destructor of ©box(T)© are also polymorphic functions, each of which require the type parameter ©T© to have an assignment operator, default and copy constructors, and destructor. When choosing an interpretation for ©S2©, one option is ©box(S3)©, for some type ©S3©.
+\item The previous step repeats until stopped, with four times as much work performed at each step.
 \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.
+This problem can occur in any resolution context where a polymorphic function that can satisfy its own type assertions is required for a possible interpretation of an expression with no constraints on its type, and is thus not limited to combinations of generic types with ©void*© conversions, though constructors for generic types often satisfy their own assertions and a polymorphic conversion such as the ©void*© conversion to a polymorphic variable can create an expression with no constraints on its type. 
+As discussed above, the \CFA expression resolver must handle this possible infinite recursion somehow, and it occurs fairly naturally in code like the above that uses generic types. 
 
 \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. 
+\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, and a function may return one. 
 Particularly relevantly for resolution, a tuple may be implicitly \emph{destructured} into a list of values, as in the call to ©swap© below:
 \begin{lstlisting}
@@ -248 +350 @@
 
 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
+// cannot 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.
@@ -256 +358 @@
 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 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, as below:
+\begin{lstlisting} 
+const int magic = 42;
+
+void inc_print( int& x ) { printf("%d\n", ++x); }
+
+print_inc( magic ); // legal; implicitly generated code in green below:
+
+¢int tmp = magic;¢ // copies to safely strip const-qualifier
+¢print_inc( tmp );¢ // tmp is incremented, magic is unchanged
+\end{lstlisting}
 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{Special Literal Types}
+Another proposal currently under consideration for the \CFA type-system is assigning special types to the literal values ©0© and ©1©. 
+Implicit conversions from these types 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 approach 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 large number of valid interpretations.
 
 \subsection{Deleted Function Declarations}
@@ -271 +383 @@
 int somefn(char) = delete;
 \end{lstlisting}
+This feature is typically used in \CCeleven to make a type non-copyable by deleting its copy constructor and assignment operator, or forbidding some interpretations of a polymorphic function by specifically deleting the forbidden overloads. 
 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}
-The expression resolution problem is essentially to determine an optimal matching between some combination of argument interpretations and the parameter list of some overloaded instance of a function; the argument interpretations are produced by recursive invocations of expression resolution, where the base case is zero-argument functions (which are, for purposes of this discussion, semantically equivalent to named variables or constant literal expressions). 
-Assuming that the matching between a function's parameter list and a combination of argument interpretations can be done in $O(p^k)$ time, where $p$ is the number of parameters and $k$ is some positive number, if there are $O(i)$ valid interpretations for each subexpression, there will be $O(i)$ candidate functions and $O(i^p)$ possible argument combinations for each expression, so a single recursive call to expression resolution will take $O(i^{p+1} \cdot p^k)$ time if it compares all combinations. 
-Given this bound, resolution of a single top-level expression tree of depth $d$ takes $O(i^{p+1} \cdot p^{k \cdot d})$ time\footnote{The call tree will have leaves at depth $O(d)$, and each internal node will have $O(p)$ fan-out, producing $O(p^d)$ total recursive calls.}.
-Expression resolution is somewhat unavoidably exponential in $p$, the number of function parameters, and $d$, the depth of the expression tree, but these values are fixed by the user programmer, and generally bounded by reasonably small constants. 
-$k$, on the other hand, is mostly dependent on the representation of types in the system and the efficiency of type assertion checking; if a candidate argument combination can be compared to a function parameter list in linear time in the length of the list (\ie $k = 1$), then the $p^{k \cdot d}$ term is linear in the input size of the source code for the expression, otherwise the resolution algorithm will exibit sub-linear performance scaling on code containing more-deeply nested expressions.
+The expression resolution problem is determining an optimal match between some combination of argument interpretations and the parameter list of some overloaded instance of a function; the argument interpretations are produced by recursive invocations of expression resolution, where the base case is zero-argument functions (which are, for purposes of this discussion, semantically equivalent to named variables or constant literal expressions). 
+Assuming that the matching between a function's parameter list and a combination of argument interpretations can be done in $\bigO{p^k}$ time, where $p$ is the number of parameters and $k$ is some positive number, if there are $\bigO{i}$ valid interpretations for each subexpression, there will be $\bigO{i}$ candidate functions and $\bigO{i^p}$ possible argument combinations for each expression, so for a single recursive call expression resolution takes $\bigO{i^{p+1} \cdot p^k}$ time if it must compare all combinations, or $\bigO{i(p+1) \cdot p^k}$ time if argument-parameter matches can be chosen independently of each other. 
+Given these bounds, resolution of a single top-level expression tree of depth $d$ takes $\bigO{i^{p+1} \cdot p^{k \cdot d}}$ time under full-combination matching, or $\bigO{i(p+1) \cdot p^{k \cdot d}}$ time for independent-parameter matching\footnote{A call tree has leaves at depth $\bigO{d}$, and each internal node has $\bigO{p}$ fan-out, producing $\bigO{p^d}$ total recursive calls.}.
+
+Expression resolution is somewhat unavoidably exponential in $d$, the depth of the expression tree, and if arguments cannot be matched to parameters independently of each other, expression resolution is also exponential in $p$. 
+However, both $d$ and $p$ are fixed by the user programmer, and generally bounded by reasonably small constants. 
+$k$, on the other hand, is mostly dependent on the representation of types in the system and the efficiency of type assertion checking; if a candidate argument combination can be compared to a function parameter list in linear time in the length of the list (\ie $k = 1$), then the $p^{k \cdot d}$ factor is linear in the input size of the source code for the expression, otherwise the resolution algorithm exibits sub-linear performance scaling on code containing more-deeply nested expressions.
 The number of valid interpretations of any subexpression, $i$, is bounded by the number of types in the system, which is possibly infinite, though practical resolution algorithms for \CFA must be able to place some finite bound on $i$, possibly at the expense of type-system completeness. 
 
