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1\chapter{Resolution Algorithms}
2\label{resolution-chap}
3
4The main task of the \CFACC{} type-checker is \emph{expression resolution}: determining which declarations the identifiers in each expression correspond to.
5Resolution is a straightforward task in C, as no simultaneously-visible declarations share identifiers, but in \CFA{}, the name overloading features discussed in Section~\ref{overloading-sec} generate multiple candidate declarations for each identifier.
6A given matching between identifiers and declarations in an expression is an \emph{interpretation}; an interpretation also includes information about polymorphic type bindings and implicit casts to support the \CFA{} features discussed in Sections~\ref{poly-func-sec} and~\ref{implicit-conv-sec}, each of which increase the number of valid candidate interpretations.
7To choose among valid interpretations, a \emph{conversion cost} is used to rank interpretations.
8This conversion cost is summed over all subexpression interpretations in the interpretation of a top-level expression.
9Hence, the expression resolution problem is to find the unique minimal-cost interpretation for an expression, reporting an error if no such unique interpretation exists.
10
11\section{Expression Resolution}
12
13The expression resolution pass in \CFACC{} must traverse the input expression, match identifiers to available declarations, rank candidate interpretations according to their conversion cost, and check type assertion satisfaction for these candidates.
14Once the set of valid interpretations for the top-level expression has been found, the expression resolver selects the unique minimal-cost candidate or reports an error.
15
16The expression resolution problem in \CFA{} is more difficult than the analogous problems in C or \CC{}.
17As mentioned above, the lack of name overloading in C makes its resolution problem substantially easier, but a comparison of the richer type systems in \CFA{} and \CC{} highlights some of the challenges in \CFA{} expression resolution.
18The key distinction between \CFA{} and \CC{} resolution is that \CC{} uses a greedy algorithm for selection of candidate functions given their argument interpretations, whereas \CFA{} allows contextual information from superexpressions to influence the choice among candidate functions.
19One key use of this contextual information is for type inference of polymorphic return types; \CC{} requires explicit specification of template type parameters which only occur in a function's return type, while \CFA{} allows the instantiation of these type parameters to be inferred from context (and in fact does not allow explicit specification of type parameters to a function).
20Similarly, while both \CFA{} and \CC{} rank candidate functions based on a cost metric for implicit conversions, \CFA{} allows a suboptimal subexpression interpretation to be selected if it allows a lower-cost overall interpretation, while \CC{} requires that each subexpression interpretation have minimal cost.
21Because of this use of contextual information, the \CFA{} expression resolver must consider multiple interpretations of each function argument, while the \CC{} compiler has only a single interpretation for each argument\footnote{With the exception of address-of operations on functions.}.
22Additionally, until the introduction of concepts in \CCtwenty{} \cite{C++Concepts}, \CC{} expression resolution has no analogue to \CFA{} assertion satisfaction checking, a further  complication for a \CFA{} compiler.
23The precise definition of \CFA{} expression resolution in this section further expands on the challenges of this problem.
24
25\subsection{Type Unification}
26
27The polymorphism features of \CFA{} require binding of concrete types to polymorphic type variables.
28Briefly, \CFACC{} keeps a mapping from type variables to the concrete types they are bound to as an auxiliary data structure during expression resolution; Chapter~\ref{env-chap} describes this \emph{environment} data structure in more detail.
29A \emph{unification} algorithm is used to simultaneously check two types for equivalence with respect to the substitutions in an environment and update that environment.
30Essentially, unification recursively traverses the structure of both types, checking them for equivalence, and when it encounters a type variable, it replaces it with the concrete type it is bound to; if the type variable has not yet been bound, the unification algorithm assigns the equivalent type as the bound type of the variable, after performing various consistency checks.
31Ditchfield~\cite{Ditchfield92} and Bilson~\cite{Bilson03} describe the semantics of \CFA{} unification in more detail.
32
33\subsection{Conversion Cost} \label{conv-cost-sec}
34
35\CFA{}, like C, allows inexact matches between the type of function parameters and function call arguments.
36Both languages insert \emph{implicit conversions} in these situations to produce an exact type match, and \CFA{} also uses the relative \emph{cost} of different conversions to select among overloaded function candidates.
37C does not have an explicit cost model for implicit conversions, but the usual arithmetic conversions'' \cite[\S{}6.3.1.8]{C11} used to decide which arithmetic operators to apply define one implicitly.
38The only context in which C has name overloading is the arithmetic operators, and the usual arithmetic conversions define a \emph{common type} for mixed-type arguments to binary arithmetic operators.
39Since for backward-compatibility purposes the conversion costs of \CFA{} must produce an equivalent result to these common type rules, it is appropriate to summarize \cite[\S{}6.3.1.8]{C11} here:
40
41\begin{itemize}
42\item If either operand is a floating-point type, the common type is the size of the largest floating-point type. If either operand is !_Complex!, the common type is also !_Complex!.
43\item If both operands are of integral type, the common type has the same size\footnote{Technically, the C standard defines a notion of \emph{rank} in \cite[\S{}6.3.1.1]{C11}, a distinct value for each \lstinline{signed} and \lstinline{unsigned} pair; integral types of the same size thus may have distinct ranks. For instance, though \lstinline{int} and \lstinline{long} may have the same size, \lstinline{long} always has greater rank. The standard-defined types are declared to have greater rank than any types of the same size added as compiler extensions.} as the larger type.
44\item If the operands have opposite signedness, the common type is !signed! if the !signed! operand is strictly larger, or !unsigned! otherwise. If the operands have the same signedness, the common type shares it.
45\end{itemize}
46
47Beginning with the work of Bilson~\cite{Bilson03}, \CFA{} defines a \emph{conversion cost} for each function call in a way that generalizes C's conversion rules.
48Loosely defined, the conversion cost counts the implicit conversions utilized by an interpretation.
49With more specificity, the cost is a lexicographically-ordered tuple, where each element corresponds to a particular kind of conversion.
