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1\chapter{Resolution Algorithms}
2\label{c:content2}
3
4The \CFA's type-system provides expressive polymorphism: variables can be overloaded, functions can be overloaded by argument and return types, tuple types, generic (polymorphic) functions and types (aggregates) can have multiple type parameters with associated restrictions, and C's multiple implicit type-conversions must be respected.
5This generality leads to internal complexity and correspondingly higher compilation cost.
6The reason is that the type resolver must analyze \emph{each} component of an expression with many possible forms of overloading, type restrictions, and conversions.
7Designing a ruleset that behaves as expected, \ie matches with C programmer expectations, and implementing an efficient algorithm is a very challenging task.
8
9My first work on the \CFA type-system was as a co-op student.
10At that time, compiling a medium-sized \CFA program using advanced polymorphism took multiple minutes (5+ minutes)~\cite[\S~5]{Yu20}.
11After finding and fixing the following resolution problems:
12\begin{enumerate}[itemsep=0pt]
13\item
14new AST data structure
15\item
16special symbol table and argument-dependent lookup
17\item
18late assertion-satisfaction
19\item
20revised function-type representation
21\item
22skip pruning on expressions for function types
23\end{enumerate}
24\VRef[Table]{t:SelectedFileByCompilerBuild} shows the improvement of selected tests with accumulated reductions in compile time across each of the 5 fixes.
25The large reduction in compilation times, significantly improved the development of the \CFA's runtime because of its frequent compilation cycles.
26
27\begin{table}[htb]
28\centering
29\caption{Compile time of selected files by compiler build in seconds}
30\label{t:SelectedFileByCompilerBuild}
31\lstset{deletekeywords={mutex,thread}}
32\setlength{\extrarowheight}{1pt}
33\vspace*{-4pt}
34\begin{tabular}{l|r|rrrrr}
35                                & \multicolumn{1}{c|}{Original} & \multicolumn{5}{c}{Accumulative fixes}        \\
36\cline{3-7}                             
37Test case               & \multicolumn{1}{c|}{resolver} & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{2} & \multicolumn{1}{c}{3} & \multicolumn{1}{c}{4} & \multicolumn{1}{c}{5} \\
38\hline
39@lib/fstream@   & 86.4  & 25.2  & 10.8  & 9.5   & 7.8   & 7.1   \\
40@lib/mutex@             & 210.4 & 77.4  & 16.7  & 15.1  & 12.6  & 11.7  \\
41@lib/vector@    & 17.2  & 8.9   & 3.1   & 2.8   & 2.4   & 2.2   \\
42@lib/stdlib@    & 16.6  & 8.3   & 3.2   & 2.9   & 2.6   & 2.4   \\
43@test/io2@              & 300.8 & 53.6  & 43.2  & 27.9  & 19.1  & 16.3  \\
44@test/thread@   & 210.9 & 73.5  & 17.0  & 15.1  & 12.6  & 11.8  \\
45\end{tabular}
46\smallskip
47\newline
48Results are average of 5 runs (3 runs if time exceeds 100 seconds)
49\end{table}
50
51Since then, many new features utilizing the expressiveness of \CFA's type system have been implemented, such as generic container types similar to those in \CC's standard template library.
52During the development of multiple \CFA programs and libraries, more weaknesses and design flaws have been discovered within the type system.
53Some of those problems arise from the newly introduced language features described in the previous chapter.
54In addition, fixing unexpected interactions within the type system has presented challenges.
55This chapter describes in detail the type-resolution rules currently in use and some major problems that have been identified.
56Not all of those problems have solutions, because fixing them may require redesigning parts of the \CFA type system at a larger scale, which correspondingly affects the language design.
57
58
59\section{Expression Cost-Model}
60
61\CFA has been using an expression cost-model to resolve ambiguity of overloaded expressions from the very beginning.
62Most \CFA operators can be overloaded in \CFA\footnote{
63Excluding the comma operator, short-circuit logical operators \lstinline{&&} and \lstinline{||} (control structures) and ternary conditional operator \lstinline{?} (control structure).
64While \CC overloads the short-circuit logical operators, the overloaded operators do not exhibit the short-circuit semantics, which is confusing.};
65hence, they are treated the same way as other function calls, and the same rules for overload resolution must apply to them as well.
66
67In \CFA, candidates of an overloaded expression are ranked by numerical cost elements, that accounts for the type conversions needed from the argument type to the corresponding declared function parameter type, as well as the polymorphic types and restrictions introduces by the @forall@ clause.
68Currently the expression cost used in \CFA has the following components, ranked from higher to lower by importance:
69\begin{enumerate}
70\item \textbf{Unsafe} cost representing a narrowing conversion of arithmetic types, e.g.
71@int@ to @short@, and qualifier-dropping conversions for pointer and reference types;
72\item \textbf{Polymorphic} cost where the function parameter type is or contains a polymorphic type variable;
73\item \textbf{Safe} cost representing a widening conversion e.g.
74@short@ to @int@, qualifier-adding conversions for pointer and reference types, and value conversion for enumeration constants.
75\item \textbf{Variable} cost that counts the number of polymorphic variables, if any, introduced by the forall clause in the function declaration;
76\item \textbf{Specialization} cost that counts the number of restrictions introduced by the type assertions.