-The research goal of this project is to develop a performant expression resolver for \CFA; this analysis suggests two primary areas of investigation to accomplish that end. 
-The first is efficient argument-parameter matching; Bilson\cite{Bilson03} mentions significant optimization opportunities available in the current literature to improve on the existing CFA compiler.
+The research goal of this project is to develop a performant expression resolver for \CFA; this analysis suggests three primary areas of investigation to accomplish that end. 
+The first area of investigation is efficient argument-parameter matching; Bilson~\cite{Bilson03} mentions significant optimization opportunities available in the current literature to improve on the existing CFA compiler.
 %TODO: look up and lit review 
-The second, and likely more fruitful, area of investigation is heuristics and algorithmic approaches to reduce the number of argument interpretations considered in the common case; given the large ($p+1$) exponent on number of interpretations considered in the runtime analysis, even small reductions here could have a significant effect on overall resolver runtime. 
+The second area of investigation is minimizing dependencies between argument-parameter matches; the current CFA compiler attempts to match entire argument combinations against functions at once, potentially attempting to match the same argument against the same parameter multiple times. 
+Whether the feature set of \CFA admits an expression resolution algorithm where arguments can be matched to parameters independently of other arguments in the same function application is an area of open research; polymorphic type paramters produce enough of a cross-argument dependency that the problem is not trivial. 
+If cross-argument resolution dependencies cannot be completely eliminated, effective caching strategies to reduce duplicated work between equivalent argument-parameter matches in different combinations may mitigate the asymptotic defecits of the whole-combination matching approach. 
+The final area of investigation is heuristics and algorithmic approaches to reduce the number of argument interpretations considered in the common case; if argument-parameter matches cannot be made independent, even small reductions in $i$ should yield significant reductions in the $i^{p+1}$ resolver runtime factor. 
 The discussion below presents a number of largely orthagonal axes for expression resolution algorithm design to be investigated, noting prior work where applicable.