50In Bilson's design, conversion cost is a 3-tuple, $(unsafe, poly, safe)$, where $unsafe$ is the count of unsafe (narrowing) conversions, $poly$ is the count of polymorphic type bindings, and $safe$ is the sum of the degree of safe (widening) conversions.
51Degree of safe conversion is calculated as path weight in a directed graph of safe conversions between types; Bilson's version of this graph is in Figure~\ref{bilson-conv-fig}.
52The safe conversion graph is designed such that the common type $c$ of two types $u$ and $v$ is compatible with the C standard definitions from \cite[\S{}6.3.1.8]{C11} and can be calculated as the unique type minimizing the sum of the path weights of $\overrightarrow{uc}$ and $\overrightarrow{vc}$.
53The following example lists the cost in the Bilson model of calling each of the following functions with two !int! parameters, where the interpretation with the minimum total cost will be selected:
54
55\begin{cfa}
56void f$$$_1$$$(char, long); $\C{// (1,0,1)}$
57void f$$$_2$$$(short, long); $\C{// (1,0,1)}$
58forall(otype T) void f$$$_3$$$(T, long); $\C{// (0,1,1)}$
59void f$$$_4$$$(long, long); $\C{// (0,0,2)}$
60void f$$$_5$$$(int, unsigned long); $\C{// (0,0,2)}$
61void f$$$_6$$$(int, long); $\C{// (0,0,1)}$
62\end{cfa}
63
64Note that safe and unsafe conversions are handled differently; \CFA{} counts distance of safe conversions (\eg{} !int! to !long! is cheaper than !int! to !unsigned long!), while only counting the number of unsafe conversions (\eg{} !int! to !char! and !int! to !short! both have unsafe cost 1, as in !f!$_1$ and !f!$_2$ above).
65These costs are summed over the parameters in a call; in the example above, the cost of the two !int! to !long! conversions for !f!$_4$ sum equal to the one !int! to !unsigned long! conversion for !f!$_5$.
66
67\begin{figure}
68        \centering
69        \begin{subfigure}[h]{3in}
70                \includegraphics{figures/bilson-conv-graph}
71                \caption{Bilson} \label{bilson-conv-fig}
72        \end{subfigure}~\begin{subfigure}[h]{3in}
73                \includegraphics{figures/extended-conv-graph}
74                \caption{Extended} \label{extended-conv-fig}
75        \end{subfigure}
76        % \includegraphics{figures/safe-conv-graph}
77        \caption[Safe conversion graphs.]{Safe conversion graphs. In both graphs, plain arcs have cost $safe = 1, sign = 0$ while dashed sign-conversion arcs have cost $safe = 1, sign = 1$. As per \cite[\S{}6.3.1.8]{C11}, types promote to types of the same signedness with greater rank, from \lstinline{signed} to \lstinline{unsigned} with the same rank, and from \lstinline{unsigned} to \lstinline{signed} with greater size. The arc from \lstinline{unsigned long} to \lstinline{long long} (highlighted in red in \ref{bilson-conv-fig}) is deliberately omitted in \ref{extended-conv-fig}, as on the presented system \lstinline{sizeof(long) == sizeof(long long)}.}
78        \label{safe-conv-fig}
79\end{figure}
80
81As part of adding reference types to \CFA{} (see Section~\ref{type-features-sec}), Schluntz added a new $reference$ element to the cost tuple, which counts the number of implicit reference-to-rvalue conversions performed so that candidate interpretations can be distinguished by how closely they match the nesting of reference types; since references are meant to act almost indistinguishably from lvalues, this $reference$ element is the least significant in the lexicographic comparison of cost tuples.
82
83I also refined the \CFA{} cost model as part of this thesis work.
84Bilson's \CFA{} cost model includes the cost of polymorphic type bindings from a function's type assertions in the $poly$ element of the cost tuple; this has the effect of making more-constrained functions more expensive than less-constrained functions, as in the following example, based on differing iterator types:
85
86\begin{cfa}
87forall(dtype T | { T& ++?(T&); }) T& advance$$$_1$$$(T& i, int n);
88forall(dtype T | { T& ++?(T&); T& ?+=?(T&, int)}) T& advance$$$_2$$$(T& i, int n);
89\end{cfa}
90
91In resolving a call to !advance!, the binding to the !T&! parameter in the assertions is added to the $poly$ cost in Bilson's model.
92However, type assertions actually make a function \emph{less} polymorphic, and as such functions with more type assertions should be preferred in type resolution.
93In the example above, if the meaning of !advance! is increment !i! !n! times'', !advance!$_1$ requires an !n!-iteration loop, while !advance!$_2$ can be implemented more efficiently with the !?+=?! operator; as such, !advance!$_2$ should be chosen over !advance!$_1$ whenever its added constraint can be satisfied.
94Accordingly, a $specialization$ element is now included in the \CFA{} cost tuple, the values of which are always negative.
95Each type assertion subtracts 1 from $specialization$, so that more-constrained functions cost less, and thus are chosen over less-constrained functions, all else being equal.
96A more sophisticated design would define a partial order over sets of type assertions by set inclusion (\ie{} one function would only cost less than another if it had a strict superset of assertions,  rather than just more total assertions), but I did not judge the added complexity of computing and testing this order to be worth the gain in specificity.
97
98I also incorporated an unimplemented aspect of Ditchfield's earlier cost model.
99In the example below, adapted from \cite[p.89]{Ditchfield92}, Bilson's cost model only distinguished between the first two cases by accounting extra cost for the extra set of !otype! parameters, which, as discussed above, is not a desirable solution:
100
101\begin{cfa}
102forall(otype T, otype U) void f$$$_1$$$(T, U)$\C[3.25in]{// polymorphic}$
103forall(otype T) void f$$$_2$$$(T, T)$\C[3.25in]{// less polymorphic}$
104forall(otype T) void f$$$_3$$$(T, int)$\C[3.25in]{// even less polymorphic}$
105forall(otype T) void f$$$_4$$$(T*, int); $\C[3.25in]{// least polymorphic}$
106\end{cfa}
107
108The new cost model accounts for the fact that functions with more polymorphic variables are less constrained by introducing a $var$ cost element that counts the number of type variables on a candidate function.
