# Changeset 4b1c8da for doc

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Jan 19, 2021, 9:01:55 AM (7 months ago)
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 rfcd0b9d7 \renewcommand{\subsectionmark}[1]{\markboth{\thesubsection\quad #1}{\thesubsection\quad #1}} \pagenumbering{roman} \linenumbers                                            % comment out to turn off line numbering %\linenumbers                                            % comment out to turn off line numbering \maketitle \pdfbookmark[1]{Contents}{section} \tableofcontents \clearpage \thispagestyle{plain} \pagenumbering{arabic} \begin{abstract} \CFA is an evolutionary extension to the C programming language, featuring a parametric type system, and is currently under active development. The reference compiler for \CFA language, @cfa-cc@, has some of its major components dated back to early 2000s, and is based on inefficient data structures and algorithms. Some improvements targeting the expression resolution algorithm, suggested by a recent prototype experiment on a simplified model, are implemented in @cfa-cc@ to support the full \CFA language. These optimizations speed up the compiler significantly by a factor of 20 across the existing \CFA codebase, bringing the compilation time of a mid-sized \CFA source file down to 10-second level. A few cases derived from realistic code examples that causes trouble to the compiler are analyzed in detail, with proposed solutions. This step of \CFA project development is critical to its eventual goal to be used alongside C for large software systems. \CFA is an evolutionary, non-object-oriented extension of the C programming language, featuring a parametric type-system, and is currently under active development. The reference compiler for the \CFA language, @cfa-cc@, has some of its major components dated back to the early 2000s, which are based on inefficient data structures and algorithms. This report introduces improvements targeting the expression resolution algorithm, suggested by a recent prototype experiment on a simplified model, which are implemented in @cfa-cc@ to support the full \CFA language. These optimizations speed up the compiler by a factor of 20 across the existing \CFA codebase, bringing the compilation time of a mid-sized \CFA source file down to the 10-second level. A few problem cases derived from realistic code examples are analyzed in detail, with proposed solutions. This work is a critical step in the \CFA project development to achieve its eventual goal of being used alongside C for large software systems. \end{abstract} \section*{Acknowledgements} Thanks Mum. \clearpage \tableofcontents \section{Introduction} \CFA language, developed by the Programming Language Group at University of Waterloo, has a long history, with the first proof-of-concept compiler built in 2003 by Richard Bilson~\cite{Bilson03}. Many new features are added to the language over time, but the core of \CFA, parametric functions introduced by the @forall@ clause (hence the name of the language), with the type system supporting parametric overloading, remains mostly unchanged. The current \CFA reference compiler @cfa-cc@ still includes many parts taken directly from the original Bilson's implementation, and serves as a starting point for the enhancement work to the type system. Unfortunately, it does not provide the efficiency required for the language to be used practically: a \CFA source file of approximately 1000 lines of code can take a few minutes to compile. The cause of the problem is that the old compiler used inefficient data structures and algorithms for expression resolution, which involved a lot of copying and redundant work. This paper presents a series of optimizations to the performance-critical parts of the resolver, with a major rework of the data structure used by the compiler, using a functional programming approach to reduce memory complexity. Subsequent improvements are mostly suggested by running the compiler builds with a performance profiler against the \CFA standard library source code and a test suite to find the most underperforming components in the compiler algorithm. The \CFA team endorses a pragmatic philosophy in work that mostly focuses on practical implications of language design and implementation, rather than the theoretical limits. In particular, the compiler is designed to work on production \CFA code efficiently and keep type safety, while sometimes making compromises to expressiveness in extreme corner cases. However, when these corner cases do appear in actual usage, they need to be thoroughly investigated. Analysis presented in this paper, therefore, are conducted on a case-by-case basis. Some of them eventually point to certain weaknesses in the language design and solutions are proposed based on experimental results. \section{Completed work} \CFA language, developed by the Programming Language Group at the University of Waterloo, has a long history, with the initial language design in 1992 by Glen Ditchfield~\cite{Ditchfield92} and the first proof-of-concept compiler built in 2003 by Richard Bilson~\cite{Bilson03}. Many new features have been added to the language over time, but the core of \CFA's type-system --- parametric functions introduced by the @forall@ clause (hence the name of the language) providing parametric overloading --- remains mostly unchanged. The current \CFA reference compiler, @cfa-cc@, is designed using the visitor pattern~\cite{vistorpattern} over an abstract syntax tree (AST), where multiple passes over the AST modify it for subsequent passes. @cfa-cc@ still includes many parts taken directly from the original Bilson implementation, which served as the starting point for this enhancement work to the type system. Unfortunately, the prior implementation did not provide the efficiency required for the language to be practical: a \CFA source file of approximately 1000 lines of code can take a multiple minutes to compile. The cause of the problem is that the old compiler used inefficient data structures and algorithms for expression resolution, which involved significant copying and redundant work. This report presents a series of optimizations to the performance-critical parts of the resolver, with a major rework of the compiler data-structures using a functional-programming approach to reduce memory complexity. The improvements were suggested by running the compiler builds with a performance profiler against the \CFA standard-library source-code and a test suite to find the most underperforming components in the compiler algorithm. The \CFA team endorses a pragmatic philosophy that focuses on practical implications of language design and implementation rather than theoretical limits. In particular, the compiler is designed to be expressive with respect to code reuse while maintaining type safety, but compromise theoretical soundness in extreme corner cases. However, when these corner cases do appear in actual usage, they need to be thoroughly investigated. A case-by-case analysis is presented for several of these corner cases, some of which point to certain weaknesses in the language design with solutions proposed based on experimental results. \section{AST restructuring} \subsection{Memory model with sharing} A major rework of the abstract syntax tree (AST) data structure in the compiler is completed as the first step of the project. The majority of work were documented in the reference manual of the compiler~\cite{cfa-cc}. To summarize: \begin{itemize} \item AST nodes (and therefore subtrees) can be shared without copying when reused. \item Modifications apply the functional programming principle, making copies for local changes without affecting the original data shared by other owners. In-place mutations are permitted as a special case when sharing does not happen. The logic is implemented by reference counting. \item Memory allocation and freeing are performed automatically using smart pointers. \end{itemize} The resolver algorithm designed for overload resolution naturally introduces a significant amount of reused intermediate representations, especially in the following two places: \begin{itemize} \item Function overload candidates are computed by combining the argument candidates bottom-up, with many of them being a common term. For example, if $n$ overloads of a function @f@ all take an integer for the first parameter but different types for the second (@f( int, int )@, @f( int, double )@, etc.) the