109In the example above, !f!$_1$ has $var = 2$, while the others have $var = 1$.
110
111The new cost model also accounts for a nuance unhandled by Ditchfield or Bilson, in that it makes the more specific !f!$_4$ cheaper than the more generic !f!$_3$; !f!$_4$ is presumably somewhat optimized for handling pointers, but the prior \CFA{} cost model could not account for the more specific binding, as it simply counted the number of polymorphic unifications.
112In the modified model, each level of constraint on a polymorphic type in the parameter list results in a decrement of the $specialization$ cost element, which is shared with the count of assertions due to their common nature as constraints on polymorphic type bindings.
113Thus, all else equal, if both a binding to !T! and a binding to !T*! are available, the model chooses the more specific !T*! binding with $specialization = -1$.
114This process is recursive, such that !T**! has $specialization = -2$.
115This calculation works similarly for generic types, \eg{} !box(T)! also has specialization cost $-1$.
116For multi-argument generic types, the least-specialized polymorphic parameter sets the specialization cost, \eg{} the specialization cost of !pair(T, S*)! is $-1$ (from !T!) rather than $-2$ (from !S!).
117Specialization cost is not counted on the return type list; since $specialization$ is a property of the function declaration, a lower specialization cost prioritizes one declaration over another.
118User programmers can choose between functions with varying parameter lists by adjusting the arguments, but the same is not true in general of varying return types\footnote{In particular, as described in Section~\ref{expr-cost-sec}, cast expressions take the cheapest valid and convertable interpretation of the argument expression, and expressions are resolved as a cast to \lstinline{void}. As a result of this, including return types in the $specialization$ cost means that a function with return type \lstinline{T*} for some polymorphic type \lstinline{T} would \emph{always} be chosen over a function with the same parameter types returning \lstinline{void}, even for \lstinline{void} contexts, an unacceptably counter-intuitive result.}, so the return types are omitted from the $specialization$ element.
119Since both $vars$ and $specialization$ are properties of the declaration rather than any particular interpretation, they are prioritized less than the interpretation-specific conversion costs from Bilson's original 3-tuple.
120
121A final refinement I have made to the \CFA{} cost model is with regard to choosing among arithmetic conversions.
122The C standard \cite[\S{}6.3.1.8]{C11} states that the common type of !int! and !unsigned int! is !unsigned int! and that the common type of !int! and !long! is !long!, but does not provide guidance for making a choice among conversions.
123Bilson's \CFACC{} uses conversion costs based off Figure~\ref{bilson-conv-fig}.
124However, Bilson's design results in inconsistent and somewhat surprising costs, with conversion to the next-larger same-sign type generally (but not always) double the cost of conversion to the !unsigned! type of the same size.
125In the redesign, for consistency with the approach of the usual arithmetic conversions, which select a common type primarily based on size, but secondarily on sign, arcs in the new graph are annotated with whether they represent a sign change, and such sign changes are summed in a new $sign$ cost element that lexicographically succeeds $safe$.
126This means that sign conversions are approximately the same cost as widening conversions, but slightly more expensive (as opposed to less expensive in Bilson's graph), so maintaining the same signedness is consistently favoured.
127This refined conversion graph is shown in Figure~\ref{extended-conv-fig}.
128
129With these modifications, the current \CFA{} cost tuple is as follows:
130
131\begin{equation*}
132        (unsafe, poly, safe, sign, vars, specialization, reference)
133\end{equation*}
134
135\subsection{Expression Cost} \label{expr-cost-sec}
136
137The mapping from \CFA{} expressions to cost tuples is described by Bilson in \cite{Bilson03}, and remains effectively unchanged with the exception of the refinements to the cost tuple described above.
138Nonetheless, some salient details are repeated here for the sake of completeness.
139
140On a theoretical level, the resolver treats most expressions as if they were function calls.
141Operators in \CFA{} (both those existing in C and added features like constructors) are all modelled as function calls.
142In terms of the core argument-parameter matching algorithm, overloaded variables are handled the same as zero-argument function calls, aside from a different pool of candidate declarations and setup for different code generation.
143Similarly, an aggregate member expression !a.m! can be modelled as a unary function !m! that takes one argument of the aggregate type.
144Literals do not require sophisticated resolution, as in C the syntactic form of each implies their result types: !42! is !int!, !"hello"! is !char*!, \etc{}\footnote{Struct literals (\eg{} \lstinline|(S)\{ 1, 2, 3 \}| for some struct \lstinline{S}) are a somewhat special case, as they are known to be of type \lstinline{S}, but require resolution of the implied constructor call described in Section~\ref{ctor-sec}.}.
145
146Since most expressions can be treated as function calls, nested function calls are the primary component of complexity in expression resolution.
147Each function call has an \emph{identifier} that must match the name of the corresponding declaration, and a possibly-empty list of \emph{arguments}.
148These arguments may be function call expressions themselves, producing a tree of function-call expressions to resolve, where the leaf expressions are generally nullary functions, variable expressions, or literals.
149A single instance of expression resolution consists of matching declarations to all the identifiers in the expression tree of a top-level expression, along with inserting any conversions and satisfying all assertions necessary for that matching.
150The cost of a function-call expression is the sum of the conversion costs of each argument type to the corresponding parameter and the total cost of each subexpression, recursively calculated.
151\CFA{} expression resolution must produce either the unique lowest-cost interpretation of the top-level expression, or an appropriate error message if none exists.