first term is reused $n$ times for each of the generated candidate expressions. This effect is particularly bad for deep expression trees. \item In the unification algorithm and candidate elimination step, actual types are obtained by substituting the type parameters by their bindings. Let $n$ be the complexity (\ie number of nodes in representation) of the original type, $m$ be the complexity of bound type for parameters, and $k$ be the number of occurrences of type parameters in the original type. If everything needs to be deep-copied, the substitution step takes $O(n+mk)$ time and memory, while using shared nodes it is reduced to $O(n)$ time and $O(k)$ memory. \end{itemize} One of the worst examples for the old compiler is a long chain of I/O operations \begin{cfa} sout | 1 | 2 | 3 | 4 | ... \end{cfa} The pipe operator is overloaded by \CFA I/O library for every primitive type in C language, as well as I/O manipulators defined by the library. In total there are around 50 overloads for the output stream operation. On resolving the $n$-th pipe operator in the sequence, the first term, which is the result of sub-expression containing $n-1$ pipe operators, is reused to resolve every overload. Therefore at least $O(n^2)$ copies of expression nodes are made during resolution, not even counting type unification cost; combined with two large factors from number of overloads of pipe operators, and that the output stream type'' in \CFA is a trait with 27 assertions (which adds to complexity of the pipe operator's type) this makes compiling a long output sequence extremely slow. In new AST representation only $O(n)$ copies are required and type of pipe operator is not copied at all. Reduction in space complexity is especially important, as preliminary profiling result on the old compiler build shows that over half of time spent in expression resolution are on memory allocations. A major rework of the AST data-structure in the compiler was completed as the first step of the project. The majority of this work is documented in my prior report documenting the compiler reference-manual~\cite{cfa-cc}. To summarize: \begin{itemize} \item AST nodes (and therefore subtrees) can be shared without copying. \item Modifications are performed using functional-programming principles, making copies for local changes without affecting the original data shared by other owners. In-place mutations are permitted as a special case when there is no sharing. The logic is implemented by reference counting. \item Memory allocation and freeing are performed automatically using smart pointers~\cite{smartpointers}. \end{itemize} The resolver algorithm, designed for overload resolution, uses a significant amount of reused, and hence copying, for the intermediate representations, especially in the following two places: \begin{itemize} \item Function overload candidates are computed by combining the argument candidates bottom-up, with many being a common term. For example, if $n$ overloads of a function @f@ all take an integer for the first parameter but different types for the second, \eg @f( int, int )@, @f( int, double )@, etc., the first term is copied $n$ times for each of the generated candidate expressions. This copying is particularly bad for deep expression trees. \item In the unification algorithm and candidate elimination step, actual types are obtained by substituting the type parameters by their bindings. Let $n$ be the complexity (\ie number of nodes in representation) of the original type, $m$ be the complexity of the bound type for parameters, and $k$ be the number of occurrences of type parameters in the original type. If every substitution needs to be deep-copied, these copy step takes $O(n+mk)$ time and memory, while using shared nodes it is reduced to $O(n)$ time and $O(k)$ memory. \end{itemize} One of the worst examples for the old compiler is a long chain of I/O operations: \begin{cfa} sout | 1 | 2 | 3 | 4 | ...;   // print integer constants \end{cfa} The pipe operator is overloaded by the \CFA I/O library for every primitive type in the C language, as well as I/O manipulators defined by the library. In total, there are around 50 overloads for the output stream operation. On resolving the $n$-th pipe operator in the sequence, the first term, which is the result of sub-expression containing $n-1$ pipe operators, is reused to resolve every overload. Therefore at least $O(n^2)$ copies of expression nodes are made during resolution, not even counting type unification cost; combined with the two large factors from number of overloads of pipe operators, and that the output stream type'' in \CFA is a trait with 27 assertions (which adds to complexity of the pipe operator's type) this makes compiling a long output sequence extremely slow. In the new AST representation, only $O(n)$ copies are required and the type of the pipe operator is not copied at all. Reduction in space complexity is especially important, as preliminary profiling results on the old compiler build showed over half of the time spent in expression resolution is on memory allocations. Since the compiler codebase is large and the new memory model mostly benefits expression resolution, some of the old data structures are still kept, and a conversion pass happens before and after the general resolve phase. Rewriting every compiler module will take longer, and whether the new model is correct was unknown when this project started, therefore only the resolver is currently implemented with the new data structure. \subsection{Merged resolver calls} The pre-resolve phase of compilation, inadequately called validate'' in the compiler source code, does more than just simple syntax validation, as it also normalizes input program. Some of them, however, requires type information on expressions and therefore needs to call the resolver before the general resolve phase. There are three notable places where the resolver is invoked: \begin{itemize} \item Attempt to generate default constructor, copy constructor and destructor for user-defined @struct@ types \item Resolve @with@ statements (the same as in Python, which introduces fields of a structure directly in scope) The pre-resolve phase of compilation, inappropriately called validate'' in the compiler source code, has a number of passes that do more than simple syntax and semantic validation; some passes also normalizes the input program. A few of these passes require type information for expressions, and therefore, need to call the resolver before the general resolve phase. There are three notable places where the resolver is invoked: \begin{itemize} \item Generate default constructor, copy constructor and destructor for user-defined @struct@ types. \item Resolve @with@ statements (the same as in Pascal~\cite{pascal}), which introduces fields of a structure directly into a scope. \item Resolve @typeof@ expressions (cf. @decltype@ in \CC); note that this step may depend on symbols introduced by @with@ statements. \end{itemize} Since the compiler codebase is large and the new memory model mostly only benefits expression resolution, the old data structure is still kept, and a conversion pass happens before and after resolve phase. Rewriting every compiler module will take a long time, and whether the new model is correct is still unknown when started, therefore only the resolver is implemented with the new data structure. Since the constructor calls were one of the most expensive to resolve (reason will be shown in the next section), pre-resolve phase were taking more time after resolver moves to the more efficient new implementation. To better facilitate the new resolver, every step that requires type information are reintegrated as part of resolver. A by-product of this work is that the reversed dependence of @with@ statement and @typeof@ can now be handled. Previously, the compiler is unable to handle cases such as Since the constructor calls are one of the most expensive to resolve (reason given in~\VRef{s:SpecialFunctionLookup}), this pre-resolve phase was taking a large amount of time even after the resolver was changed to the more efficient new implementation. The problem is that multiple resolutions repeat a significant amount of work. Therefore, to better facilitate the new resolver, every step that requires type information should be integrated as part of the general resolver phase. A by-product of this work is that reversed dependence between @with@ statement and @typeof@ can now be handled. Previously, the compiler was unable to handle cases such as: \begin{cfa} struct S { int x; }; S foo(); typeof( foo() ) s; // type is S with (s) { with (s) { x; // refers to s.x } \end{cfa} since type of @s@ is still unresolved when handling @with@ expressions. Instead, the new (and correct) approach is to evaluate @typeof@ expressions when the declaration is first seen, and it suffices because of the declaration-before-use rule. since the type of @s@ is unresolved when handling @with@ expressions because the @with@ pass follows the @typeof@ pass (interchanging passes only interchanges the problem). Instead, the new (and correct) approach is to evaluate @typeof@ expressions when the declaration is first seen during resolution, and it suffices because of the declaration-before-use rule. \subsection{Special function lookup} Reducing the number of functions looked up for overload resolution is an effective way to gain performance when there are many overloads but most of them are trivially wrong. In practice, most functions have few (if any) overloads but there are notable exceptions. Most importantly, constructor @?