152The cost model of \CFA{} precludes a greedy bottom-up resolution pass, as constraints and costs introduced by calls higher in the expression tree can change the interpretation of those lower in the tree, as in the following example:
153
154\begin{cfa}
155void f(int);
156double g$$$_1$$$(int);
157int g$$$_2$$$(long);
158
159f( g(42) );
160\end{cfa}
161
162Considered independently, !g!$_1$!(42)! is the cheapest interpretation of !g(42)!, with cost $(0,0,0,0,0,0,0)$ since the argument type is an exact match.
163However, in context, an unsafe conversion is required to downcast the return type of !g!$_1$ to an !int! suitable for !f!, for a total cost of $(1,0,0,0,0,0,0)$ for !f( g!$_1$!(42) )!.
164If !g!$_2$ is chosen, on the other hand, there is a safe upcast from the !int! type of !42! to !long!, but no cast on the return of !g!$_2$, for a total cost of $(0,0,1,0,0,0,0)$ for !f( g!$_2$!(42) )!; as this is cheaper, !g!$_2$ is chosen.
165Due to this design, all valid interpretations of subexpressions must in general be propagated to the top of the expression tree before any can be eliminated, a lazy form of expression resolution, as opposed to the eager expression resolution allowed by C or \CC{}, where each expression can be resolved given only the resolution of its immediate subexpressions.
166
167If there are no valid interpretations of the top-level expression, expression resolution fails and must produce an appropriate error message.
168If any subexpression has no valid interpretations, the process can be short-circuited and the error produced at that time.
169If there are multiple valid interpretations of a top-level expression, ties are broken based on the conversion cost, calculated as above.
170If there are multiple minimal-cost valid interpretations of a top-level expression, that expression is said to be \emph{ambiguous}, and an error must be produced.
171Multiple minimal-cost interpretations of a subexpression do not necessarily imply an ambiguous top-level expression, however, as the subexpression interpretations may be disambiguated based on their return type or by selecting a more-expensive interpretation of that subexpression to reduce the overall expression cost, as in the example above.
172
173The \CFA{} resolver uses type assertions to filter out otherwise-valid subexpression interpretations.
174An interpretation can only be selected if all the type assertions in the !forall! clause on the corresponding declaration can be satisfied with a unique minimal-cost set of satisfying declarations.
175Type assertion satisfaction is tested by performing type unification on the type of the assertion and the type of the declaration satisfying the assertion.
176That is, a declaration that satisfies a type assertion must have the same name and type as the assertion after applying the substitutions in the type environment.
177Assertion-satisfying declarations may be polymorphic functions with assertions of their own that must be satisfied recursively.
178This recursive assertion satisfaction has the potential to introduce infinite loops into the type resolution algorithm, a situation which \CFACC{} avoids by imposing a hard limit on the depth of recursive assertion satisfaction (currently 4); this approach is also taken by \CC{} to prevent infinite recursion in template expansion, and has proven to be effective and not unduly restrictive of the expressive power of \CFA{}.
179
180Cast expressions must be treated somewhat differently than functions for backwards compatibility purposes with C.
181In C, cast expressions can serve two purposes, \emph{conversion} (\eg{} !(int)3.14!), which semantically converts a value to another value in a different type with a different bit representation, or \emph{coercion} (\eg{} !void* p; (int*)p;!), which assigns a different type to the same bit value.
182C provides a set of built-in conversions and coercions, and user programmers are able to force a coercion over a conversion if desired by casting pointers.
183The overloading features in \CFA{} introduce a third cast semantic, \emph{ascription} (\eg{} !int x; double x; (int)x;!), which selects the overload that most-closely matches the cast type.
184However, since ascription does not exist in C due to the lack of overloadable identifiers, if a cast argument has an unambiguous interpretation as a conversion argument then it must be interpreted as such, even if the ascription interpretation would have a lower overall cost.
185This is demonstrated in the following example, adapted from the C standard library:
186
187\begin{cfa}
188unsigned long long x;
189(unsigned)(x >> 32);
190\end{cfa}
191
192In C semantics, this example is unambiguously upcasting !32! to !unsigned long long!, performing the shift, then downcasting the result to !unsigned!, at total cost $(1,0,3,1,0,0,0)$.
193If ascription were allowed to be a first-class interpretation of a cast expression, it would be cheaper to select the !unsigned! interpretation of !?>>?! by downcasting !x! to !unsigned! and upcasting !32! to !unsigned!, at a total cost of $(1,0,1,1,0,0,0)$.
194However, this break from C semantics is not backwards compatibile, so to maintain C compatibility, the \CFA{} resolver selects the lowest-cost interpretation of the cast argument for which a conversion or coercion to the target type exists (upcasting to !unsigned long long! in the example above, due to the lack of unsafe downcasts), using the cost of the conversion itself only as a tie-breaker.
195For example, in !int x; double x; (int)x;!, both declarations have zero-cost interpretations as !x!, but the !int x! interpretation is cheaper to cast to !int!, and is thus selected.
196Thus, in contrast to the lazy resolution of nested function-call expressions discussed above, where final interpretations for each subexpression are not chosen until the top-level expression is reached, cast expressions introduce eager resolution of their argument subexpressions, as if that argument was itself a top-level expression.
197
198\section{Resolution Algorithms}
199
200\CFA{} expression resolution is not, in general, polynomial in the size of the input expression, as shown in Section~\ref{resn-analysis-sec}.
201While this theoretical result is daunting, its implications can be mitigated in practice.
202\CFACC{} does not solve one instance of expression resolution in the course of compiling a program, but rather thousands; therefore, if the worst case of expression resolution is sufficiently rare, worst-case instances can be amortized by more-common easy instances for an acceptable overall runtime, as shown in Section~\ref{instance-expr-sec}.
203Secondly, while a programmer \emph{can} deliberately generate a program designed for inefficient compilation\footnote{See for instance \cite{Haberman16}, which generates arbitrarily large \CC{} template expansions from a fixed-size source file.}, source code tends to follow common patterns.