{}@, destructor @^?{}@, and assignment @?=?@ are generated for every user-defined type, and in a large source file there can be hundreds of them. Furthermore, many calls to them are generated for initializing variables and passing arguments. This fact makes them the most overloaded and most called functions. In an object-oriented programming language, object has methods declared with their types, so a call such as @obj.f()@ only needs to perform lookup in the method table corresponding to type of @obj@. \CFA on the other hand, does not have methods, and all types are open (\ie new operations can be defined on them), so a similar approach will not work in general. However, the big 3'' operators have a unique property enforced by the language rules, such that the first parameter must have a reference type. Since \CFA does not have class inheritance, reference type must always match exactly. Therefore, argument-dependent lookup can be implemented for these operators, by using a dedicated symbol table. The lookup key used for the special functions is the mangled type name of the first parameter, which acts as the @this@ parameter in an object-oriented language. To handle generic types, the type parameters are stripped off, and only the base type is matched. Note that a constructor (destructor, assignment operator) taking arbitrary @this@ argument, for example @forall( dtype T ) void ?{}( T & );@ is not allowed, and it guarantees that if the @this@ type is known, all possible overloads can be found by searching with the given type. In case that the @this@ argument itself is overloaded, it is resolved first and all possible result types are used for lookup. Note that for the generated expressions, the particular variable for @this@ argument is fully known, without overloads, so the majority of constructor call resolutions only need to check for one given object type. Explicit constructor calls and assignment statements sometimes may require lookup for multiple types. In the extremely rare case that type of @this@ argument is yet unbound, everything will have to be checked, just like without the argument-dependent lookup algorithm; fortunately, this case almost never happens in practice. An example is found in the library function @new@: \label{s:SpecialFunctionLookup} Reducing the number of function looked ups for overload resolution is an effective way to gain performance when there are many overloads but most of them are trivially wrong. In practice, most functions have few (if any) overloads but there are notable exceptions. Most importantly, constructor @?{}@, destructor @^?{}@, and assignment @?=?@ are generated for every user-defined type (@struct@ and @union@ in C), and in a large source file there can be hundreds of them. Furthermore, many calls are generated for initializing variables, passing arguments and copying values. This fact makes them the most overloaded and most called functions. In an object-oriented programming language, the object-method types are scoped within a class, so a call such as @obj.f()@ only needs to perform lookup in the method table corresponding to the type of @obj@. \CFA on the other hand, does not have methods, and all types are open, \ie new operations can be defined on them without inheritance; at best a \CFA type can be constrained by a translation unit. However, the big 3'' operators have a unique property enforced by the language rules: the first parameter must be a reference to its associated type, which acts as the @this@ parameter in an object-oriented language. Since \CFA does not have class inheritance, the reference type must always match exactly. Therefore, argument-dependent lookup can be implemented for these operators by using a dedicated, fast symbol-table. The lookup key for the special functions is the mangled type name of the first parameter. To handle generic types, the type parameters are stripped off, and only the base type is matched. Note a constructor (destructor, assignment operator) may not take an arbitrary @this@ argument, \eg @forall( dtype T ) void ?{}( T & )@, thus guaranteeing that if the @this@ type is known, all possible overloads can be found by searching with this given type. In the case where the @this@ argument itself is overloaded, it is resolved first and all possible result types are used for lookup. Note that for a generated expression, the particular variable for the @this@ argument is fully known, without overloads, so the majority of constructor-call resolutions only need to check for one given object type. Explicit constructor calls and assignment statements sometimes require lookup for multiple types. In the extremely rare case that the @this@-argument type is unbound, all necessary types are guaranteed to be checked, as for the previous lookup without the argument-dependent lookup; fortunately, this complex case almost never happens in practice. An example is found in the library function @new@: \begin{cfa} forall( dtype T | sized( T ), ttype TT | { void ?{}( T &, TT ); } ) T * new( TT p ) { return &(*malloc()){ p }; } \end{cfa} as @malloc@ may return a pointer to any type, depending on context. Interestingly, this particular line of code actually caused another complicated issue, where the unusually massive work of checking every constructor in presence makes the case even worse. Section~\ref{s:TtypeResolutionInfiniteRecursion} presents a detailed analysis for the problem. The callable'' operator @?()@ (cf. @operator()@ in \CC) could also be included in the special operator list, as it is usually only on user-defined types, and the restriction that first argument must be a reference seems reasonable in this case. as @malloc@ may return a pointer to any type, depending on context. Interestingly, this particular declaration actually causes another complicated issue, making the complex checking of every constructor even worse. \VRef[Section]{s:TtypeResolutionInfiniteRecursion} presents a detailed analysis of this problem. The callable'' operator @?()@ (cf. @operator()@ in \CC) can also be included in this special operator list, as it is usually only on user-defined types, and the restriction that the first argument must be a reference seems reasonable in this case. \subsection{Improvement of function type representation} Since substituting type parameters with their bound types is one fundamental operation in many parts of resolver algorithm (particularly unification and environment binding), making as few copies of type nodes as possible helps reducing memory complexity. Even with the new memory management model, allocation is still a significant factor of resolver performance. Conceptually, operations on type nodes of AST should be performed in functional programming style, treating the data structure as immutable and only copy when necessary. The in-place mutation is a mere optimization that does not change logic of operations. The model was broken on function types by an inappropriate design. Function types require some special treatment due to the existence of assertions. In particular, it must be able to distinguish two different kinds of type parameter usage: Since substituting type parameters with their bound types is one fundamental operation in many parts of resolver algorithm (particularly unification and environment binding), making as few copies of type nodes as possible helps reducing memory complexity. Even with the new memory management model, allocation is still a significant factor of resolver performance. Conceptually, operations on type nodes of the AST should be performed in functional-programming style, treating the data structure as immutable and only copying when necessary. The in-place mutation is a mere optimization that does not change the logic for operations. However, the model was broken for function types by an inappropriate design. Function types require special treatment due to the existence of assertions that constrain the types it supports. Specifically, it must be possible to distinguish two different kinds of type parameter usage: \begin{cfa} forall( dtype T ) void foo( T * t ) { forall( dtype U ) void bar( T * t, U * u ) { ... } } \end{cfa} Here, only @U@ is a free parameter in declaration of @bar@, as it appears in the function's own forall clause; while @T@ is not free. Moreover, the resolution algorithm also has to distinguish type bindings of multiple calls to the same function, for example with forall( dtype U ) void bar( @T@ * t, @U@ * u ) { ... } } \end{cfa} Here, only @U@ is a free parameter in the nested declaration of function @bar@, as @T@ must be bound at the call site when resolving @bar@. Moreover, the resolution algorithm also has to distinguish type bindings of multiple calls to the same function, \eg: \begin{cfa} forall( dtype T ) int foo( T x ); foo( foo( 1.0 ) ); \end{cfa} The inner call has binding (T: double) while the outer call has binding (T: int). Therefore a unique representation of free parameters in each expression is required. This was previously done by creating a copy of the parameter declarations inside function type, and fixing references afterwards. However, fixing references is an inherently deep operation that does not work well with functional programming model, as it must be evaluated eagerly on the entire syntax tree representing the function type. The revised approach generates a unique ID value for each function call expression instance and represents an occurrence of free parameter type with a pair of generated ID and the original parameter declaration, so that references do not need to be fixed, and a shallow copy of function type is possible. Note that after the change, all declaration nodes in syntax tree representation maps one-to-one with the actual declarations in the program, and therefore are guaranteed to be unique. Such property can potentially enable more optimizations, and some related ideas are presented after Section~\ref{s:SharedSub-ExpressionCaseUniqueExpressions}. int i = foo( foo( 1.0 ) ); \end{cfa} The inner call has binding (T: double) while the outer call has binding (T: int). Therefore a unique representation for the free parameters is required in each expression. This type binding was previously done by creating a copy of the parameter declarations inside the function type and fixing references afterwards. However, fixing references is an inherently deep operation that does not work well with the functional-programming style, as it forces eager evaluation on the entire syntax tree representing the function type. The revised approach generates a unique ID value for each function call expression instance and represents an occurrence of a free-parameter type with a pair of generated ID and original parameter declaration, so references are unique and a shallow copy of the function type is possible. Note that after the change, all declaration nodes in the syntax-tree representation now map one-to-one with the actual declarations in the program, and therefore are guaranteed to be unique. This property can potentially enable more optimizations, and some related ideas are presented at the end of \VRef{s:SharedSub-ExpressionCaseUniqueExpressions}. \subsection{Improvement of pruning steps} A minor improvement for candidate elimination is to skip the step on the function overloads themselves and only perform on results of function application. As function calls are usually by name, the name resolution rule dictates that every function candidate necessarily has a different type; indirect function calls are rare, and when they do appear, they usually will not have many possible interpretations, and those rarely matches exactly in argument type. Since function types have a much more complex representation than data types (with multiple parameters and assertions), checking equality on them also takes longer. A brief test of this approach shows that the number of function overloads considered in expression resolution increases by a negligible amount of less than 1 percent, while type comparisons in candidate elimination are cut by more than half. Improvement is consistent over all \CFA source files in the test suite. A minor improvement for candidate elimination is to skip the step on the function overloads and only check the results of function application. As function calls are usually by name (versus pointers to functions), the name resolution rule dictates that every function candidate necessarily has a different type; indirect function calls are rare, and when they do appear, there are even fewer cases with multiple interpretations, and these rarely match exactly in argument type. Since function types have a much more complex representation (with multiple parameters and assertions) than data types, checking equality on them also takes longer. A brief test of this approach shows that the number of function overloads considered in expression resolution increases by an amount of less than 1 percent, while type comparisons in candidate elimination are reduced by more than half. This improvement is consistent over all \CFA source files in the test suite. \label{s:SharedSub-ExpressionCaseUniqueExpressions} Unique expression denotes an expression that must be evaluated only once, to prevent unwanted side effects. It is currently only a compiler artifact, generated on tuple member expression of the form Unique expression denotes an expression evaluated only once to prevent unwanted side effects. It is currently only a compiler artifact, generated for tuple-member expression of the form: \begin{cfa} struct S { int a; int b; }; s.[a, b]; // tuple member expression, type is [int, int] \end{cfa} If the aggregate expression contains function calls, it cannot be evaluated multiple times: If the aggregate expression is function call, it cannot be evaluated multiple times: \begin{cfa} S makeS(); makeS().[a, b]; // this should only make one S makeS().[a, b]; // this should only generate a unique S \end{cfa} Before code generation, the above expression is internally represented as \end{cfa} at code generation, where @_unique_var@ and @_unique_var_evaluated@ are generated variables whose scope covers all appearances of the same expression. Note that although the unique expression is only used for tuple expansion now, it is a generally useful construction, and can be seen in other languages, such as Scala's @lazy val@~\cite{Scala}; therefore it could be worthwhile to introduce the unique expression to a broader context in \CFA and even make it directly available to programmers. In the compiler's visitor pattern, however, this creates a problem where multiple paths to a logically unique expression exist, so it may be modified more than once and become ill-formed; some specific intervention is required to ensure that unique expressions are only visited once. Furthermore, a unique expression appearing in more than one places will be copied on mutation so its representation is no longer unique. Some hacks are required to keep it in sync, and the methods are different when mutating the unique expression instance itself or its underlying expression. Example when mutating the underlying expression (visit-once guard) The conditional check ensures a single call to @makeS()@ even though there are logically multiple calls because of the tuple field expansion. Note that although the unique expression is only used for tuple expansion now, it is a generally useful construction, and is seen in other programming languages, such as Scala's @lazy val@~\cite{Scala}; therefore it may be worthwhile to introduce the unique expression to a broader context in \CFA and even make it directly available to programmers. In the compiler's visitor pattern, however, this creates a problem where multiple paths to a logically unique expression exist, so it may be modified more than once and become ill-formed; some specific intervention is