204Programmers generally do not want to run the full compiler algorithm in their heads, and as such keep mental shortcuts in the form of language idioms.
205If the compiler can be tuned to handle idiomatic code more efficiently, then the reduction in runtime for idiomatic (but otherwise difficult) resolution instances can make a significant difference in total compiler runtime.
206
207\subsection{Worst-case Analysis} \label{resn-analysis-sec}
208
209Expression resolution has a number of components that contribute to its runtime, including argument-parameter type unification, recursive traversal of the expression tree, and satisfaction of type assertions.
210
211If the bound type for a type variable can be looked up or mutated in constant time (as asserted in Table~\ref{env-bounds-table}), then the runtime of the unification algorithm to match an argument to a parameter is usually proportional to the complexity of the types being unified.
212In C, complexity of type representation is bounded by the most-complex type explicitly written in a declaration, effectively a small constant; in \CFA{}, however, polymorphism can generate more-complex types:
213
214\begin{cfa}
215        forall(otype T) pair(T) wrap(T x, T y);
216
217        wrap(wrap(wrap(1, 2), wrap(3, 4)), wrap(wrap(5, 6), wrap(7, 8)));
218\end{cfa}
219
220To resolve the outermost !wrap!, the resolver must check that !pair(pair(int))! unifies with itself, but at three levels of nesting, !pair(pair(int))! is more complex than either !pair(T)! or !T!, the types in the declaration of !wrap!.
221Accordingly, the cost of a single argument-parameter unification is $O(d)$, where $d$ is the depth of the expression tree, and the cost of argument-parameter unification for a single candidate for a given function call expression is $O(pd)$, where $p$ is the number of parameters.
222This bound does not, however, account for the higher costs of unifying two polymorphic type variables, which may in the worst case result in a recursive unification of all type variables in the expression (as discussed in Chapter~\ref{env-chap}).
223Since this recursive unification reduces the number of type variables, it may happen at most once, for an added $O(p^d)$ cost for a top-level expression with $O(p^d)$ type variables.
224
225Implicit conversions are also checked in argument-parameter matching, but the cost of checking for the existence of an implicit conversion is again proportional to the complexity of the type, $O(d)$.
226Polymorphism also introduces a potential expense here; for a monomorphic function there is only one potential implicit conversion from argument type to parameter type, while if the parameter type is an unbound polymorphic type-variable then any implicit conversion from the argument type could potentially be considered a valid binding for that type variable.
227\CFA{}, however, requires exact matches for the bound type of polymorphic parameters, removing this problem.
228An interesting question for future work is whether loosening this requirement incurs a significant compiler runtime cost in practice; preliminary results from the prototype system described in Chapter~\ref{expr-chap} suggest it does not.
229
230Considering the recursive traversal of the expression tree, polymorphism again greatly expands the worst-case runtime.
231Let $i$ be the number of candidate declarations for each function call; if all of these candidates are monomorphic, then there are no more than $i$ unambiguous interpretations of the subexpression rooted at that function call.
232Ambiguous minimal-cost subexpression interpretations may also be collapsed into a single \emph{ambiguous interpretation}, as the presence of such a subexpression interpretation in the final solution is an error condition.
233One safe pruning operation during expression resolution is to discard all subexpression interpretations with greater-than-minimal cost for their return type, as such interpretations cannot beat the minimal-cost interpretation with their return type for the overall optimal solution.
234As such, with no polymorphism, each declaration can generate no more than one minimal-cost interpretation with its return type, so the number of possible subexpression interpretations is $O(i)$ (note that in C, which lacks overloading, $i \leq 1$).
235With polymorphism, however, a single declaration (like !wrap! above) can have many concrete return types after type variable substitution, and could in principle have a different concrete return type for each combination of argument interpretations.
236Calculated recursively, the bound on the total number of candidate interpretations is $O(i^{p^d})$, each with a distinct type.
237
238Given these calculations of number of subexpression interpretations and matching costs, the upper bound on runtime for generating candidates for a single subexpression $d$ levels up from the leaves is $O( i^{p^d} \cdot pd )$.
239Since there are $O(p^d)$ subexpressions in a single top-level expression, the total worst-case cost of argument-parameter matching with the overloading and polymorphism features of \CFA{} is $O( i^{p^d} \cdot pd \cdot p^d )$.
240Since the size of the expression is $O(p^d)$, letting $n = p^d$ this simplifies to $O(i^n \cdot n^2)$
241
242This bound does not yet account for the cost of assertion satisfaction, however.
243\CFA{} uses type unification on the assertion type and the candidate declaration type to test assertion satisfaction; this unification calculation has cost proportional to the complexity of the declaration type after substitution of bound type variables; as discussed above, this cost is $O(d)$.
244If there are $O(a)$ type assertions on each declaration, there are $O(i)$ candidates to satisfy each assertion, for a total of $O(ai)$ candidates to check for each declaration.
245However, each assertion candidate may generate another $O(a)$ assertions, recursively until the assertion recursion limit $r$ is reached, for a total cost of $O((ai)^r \cdot d)$.
246Now, $a$ and $i$ are properties of the set of declarations in scope, while $r$ is defined by the language spec, so $(ai)^r$ is essentially a constant for purposes of expression resolution, albeit a very large one.
247It is not uncommon in \CFA{} to have functions with dozens of assertions, and common function names (\eg{} !?{}!, the constructor) can have hundreds of overloads.
248
249It is clear that assertion satisfaction costs can be very large, and in fact a method for heuristically reducing these costs is one of the key contributions of this thesis, but it should be noted that the worst-case analysis is a particularly poor match for actual code in the case of assertions.
250It is reasonable to assume that most code compiles without errors, as an actively-developed project is compiled many times, generally with relatively few new errors introduced between compiles.
251However, the worst-case bound for assertion satisfaction is based on recursive assertion satisfaction calls exceeding the limit, which is an error case.
252In practice, then, the depth of recursive assertion satisfaction should be bounded by a small constant for error-free code, which accounts for the vast majority of problem instances.
253
254Similarly, uses of polymorphism like those that generate the $O(d)$ bound on unification or the $O(i^{p^d})$ bound on number of candidates are rare, but not completely absent.
255This analysis points to type unification, argument-parameter matching, and assertion satisfaction as potentially costly elements of expression resolution, and thus profitable targets for algorithmic investigation.
256Type unification is discussed in Chapter~\ref{env-chap}, while the other aspects are covered below.
257
258\subsection{Argument-Parameter Matching} \label{arg-parm-matching-sec}
259
260Pruning possible interpretations as early as possible is one way to reduce the real-world cost of expression resolution, provided that a sufficient proportion of interpretations are pruned to pay for the cost of the pruning algorithm.
261One opportunity for interpretation pruning is by the argument-parameter type matching, but the literature \cite{Baker82,Bilson03,Cormack81,Ganzinger80,Pennello80,PW:overload} provides no clear answers on whether candidate functions should be chosen according to their available arguments, or whether argument resolution should be driven by the available function candidates.
262For programming languages without implicit conversions, argument-parameter matching is essentially the entirety of the expression resolution problem, and is generally referred to as overload resolution'' in the literature.
263All expression-resolution algorithms form a DAG of interpretations, some explicitly, some implicitly; in this DAG, arcs point from function-call interpretations to argument interpretations, as in Figure~\ref{res-dag-fig}
264
265\begin{figure}[h]
266        \centering
267        \begin{subfigure}[h]{3in}
268        \begin{cfa}
269        char *p$$$_1$$$;
270        int *p$$$_2$$$;
271
272        char *f$$$_1$$$(char*, int*);
273        double *f$$$_2$$$(int*, int*);
274
275        f$$$_A$$$( f$$$_B$$$( p$$$_A$$$, p$$$_B$$$ ), p$$$_C$$$ );
276        \end{cfa}
277        \end{subfigure}~\begin{subfigure}[h]{2.5in}
278        \includegraphics{figures/resolution-dag}
279        \end{subfigure}
280        \caption[Resolution DAG for a simple expression.]{Resolution DAG for a simple expression, annotated with explanatory subscripts. Functions that do not have a valid argument matching are covered with an \textsf{X}.} \label{res-dag-fig}
281\end{figure}
282
283Note that some interpretations may be part of more than one super-interpretation, as with the !p!$_2$ interpretation of !p!$_B$, while some valid subexpression interpretations, like the !f!$_2$ interpretation of !f!$_B$, are not used in any interpretation of their superexpression.
284
285Overload resolution was first seriously considered in the development of compilers for the Ada programming language, with different algorithms making various numbers of passes over the expression DAG, these passes being either top-down or bottom-up.
286Baker's algorithm~\cite{Baker82} takes a single pass from the leaves of the expression tree up, pre-computing argument candidates at each step.
287For each candidate function, Baker attempts to match argument types to parameter types in sequence, failing if any parameter cannot be matched.
288
289Bilson~\cite{Bilson03} similarly pre-computes argument-candidates in a single bottom-up pass in the original \CFACC{}, but then explicitly enumerates all possible argument combinations for a multi-parameter function.
290These argument combinations are matched to the parameter types of the candidate function as a unit rather than individual arguments.
291Bilson's approach is less efficient than Baker's, as the same argument may be compared to the same parameter many times, but does allow a more straightforward handling of polymorphic type-binding and tuple-typed expressions.
292
293Unlike Baker and Bilson, Cormack's algorithm~\cite{Cormack81} requests argument candidates that match the type of each parameter of each candidate function, in a single pass from the top-level expression down; memoization of these requests is presented as an optimization.
294As presented, this algorithm requires the parameter to have a known type, which is a poor fit for polymorphic type parameters in \CFA{}.
295Cormack's algorithm can be modified to request argument interpretations of \emph{any} type when provided an unbound parameter type, but this eliminates any pruning gains that could be provided by the algorithm.
296
297Ganzinger and Ripken~\cite{Ganzinger80} propose an approach (later refined by Pennello~\etal{}~\cite{Pennello80}) that uses a top-down filtering pass followed by a bottom-up filtering pass to reduce the number of candidate interpretations; they prove that a small number of such iterations is sufficient to converge to a solution for the overload resolution problem in the Ada programming language.
298Persch~\etal{}~\cite{PW:overload} developed a similar two-pass approach where the bottom-up pass is followed by the top-down pass.
299These approaches differ from Baker, Bilson, or Cormack in that they take multiple passes over the expression tree to yield a solution by applying filtering heuristics to all expression nodes.
300This approach of filtering out invalid types is unsuited to \CFA{} expression resolution, however, due to the presence of polymorphic functions and implicit conversions.
301
302Some other language designs solve the matching problem by forcing a bottom-up order.
303\CC{}, for instance, defines its overload-selection algorithm in terms of a partial order between function overloads given a fixed list of argument candidates, implying that the arguments must be selected before the function.
304This design choice improves worst-case expression resolution time by only propagating a single candidate for each subexpression, but type annotations must be provided for any function call which is polymorphic in its return type, and these annotations are often redundant.
305\CFA{}, by contrast, saves programmers from redundant annotations with its richer inference:
306
307\begin{cfa}
308forall(dtype T | sized(T)) T* malloc();
309
310int* p = malloc(); $\C{// Infers T = int from left-hand side}$
311\end{cfa}
312
313Baker~\cite{Baker82} left empirical comparison of different overload resolution algorithms to future work; Bilson~\cite{Bilson03} described an extension of Baker's algorithm to handle implicit conversions and polymorphism, but did not further explore the space of algorithmic approaches to handle both overloaded names and implicit conversions.
314This thesis closes that gap in the literature by performing performance comparisons of both top-down and bottom-up expression resolution algorithms, with results reported in Chapter~\ref{expr-chap}.