required to ensure unique expressions are only visited once. Furthermore, a unique expression appearing in more than one places is copied on mutation so its representation is no longer unique. Currently, special cases are required to keep everything synchronized, and the methods are different when mutating the unique expression instance itself or its underlying expression: \begin{itemize} \item When mutating the underlying expression (visit-once guard) \begin{cfa} void InsertImplicitCalls::previsit( const ast::UniqueExpr * unqExpr ) { if ( visitedIds.count( unqExpr->id ) ) visit_children = false; @if ( visitedIds.count( unqExpr->id ) ) visit_children = false;@ else visitedIds.insert( unqExpr->id ); } \end{cfa} Example when mutating the unique instance itself, which actually creates copies \item When mutating the unique instance itself, which actually creates copies \begin{cfa} auto mutExpr = mutate( unqExpr ); // internally calls copy when shared if ( ! unqMap.count( unqExpr->id ) ) { @if ( ! unqMap.count( unqExpr->id ) ) {@ ... } else { } \end{cfa} Such workaround seems difficult to be fit into a common visitor template. This suggests the memory model may need different kinds of nodes to accurately represent the syntax tree. Together with the fact that declaration nodes are always unique, it is possible that AST nodes can be classified by three different types: \begin{itemize} \item \textbf{Strictly unique} with only one owner (declarations); \item \textbf{Logically unique} with (possibly) many owners but should not be copied (unique expression example presented here); \item \textbf{Shared} by functional programming model, which assume immutable data structure and are copied on mutation. \end{itemize} Such workarounds are difficult to fit into the common visitor pattern, which suggests the memory model may need different kinds of nodes to accurately represent this feature in the AST. Given that declaration nodes are unique, it is possible for AST nodes to be divided into three different types: \begin{itemize} \item \textbf{Singleton} with only one owner (declarations); \item \textbf{No-copy} with multiple owners but cannot be copied (unique expression example presented here); \item \textbf{Copy} by functional-programming style, which assumes immutable data structures that are copied on mutation. \end{itemize} The boilerplate code can potentially handle these three cases differently. \section{Analysis of resolver algorithm complexity} The focus of this chapter is to identify and analyze some realistic cases that cause resolver algorithm to have an exponential run time. As previous work has shown [3], the overload resolution problem in \CFA has worst-case exponential complexity; however, only few specific patterns can trigger the exponential complexity in practice. Implementing heuristic-based optimization for those selected cases is helpful to alleviate the problem. The focus of this section is to identify and analyze some realistic cases that cause the resolver algorithm to have an exponential runtime. As previous work has shown~\cite[\S~4.2.1]{Moss19}, the overload resolution problem in \CFA has worst-case exponential complexity; however, only few specific patterns can trigger the exponential complexity in practice. Implementing heuristic-based optimization for those selected cases is helpful to alleviate the problem. \label{s:UnboundReturnType} The interaction of return type overloading and polymorphic functions creates this problem of function calls with unbound return type, and is further complicated by the presence of assertions. The interaction of return-type overloading and polymorphic functions creates function calls with unbounded return-type, and is further complicated by the presence of assertions. The prime example of a function with unbound return type is the type-safe version of C @malloc@: \begin{cfa} // size deduced from type, so no need to provide the size argument forall( dtype T | sized( T ) ) T * malloc( void ); \end{cfa} Unbound return type can be problematic in resolver algorithm complexity because a single match of function call with unbound return type may create multiple candidates. In the worst case, consider a function declared to return any @otype@: forall( dtype T | sized( T ) ) T * malloc( void ) { return (T *)malloc( sizeof(T) ); } // call C malloc int * i = malloc();  // type deduced from left-hand size $\Rightarrow$ no size argument or return cast \end{cfa} An unbound return-type is problematic in resolver complexity because a single match of a function call with an unbound return type may create multiple candidates. In the worst case, consider a function declared that returns any @otype@ (defined \VPageref{otype}): \begin{cfa} forall( otype T ) T anyObj( void ); \end{cfa} As the resolver attempts to satisfy the otype constraint on @T@, a single call to @anyObj()@ without the result type known creates at least as many candidates as the number of complete types currently in scope; with generic types it becomes even worse, for example, assuming a declaration of generic pair is available at that point: As the resolver attempts to satisfy the otype constraint on @T@, a call to @anyObj()@ in an expression, without the result type known, creates at least as many candidates as the number of complete types currently in scope; with generic types it becomes even worse, \eg assuming a declaration of a generic @pair@ is available at that point: \begin{cfa} forall( otype T, otype U ) struct pair { T first; U second; }; \end{cfa} Then an @anyObj()@ call can result in arbitrarily complex types, such as @pair( pair( int,int ), pair( int,int ) )@, and the depth can grow indefinitely until the specified parameter depth limit, thus creating exponentially many candidates. However, the expected types allowed by parent expressions are practically very few, so most of those interpretations are invalid; if the result type is never bound up to top level, by the semantic rules it is ambiguous if there are more than one valid bindings, and resolution can fail fast. It is therefore reasonable to delay resolving assertions on an unbound parameter in return type; however, with the current cost model, such behavior may further cause irregularities in candidate selection, such that the presence of assertions can change the preferred candidate, even when order of expression costs are supposed to stay the same. Detailed analysis of this issue will be presented later, in the correctness part. Then an @anyObj()@ call can result in arbitrarily complex types, such as @pair( pair( int, int ), pair( int, int ) )@, and the depth can grow indefinitely until a specified parameter-depth limit, thus creating exponentially many candidates. However, the expected types allowed by parent expressions are practically very few, so most of those interpretations are invalid; if the result type is never bound up to the top level, by the semantic rules it is ambiguous if there is more than one valid binding and resolution fails quickly. It is therefore reasonable to delay resolving assertions on an unbound parameter in a return type; however, with the current cost model, such behavior may further cause irregularities in candidate selection, such that the presence of assertions can change the preferred candidate, even when order of expression costs are supposed to stay the same. A detailed analysis of this issue is presented in \VRef{s:AnalysisTypeSystemCorrectness}. \label{s:TtypeResolutionInfiniteRecursion} @ttype@ (tuple type'') is a relatively new addition to the language that attempts to provide type-safe variadic argument semantics. Unlike regular @dtype@ parameters, @ttype@ is only valid in function parameter list, and may only appear once as the type of last parameter. At the call site, a @ttype@ parameter is bound to the tuple type of all remaining function call arguments. @ttype@ (tuple type'') is a relatively new addition to the language that attempts to provide type-safe variadic argument semantics. Unlike regular @dtype@ parameters, @ttype@ is only valid in a function parameter-list, and may only appear once as the last parameter type. At the call site, a @ttype@ parameter is bound to the tuple type of all remaining function-call arguments. There are two kinds of idiomatic @ttype@ usage: one is to provide flexible argument forwarding, similar to the variadic template in \CC (\lstinline[language=C++]|template|), as shown below in the implementation of @unique_ptr@ T * data; }; forall( dtype T | sized( T ), ttype Args | { void ?