315
316\subsection{Assertion Satisfaction} \label{assn-sat-sec}
317
318The assertion satisfaction algorithm designed by Bilson~\cite{Bilson03} for the original \CFACC{} is the most-relevant prior work to this project.
319Before accepting any subexpression candidate, Bilson first checks that that candidate's assertions can all be resolved; this is necessary due to Bilson's addition of assertion satisfaction costs to candidate costs (discussed in Section~\ref{conv-cost-sec}).
320If this subexpression interpretation ends up not being used in the final resolution, then the (sometimes substantial) work of checking its assertions ends up wasted.
321Bilson's assertion checking function recurses on two lists, !need! and !newNeed!, the current declaration's assertion set and those implied by the assertion-satisfying declarations, respectively, as detailed in the pseudo-code below (ancillary aspects of the algorithm are omitted for clarity):
322
323\begin{cfa}
324List(List(Declaration)) checkAssertions(
325                List(Assertion) need, List(Assertion) newNeed, List(Declaration) have,
326                Environment env ) {
327        if ( is_empty(need) ) {
328                if ( is_empty( newNeed ) ) return { have };
329                else return checkAssertions( newNeed, {}, have, env );
330        }
331
333        Type adjType = substitute( a.type, env );
334        List(Declaration) candidates = decls_matching( a.name );
335        List(List(Declaration)) alternatives = {}
336        for ( Declaration c : candidates ) {
337                Environment newEnv = env;
338                if ( unify( adjType, c.type, newEnv ) ) {
339                        append( alternatives,
340                                checkAssertions(
341                                        tail(need), append(newNeed, c.need), append(have, c), newEnv ) );
342                }
343        }
344        return alternatives;
345}
346\end{cfa}
347
348One shortcoming of this approach is that if an earlier assertion has multiple valid candidates, later assertions may be checked many times due to the structure of the recursion.
349Satisfying declarations for assertions are not completely independent of each other, since the unification process may produce new type bindings in the environment, and these bindings may not be compatible between independently-checked assertions.
350Nonetheless, with the environment data-structures discussed in Chapter~\ref{env-chap}, I have found it more efficient to produce a list of possibly-satisfying declarations for each assertion once, then check their respective environments for mutual compatibility when combining lists of assertions together.
351
352Another improvement I have made to the assertion resolution scheme in \CFACC{} is to consider all assertion-satisfying combinations at one level of recursion before attempting to recursively satisfy any !newNeed! assertions.
353Monomorphic functions are cheaper than polymorphic functions for assertion satisfaction because they are an exact match for the environment-adjusted assertion type, whereas polymorphic functions require an extra type binding.
354Thus, if there is any mutually-compatible set of assertion-satisfying declarations that does not include any polymorphic functions (and associated recursive assertions), then the optimal set of assertions does not require any recursive !newNeed! satisfaction.
355More generally, due to the \CFA{} cost-model changes I introduced in Section~\ref{conv-cost-sec}, the conversion cost of an assertion-satisfying declaration is no longer dependent on the conversion cost of its own assertions.
356As such, all sets of mutually-compatible assertion-satisfying declarations can be sorted by their summed conversion costs, and the recursive !newNeed! satisfaction pass is required only to check the feasibility of the minimal-cost sets.
357This optimization significantly reduces wasted work relative to Bilson's approach, as well as avoiding generation of deeply-recursive assertion sets, for a significant performance improvement relative to Bilson's \CFACC{}.
358
359Making the conversion cost of an interpretation independent of the cost of satisfying its assertions has further benefits.
360Bilson's algorithm checks assertions for all subexpression interpretations immediately, including those that are not ultimately used; I have developed a \emph{deferred} variant of assertion checking that waits until a top-level interpretation has been generated to check any assertions.
361If the assertions of the minimal-cost top-level interpretation cannot be satisfied then the next-most-minimal-cost interpretation's assertions are checked, and so forth until a minimal-cost satisfiable interpretation (or ambiguous set thereof) is found, or no top-level interpretations are found to have satisfiable assertions.
362In the common case where the code actually does compile, this saves the work of checking assertions for ultimately-rejected interpretations, though it does rule out some pruning opportunities for subinterpretations with unsatisfiable assertions or which are more expensive than a minimal-cost polymorphic function with the same return type.
363The experimental results in Chapter~\ref{expr-chap} indicate that this is a worthwhile trade-off.
364
365Optimizing assertion satisfaction for common idioms has also proved effective in \CFA{}; the code below is an unexceptional print statement derived from the \CFA{} test suite that nonetheless is a very difficult instance of expression resolution:
366
367\begin{cfa}
368sout | "one" | 1 | "two" | 2 | "three" | 3 | "four" | 4 | "five" | 5 | "six" | 6
369        | "seven" | 7 | "eight" | 8 | "nine" | 9 | "ten" | 10 | "end" | nl | nl;
370\end{cfa}
371
372The first thing that makes this expression so difficult is that it is 23 levels deep; Section~\ref{resn-analysis-sec} indicates that the worst-case bounds on expression resolution are exponential in expression depth.
373Secondly, the !?|?! operator is significantly overloaded in \CFA{} --- there are 74 such operators in the \CFA{} standard library, and while 9 are arithmetic operators inherited from C, the rest are polymorphic I/O operators that look similar to:
374
375\begin{cfa}
376forall( dtype ostype | ostream( ostype ) )
377ostype& ?|? ( ostype&, int );
378\end{cfa}
379
380Note that !ostream! is a trait with 25 type assertions, and that the output operators for the other arithmetic types are also valid for the !int!-type parameters due to implicit conversions.
381On this instance, deferred assertion satisfaction saves wasted work checking assertions on the wrong output operators, but does nothing about the 23 repeated checks of the 25 assertions to determine that !ofstream! (the type of !sout!) satisfies !ostream!.
382
383To solve this problem, I have developed a \emph{cached} variant of assertion checking.