{}( T &, Args ); }) void ?{}( unique_ptr( T ) & this, Args args ) { this.data = new( args ); } \end{cfa} the other is to implement structural recursion in the first-rest manner: \begin{cfa} forall( otype T, ttype Params | { void process( T ); void func( Params ); }) forall( dtype T | sized( T ), @ttype Args@ | { void ?{}( T &, Args ); }) void ?{}( unique_ptr( T ) & this, Args @args@ ) { this.data = new( @args@ );  // forward constructor arguments to dynamic allocator } \end{cfa} The other usage is to implement structural recursion in the first-rest pattern: \begin{cfa} forall( otype T, @ttype Params@ | { void process( T ); void func( Params ); }) void func( T arg1, Params p ) { process( arg1 ); func( p ); } \end{cfa} For the second use case, it is important that the number of parameters in the recursive call go down, since the call site must deduce all assertion candidates, and that is only possible if by just looking at argument types (and not their values), the recursion is known to be completed in a finite number of steps. In recent experiments, however, some flaw in the type binding rules can lead to the first kind of @ttype@ use case produce an invalid candidate that the resolver enters an infinite loop. This bug was discovered in an attempt to raise assertion recursive depth limit and one of the library program takes exponentially longer time to compile. The cause of the problem is identified to be the following set of functions. File @memory.cfa@ contains \begin{cfa} #include "memory.hfa" #include "stdlib.hfa" \end{cfa} where file @memory.hfa@ contains the @unique_ptr@ declaration above, and two other similar functions with @ttype@ parameter: \begin{cfa} forall( dtype T | sized( T ), ttype Args | { void ?{}( T &, Args ); }) { func( @p@ );  // recursive call until base case of one argument } \end{cfa} For the second use case, it is imperative the number of parameters in the recursive call goes down, since the call site must deduce all assertion candidates, and that is only possible if by observation of the argument types (and not their values), the recursion is known to be completed in a finite number of steps. In recent experiments, however, a flaw in the type-binding rules can lead to the first kind of @ttype@ use case producing an invalid candidate and the resolver enters an infinite loop. This bug was discovered in an attempt to raise the assertion recursive-depth limit and one of the library programs took exponentially longer to compile. The cause of the problem is the following set of functions: \begin{cfa} // unique_ptr  declaration from above forall( dtype T | sized( T ), ttype Args | { void ?{}( T &, Args ); } ) { // distribute forall clause void ?{}( counter_data( T ) & this, Args args ); void ?{}( counter_ptr( T ) & this, Args args ); void ?{}( unique_ptr( T ) & this, Args args ); } \end{cfa} File @stdlib.hfa@ contains \begin{cfa} forall( dtype T | sized( T ), ttype TT | { void ?{}( T &, TT ); } ) T * new( TT p ) { return &(*malloc()){ p }; } \end{cfa} In the expression @(*malloc()){p}@, the type of object being constructed is yet unknown, since the return type information is not immediately provided. That caused every constructor to be searched, and while normally a bound @ttype@ cannot be unified with any free parameter, it is possible with another free @ttype@. Therefore in addition to the correct option provided by assertion, 3 wrong options are examined, each of which again requires the same assertion, for an unknown base type T and @ttype@ arguments, and that becomes an infinite loop, until the specified recursion limit and resolution is forced to fail. Moreover, during the recursion steps, number of candidates grows exponentially, since there are always 3 options at each step. Unfortunately, @ttype@ to @ttype@ binding is necessary, to allow calling the function provided by assertion indirectly. \begin{cfa} forall( dtype T | sized( T ), ttype Args | { void ?{}( T &, Args ); }) void ?{}( unique_ptr( T ) & this, Args args ) { this.data = (T * )new( args ); } \end{cfa} Here the constructor assertion is used for the @new( args )@ call. T * new( TT p ) { return @&(*malloc()){ p };@ } \end{cfa} In the expression @(*malloc()){p}@, the type of the object being constructed is unknown, since the return-type information is not immediately available. That causes every constructor to be searched, and while normally a bound @ttype@ cannot be unified with any free parameter, it is possible with another free @ttype@. Therefore, in addition to the correct option provided by the assertion, 3 wrong options are examined, each of which again requires the same assertion, for an unknown base-type @T@ and @ttype@ argument, which becomes an infinite loop until the specified recursion limit and resolution is fails. Moreover, during the recursion steps, the number of candidates grows exponentially, since there are always 3 options at each step. Unfortunately, @ttype@ to @ttype@ binding is necessary, to allow indirectly calling a function provided in an assertion. \begin{cfa} forall( dtype T | sized( T ), ttype Args | { @void ?{}( T &, Args );@ }) void ?{}( unique_ptr( T ) & this, Args args ) { this.data = (T *)@new( args )@; } // constructor call \end{cfa} Here the constructor assertion is used by the @new( args )@ call to indirectly call the constructor on the allocated storage. Therefore, it is hard, perhaps impossible, to solve this problem by tweaking the type binding rules. An assertion caching algorithm can help improve this case by detecting cycles in recursion. Meanwhile, without the caching algorithm implemented, some changes in the \CFA source code are enough to eliminate this problem, at least in the current codebase. Note that the issue only happens with an overloaded variadic function, which rarely appears in practice, since the idiomatic use cases are for argument forwarding and self-recursion. The only overloaded @ttype@ function so far discovered in all of \CFA standard library code is the constructor, and by utilizing the argument-dependent lookup process described in Section~\ref{s:UnboundReturnType}, adding a cast before constructor call gets rid of the issue. \begin{cfa} T * new( TT p ) { return &(*(T * )malloc()){ p }; } Meanwhile, without a caching algorithm implemented, some changes in the \CFA source code are enough to eliminate this problem, at least in the current codebase. Note that the issue only happens with an overloaded variadic function, which rarely appears in practice, since the idiomatic use cases are for argument forwarding and self-recursion. The only overloaded @ttype@ function so far discovered in all of \CFA standard library is the constructor, and by utilizing the argument-dependent lookup process described in \VRef{s:UnboundReturnType}, adding a cast before the constructor call removes the issue. \begin{cfa} T * new( TT p ) { return &(*@(T * )@malloc()){ p }; } \end{cfa} \subsection{Reused assertions in nested generic type} The following test of deeply nested dynamic generic type reveals that locally caching reused assertions is necessary, rather than just a resolver optimization, because recomputing assertions can result in bloated generated code size: The following test of deeply nested, dynamic generic type reveals that locally caching reused assertions is necessary, rather than just a resolver optimization, because recomputing assertions can result in bloated generated code size: \begin{cfa} struct nil {}; int main() { #if   N==0 nil x; nil @x@; #elif N==1 cons( size_t, nil ) x; cons( size_t, nil ) @x@; #elif N==2 cons( size_t, cons( size_t, nil ) ) x; cons( size_t, cons( size_t, nil ) ) @x@; #elif N==3 cons( size_t, cons( size_t, cons( size_t, nil ) ) ) x; cons( size_t, cons( size_t, cons( size_t, nil ) ) ) @x@; // similarly for N=4,5,6 #endif } \end{cfa} At the declaration of @x@, it is implicitly initialized by generated constructor call, whose signature is given by At the declaration of @x@, it is implicitly initialized by generated constructor call, with signature: \begin{cfa} forall( otype L, otype R ) void ?{}( cons( L, R ) & ); \end{cfa} Note that the @otype@ constraint contains 4 assertions: where the @otype@ constraint contains the 4 assertions:\label{otype} \begin{cfa} void ?{}( L & ); // default constructor L & ?=?