384During the course of checking the assertions of a single top-level expression, the results are cached for each assertion checked.
385The search key for this cache is the assertion declaration with its type variables substituted according to the type environment to distinguish satisfaction of the same assertion for different types.
386This adjusted assertion declaration is then run through the \CFA{} name-mangling algorithm to produce an equivalent string-type key.
387
388One superficially-promising optimization which I did not pursue is caching assertion-satisfaction judgements between top-level expressions.
389This approach would be difficult to correctly implement in a \CFA{} compiler, due to the lack of a closed set of operations for a given type.
390New declarations related to a particular type can be introduced in any lexical scope in \CFA{}, and these added declarations may cause an assertion that was previously satisfiable to fail due to an introduced ambiguity.
391Furthermore, given the recursive nature of assertion satisfaction and the possibility of this satisfaction judgement depending on an inferred type, an added declaration may break satisfaction of an assertion with a different name and that operates on different types.
392Given these concerns, correctly invalidating a cross-expression assertion satisfaction cache for \CFA{} is a non-trivial problem, and the overhead of such an approach may possibly outweigh any benefits from such caching.
393
394The assertion satisfaction aspect of \CFA{} expression resolution bears some similarity to satisfiability problems from logic, and as such other languages with similar trait and assertion mechanisms make use of logic-program solvers in their compilers.
395For instance, Matsakis~\cite{Matsakis17} and the Rust team have developed a PROLOG-based engine to check satisfaction of Rust traits.
396The combination of the assertion satisfaction elements of the problem with the conversion-cost model of \CFA{} makes this logic-solver approach difficult to apply in \CFACC{}, however.
397Expressing assertion resolution as a satisfiability problem ignores the cost optimization aspect, which is necessary to decide among what are often many possible satisfying assignments of declarations to assertions.
398On the other hand, the deeply-recursive nature of the satisfiability problem makes it difficult to adapt to optimizing solver approaches such as linear programming.
399To maintain a well-defined programming language, any optimization algorithm used must provide an exact (rather than approximate) solution; this constraint also rules out a whole class of approximately-optimal generalized solvers.
400As such, I opted to continue Bilson's approach of designing a bespoke solver for \CFA{} assertion satisfaction, rather than attempting to re-express the problem in some more general formalism.
401
402\section{Conclusion \& Future Work} \label{resn-conclusion-sec}
403
404As the results in Chapter~\ref{expr-chap} show, the algorithmic approaches I have developed for \CFA{} expression resolution are sufficient to build a practically-performant \CFA{} compiler.
405This work may also be of use to other compiler construction projects, notably to members of the \CC{} community as they implement the new Concepts \cite{C++Concepts} standard, which includes type assertions similar to those used in \CFA{}, as well as the C-derived implicit conversion system already present in \CC{}.
406
407I have experimented with using expression resolution rather than type unification to check assertion satisfaction; this variant of the expression resolution problem should be investigated further in future work.
408This approach is more flexible than type unification, allowing for conversions to be applied to functions to satisfy assertions.
409Anecdotally, this flexibility matches user-programmer expectations better, as small type differences (\eg{} the presence or absence of a reference type, or the usual conversion from !int! to !long!) no longer break assertion satisfaction.
410Practically, the resolver prototype discussed in Chapter~\ref{expr-chap} uses this model of assertion satisfaction, with no apparent deficit in performance; the generated expressions that are resolved to satisfy the assertions are easier than the general case because they never have nested subexpressions, which eliminates much of the theoretical differences between unification and resolution.
411The main challenge to implement this approach in \CFACC{} is applying the implicit conversions generated by the resolution process in the code-generation for the thunk functions that \CFACC{} uses to pass type assertions to their requesting functions with the proper signatures.
412
413One \CFA{} feature that could be added to improve the ergonomics of overload selection is an \emph{ascription cast}; as discussed in Section~\ref{expr-cost-sec}, the semantics of the C cast operator are to choose the cheapest argument interpretation which is convertable to the target type, using the conversion cost as a tie-breaker.
414An ascription cast would reverse these priorities, choosing the argument interpretation with the cheapest conversion to the target type, only using interpretation cost to break ties\footnote{A possible stricter semantics would be to select the cheapest interpretation with a zero-cost conversion to the target type, reporting a compiler error otherwise.}.
415This would allow ascription casts to the desired return type to be used for overload selection:
416
417\begin{cfa}
418int f$$$_1$$$(int);
419int f$$$_2$$$(double);
420int g$$$_1$$$(int);
421double g$$$_2$$$(long);
422
423f((double)42)$\C[4.5in]{// select f$$_2$$ by argument cast}$
424(as double)g(42); $\C[4.5in]{// select g$$_2$$ by return ascription cast}$
425(double)g(42); $\C[4.5in]{// select g$$_1$$ NOT g$$_2$$ because of parameter conversion cost}$
426\end{cfa}
427
428Though performance of the existing resolution algorithms is promising, some further optimizations do present themselves.
429The refined cost model discussed in Section~\ref{conv-cost-sec} is more expressive, but requires more than twice as many fields; it may be fruitful to investigate more tightly-packed in-memory representations of the cost-tuple, as well as comparison operations that require fewer instructions than a full lexicographic comparison.
430Integer or vector operations on a more-packed representation may prove effective, though dealing with the negative-valued $specialization$ field may require some effort.
431
432Parallelization of various phases of expression resolution may also be useful.
433The algorithmic variants I have introduced for both argument-parameter matching and assertion satisfaction are essentially divide-and-conquer algorithms, which solve subproblem instances for each argument or assertion, respectively, then check mutual compatibility of the solutions.
434While the checks for mutual compatibility are naturally more serial, there may be some benefit to parallel resolution of the subproblem instances.
435
436The resolver prototype built for this project and described in Chapter~\ref{expr-chap} would be a suitable vehicle for many of these further experiments, and thus a technical contribution of continuing utility.
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