( L &, L & ); // assignment \end{cfa} Now since the right hand side of outermost cons is again a cons, recursive assertions are required. When the compiler cannot cache and reuse already resolved assertions, it becomes a problem, as each of those 4 pending assertions again asks for 4 more assertions one level below. Without any caching, number of resolved assertions grows exponentially, while that is obviously unnecessary since there are only $n+1$ different types involved. Even worse, this causes exponentially many wrapper functions generated later at the codegen step, and results in huge compiled binary. Now since the right hand side of outermost cons is again a cons, recursive assertions are required. \VRef[Table]{t:NestedConsTest} shows when the compiler does not cache and reuse already resolved assertions, it becomes a problem, as each of these 4 pending assertions again asks for 4 more assertions one level below. Without caching, the number of resolved assertions grows exponentially, which is unnecessary since there are only $n+1$ different types involved. Even worse, this problem causes exponentially many wrapper functions to be generated at the backend, resulting in a huge binary. As the local functions are implemented by emitting executable code on the stack~\cite{gcc-nested-func}, it means that compiled code also has exponential run time. This problem has practical implications, as nested collection types are frequently used in real production code. \begin{table}[h] \centering \caption{Compilation results of nested cons test} \label{t:NestedConsTest} \begin{tabular}{|r|r|r|} \hline \end{table} As the local functions are implemented by emitting executable code on the stack~\cite{gcc-nested-func}, it eventually means that compiled code also has exponential run time. This problem has evident practical implications, as nested collection types are frequently used in real production code. \section{Analysis of type system correctness} \label{s:AnalysisTypeSystemCorrectness} In Moss' thesis~\cite[\S~4.1.2,~p.~45]{Moss19}, the author presents the following example: From the set of candidates whose parameter and argument types have been unified and whose assertions have been satisfied, those whose sub-expression interpretations have the smallest total cost of conversion are selected ... The total cost of conversion for each of these candidates is then calculated based on the implicit conversions and polymorphism involved in adapting the types of the sub-expression interpretations to the formal parameter types. \end{quote} With this model, the algorithm picks @g1@ in resolving the @f( g( 42 ) )@ call, which seems to be undesirable. There are further evidence that shows the Bilson model is fundamentally incorrect, following the discussion of unbound return type in Section~\ref{s:UnboundReturnType}. By the conversion cost specification, a binding from a polymorphic type parameter to a concrete type incurs a polymorphic cost of 1. It remains unspecified \emph{when} the type parameters should become bound. When the parameterized types appear in the function parameters, they can be deduced from the argument type, and there is no ambiguity. In the unbound return case, however, the binding may happen at any stage in expression resolution, therefore it is impossible to define a unique local conversion cost. Note that type binding happens exactly once per parameter in resolving the entire expression, so the global binding cost is unambiguously 1. As per the current compiler implementation, it does have a notable inconsistency in handling such case. For any unbound parameter that does \emph{not} come with an associated assertion, it remains unbound to the parent expression; for those that does however, they are immediately bound in the assertion resolution step, and concrete result types are used in the parent expressions. With this model, the algorithm picks @g1@ in resolving the @f( g( 42 ) )@ call, which is undesirable. There is further evidence that shows the Bilson model is fundamentally incorrect, following the discussion of unbound return type in \VRef{s:UnboundReturnType}. By the conversion-cost specification, a binding from a polymorphic type-parameter to a concrete type incurs a polymorphic cost of 1. It remains unspecified \emph{when} the type parameters should become bound. When the parameterized types appear in function parameters, they can be deduced from the argument type, and there is no ambiguity. In the unbound return case, however, the binding may happen at any stage in expression resolution, therefore it is impossible to define a unique local conversion cost. Note that type binding happens exactly once per parameter in resolving the entire expression, so the global binding cost is unambiguously 1. In the current compiler implementation, there is a notable inconsistency in handling this case. For any unbound parameter that does \emph{not} come with an associated assertion, it remains unbound to the parent expression; for those that do, however, they are immediately bound in the assertion resolution step, and concrete result types are used in the parent expressions. Consider the following example: \begin{cfa} void h( int * ); \end{cfa} The expression @h( f() )@ eventually has a total cost of 1 from binding (T: int), but in the eager resolution model, the cost of 1 may occur either at call to @f@ or at call to @h@, and with the assertion resolution triggering a binding, the local cost of @f()@ is (0 poly, 0 spec) with no assertions, but (1 poly, -1 spec) with an assertion: \begin{cfa} forall( dtype T | { void g( T * ); } ) T * f( void ); The expression @h( f() )@ eventually has a total cost of 1 from binding (T: int), but in the eager-resolution model, the cost of 1 may occur either at the call to @f@ or at call to @h@, and with the assertion resolution triggering a binding, the local cost of @f()@ is (0 poly, 0 spec) with no assertions, but (1 poly, -1 spec) with an assertion: \begin{cfa} forall( dtype T | @{ void g( T * ); }@ ) T * f( void ); void g( int * ); void h( int * ); \end{cfa} and that contradicts the principle that adding assertions should make expression cost lower. Furthermore, the time at which type binding and assertion resolution happens is an implementation detail of the compiler, but not a part of language definition. That means two compliant \CFA compilers, one performing immediate assertion resolution at each step, and one delaying assertion resolution on unbound types, can produce different expression costs and therefore different candidate selection, making the language rule itself partially undefined and therefore unsound. By the above reasoning, the updated cost model using global sum of costs should be accepted as the standard. It also allows the compiler to freely choose when to resolve assertions, as the sum of total costs is independent of that choice; more optimizations regarding assertion resolution can also be implemented. and that contradicts the principle that adding assertions should make expression cost lower. Furthermore, the time at which type binding and assertion resolution happens is an implementation detail of the compiler, not part of the language definition. That means two compliant \CFA compilers, one performing immediate assertion resolution at each step, and one delaying assertion resolution on unbound types, can produce different expression costs and therefore different candidate selection, making the language rule itself partially undefined, and therefore, unsound. By the above reasoning, the updated cost model using global sum of costs should be accepted as the standard. It also allows the compiler to freely choose when to resolve assertions, as the sum of total costs is independent of that choice; more optimizations regarding assertion resolution can also be implemented. \section{Timing results} For the timing results presented here, the \CFA compiler is built with gcc 9.3.0, and tested on a server machine running Ubuntu 20.04, 64GB RAM and 32-core 2.2 GHz CPU, results reported by the time command, and using only 8 cores in parallel such that the time is close to the case with 100\% CPU utilization on a single thread. On the most recent build, the \CFA standard library (~1.3 MB of source code) compiles in 4 minutes 47 seconds total processor time (single thread equivalent), with the slowest file taking 13 seconds. The test suite (178 test cases, ~2.2MB of source code) completes within 25 minutes total processor time,\footnote{Including a few runtime tests; total time spent in compilation is approximately 21 minutes.} with the slowest file taking 23 seconds. In contrast, the library build on old compiler takes 85 minutes total, 5 minutes for the slowest file. Full test suite takes too long with old compiler build and is therefore not run, but the slowest test cases take approximately 5 minutes. Overall, the most recent build compared to old build in April 2020, before the project started, is consistently faster by a factor of 20. Additionally, 6 selected \CFA source files with distinct features from library and test suite are used to test compiler performance after each of the optimizations are implemented. Test files are from the most recent build and run through C preprocessor to eliminate the factor of header file changes. The selected tests are: \begin{itemize} \item @lib/fstream@ (112 KB)\footnote{File sizes are after preprocessing, with no line information (\lstinline|gcc -E -P|).}: implementation of I/O library \item @lib/mutex@ (166 KB): implementation of concurrency primitive \item @lib/vector@ (43 KB): container example, similar to \CC vector \item @lib/stdlib@ (64 KB): type-safe wrapper to @void *@-based C standard library functions \item @test/ISO2@ (55 KB): application of I/O library \item @test/thread@ (188 KB): application of threading library \end{itemize} The \CFA compiler builds are picked from git commit history that passed the test suite, and implement the optimizations incrementally: \begin{itemize} \item \#0 is the first working build of new AST data structure \item \#1 implements special symbol table and argument-dependent lookup \item \#2 implements late assertion satisfaction \item \#3 implements revised function type representation \item \#4 skips pruning on expressions with function type (most recent build) \end{itemize} The old resolver with no memory sharing and none of the optimizations above is also tested. For the timing results presented here, the \CFA compiler is built with gcc 9.3.0, and tested on a server machine running Ubuntu 20.04, 64GB RAM and 32-core 2.2 GHz CPU. Timing is reported by the @time@ command and an experiment is run using 8 cores, where each core is at 100\% CPU utilization. On the most recent build, the \CFA standard library ($\approx$1.3 MB of source code) compiles in 4 minutes 47 seconds total processor time (single thread equivalent), with the slowest file taking 13 seconds. The test suite (178 test cases, $\approx$2.2MB of source code) completes within 25 minutes total processor time, % PAB: I do not understand this footnote. %\footnote{Including a few runtime tests; total time spent in compilation is approximately 21 minutes.} with the slowest file taking 23 seconds. In contrast, the library build with the old compiler takes 85 minutes total, 5 minutes for the slowest file. The full test-suite takes too long with old compiler build and is therefore not run, but the slowest test cases take approximately 5 minutes. Overall, the most recent build compared to an old build is consistently faster by a factor of 20. \begin{table} \centering \caption{Compile time of selected files by compiler build, in seconds} \label{t:SelectedFileByCompilerBuild} \begin{tabular}{|l|r|r|r|r|r|r|} \hline \end{table} Additionally, 6 selected \CFA source files with distinct features from the library and test suite are used to illustrate the compiler performance change after each of the implemented optimizations. Test files are from the most recent build and run through the C preprocessor to expand header file, perform macro expansions, but no line number information (@gcc -E -P@). \VRef[Table]{t:SelectedFileByCompilerBuild} shows the selected tests: \begin{itemize} \item @lib/fstream@ (112 KB) \item @lib/mutex@ (166 KB): implementation of concurrency primitive \item @lib/vector@ (43 KB): container example, similar to \CC vector \item @lib/stdlib@ (64 KB): type-safe wrapper to @void *@-based C standard library functions \item @test/ISO2@ (55 KB): application of I/O library \item @test/thread@ (188 KB): application of threading library \end{itemize} versus \CFA compiler builds picked from the git commit history that implement the optimizations incrementally: \begin{itemize} \item old resolver \item \#0 is the first working build of the new AST data structure \item \#1 implements special symbol table and argument-dependent lookup \item \#2 implements late assertion-satisfaction \item \#3 implements revised function-type representation \item \#4 skips pruning on expressions for function types (most recent build) \end{itemize} Reading left to right for a test shows the benefit of each optimization on the cost of compilation. \section{Conclusion} Over the course of 8 months of active research and development in \CFA type system and compiler algorithm, performance of the reference \CFA compiler, cfa-cc, has been greatly improved, allowing mid-sized \CFA programs to be compiled and built reasonably fast. As there are also ongoing efforts in the team on building a standard library, evaluating the runtime performance, and attempting to incorporate \CFA with existing software written in C, this project is especially meaningful for practical purposes. Analysis conducted in the project were based significantly on heuristics and practical evidence, as the theoretical bounds and average cases for the expression resolution problem differ. This approach was difficult at start to follow, with an unacceptably slow compiler, since running the program through debugger and validation tools (\eg @gdb@, @valgrind@) adds another order of magnitude to run time, which was already in minutes. However, near the end of the project, many significant improvements have already been made and new optimizations can be tested immediately. The positive feedback in development cycle benefits the \CFA team as a whole, more than just for the compiler optimizations. Some potential issues of the language that may happen frequently in practice have been identified. Due to the time constraint and complex nature of these problems, a handful of them remain unsolved, but some constructive proposals are made. Notably, introducing a local assertion cache in the resolver is a common solution for a few remaining problems, so that should be the focus of work soon. The \CFA team are planning on a public alpha release of the language as the compiler performance becomes promising, and other parts of the system, such as a standard library, are also being enhanced. Ideally, the remaining problems should be resolved before release, and the solutions will also be integral to drafting a formal specification. Over the course of 8 months of active research and development of the \CFA type system and compiler algorithms, performance of the reference \CFA compiler, cfa-cc, has been greatly improved. Now, mid-sized \CFA programs are compiled reasonably fast. Currently, there are ongoing efforts by the \CFA team to augment the standard library and evaluate its runtime performance, and incorporate \CFA with existing software written in C; therefore this project is especially meaningful for these practical purposes. Accomplishing this work was difficult. Analysis conducted in the project is based significantly on heuristics and practical evidence, as the theoretical bounds and average cases for the expression resolution problem differ. As well, the slowness of the initial compiler made attempts to understand why and where problems exist extremely difficult because both debugging and validation tools (\eg @gdb@, @valgrind@, @pref@) further slowed down compilation time. However, by the end of the project, I had found and fixed several significant problems and new optimizations are easier to introduce and test. The reduction in the development cycle benefits the \CFA team as a whole. Some potential issues of the language, which happen frequently in practice, have been identified. Due to the time constraint and complex nature of these problems, a handful of them remain unsolved, but some constructive proposals are made. Notably, introducing a local assertion cache in the resolver is a reasonable solution for a few remaining problems, so that should be the focus of future work. The \CFA team are planning on a public alpha release of the language as the compiler performance, given my recent improvements, is now useable. Other parts of the system, such as the standard library, have made significant gains due to the speed up in the development cycle. Ideally, the remaining problems should be resolved before release, and the solutions will also be integral to drafting a formal specification. \addcontentsline{toc}{section}{\refname}