77\end{enumerate}
78The comparison of cost tuple is by lexicographical order, starting from the highest importance term (unsafe cost) and the lower one has lower cost, with ties going to the second term (polymorphic cost) and so on.
79At a sub-expression level, the lowest cost candidate for each result type is included as a possible interpretation of the expression; at the top level all possible interpretations of different types are considered and the overall lowest cost is selected as the final interpretation of the expression.
80
81In many languages that support function and operator overloading, such as \CC and Java, a partial ordering system decides which interpretation of the expression gets selected, which means that sometimes the candidates are incomparable (none of those are considered a better match) and only when one candidate is considered better than all others (maximal with respect to the partial order) is the expression unambiguous.
82
83In \CFA trying to use such a system is problematic because of the presence of return type overloading of functions, and overloading of variables.
84The resolution algorithms used in \CC and Java are greedy, as they select the best match for each sub-expression without considering the higher level ones, and therefore at each step of resolution, the arguments are already given unique interpretations, so the ordering only needs to consider comparing different sets of conversion targets (function parameter types) on the same set of input.
85However, in \CFA expression resolution considers multiple interpretations of argument sub-expressions with different types, so it is possible that both the selected function and the set of arguments are different, and cannot be compared if we choose to use some kind of partial ordering system.
86Since this situation can arise quite often in \CFA, even in the simplest form such as an expression \textbf{f(a)} where both the function name \textbf{f} and variable name \textbf{a} are overloaded.
87We do not want the resolution algorithm to report too many expressions as ambiguous (which would therefore be compilation errors) and restrict the flexibility of \CFA by too much.
88The previous documentations and papers on \CFA expression resolution never explained why such a cost system is used; this could be a plausible guess of the original motivation of introducing the cost system to \CFA.
89
90On the contrary, using such a cost-based model can sometimes make \CFA expression resolution too permissive; the system will always attempt to select the lowest cost option, and only when there are multiple options tied at the lowest cost it reports the expression as ambiguous.
91With so many elements in the cost tuple, ties are expected to be uncommon.
92Other than the case of multiple exact matches which would all have cost of zero, incomparable candidates under a partial ordering of being more specific can often have different expression costs since different kinds of implicit conversions are involved, resulting in seemingly arbitrary overload selections.
93
94Ada is another programming language that has overloading based on return type.
95Although Ada also allows implicit type conversions of function arguments, it is fairly conservative on resolving ambiguities.
96There are only two "preference" rules in Ada overload resolution, one for primitive arithmetic operators and one for universal access types (analogous to void* in C); any other cases where an expression have multiple legal interpretations are considered ambiguous.
97The current overload resolution system for \CFA is on the other end of the spectrum, as it tries to order every legal interpretations of an expression and chooses the best one according to cost, occasionally giving unexpected results.
98
99There are currently at least three different situations where the polymorphic cost element of the cost model does not yield a candidate selection that is clearly justifiable, and one of them is straight up wrong.
100Here are those three cases:
101\begin{enumerate}
102\item Polymorphic exact match versus non-polymorphic inexact match: consider the following declarations
103
104\begin{cfa}
105forall (T) void f (T); // 1
106void f (long); // 2
107
108f (42); // currently selects 2
109\end{cfa}
110Under the current cost model, option 1 incurs a polymorphic cost from matching the argument type \textbf{int} to type variable \textbf{T}, and option 2 incurs a safe cost from integer promotion of type \textbf{int} to \textbf{long}.
111Since polymorphic cost is ranked above safe conversion cost, option 2 is considered to have lower cost and gets selected.
112
113
114In contrast, the template deduction and overload resolution rules in \CC selects option 1 instead (converting forall to the equivalent function template declaration).
115\CC performs template argument deduction and overload candidate ranking in two separate steps: in the first step the type parameters are deduced for each primary function template, and if the corresponding template instantiation succeeds, the resulting function prototype is added to the resolution candidate set.
116In the second step the implicit conversions (if any) applied to argument types are compared after taking away top-level qualifiers and references, and it prefers an exact match, followed by basic type promotions (roughly corresponds to safe conversion in \CFA), and then other kinds of conversions (roughly corresponds to unsafe conversion in \CFA).
117Only when the type conversions are the same does it prioritize a non-template candidate.
118In this example, option 1 produces the prototype void f(int) which gives an exact match and therefore takes priority.
119The \CC resolution rules effectively makes option 2 a specialization that only applies to type \textbf{long} exactly,\footnote{\CC does have explicit template specializations, however they do not participate directly in overload resolution and can sometimes lead to unintuitive results.} while the current \CFA rules make option 2 apply for all integral types below \textbf{long}.
120Such a discrepancy could be explained as a design decision that since \CFA polymorphic functions are real entities and are separately compiled, calling them would require passing type information and thus have an actual runtime cost.
121
122\item Having a lower total polymorphic cost does not always mean a function is more specialized.
123The following example is taken from Aaron Moss's thesis, which discusses some improvements to the \CFA expression cost model, where he claims the following function prototypes are increasingly more constrained:
124
125\begin{cfa}
126forall(T, U) void f(T, U); //1, polymorphic
127forall(T) void f(T, T); //2, less polymorphic
128forall(T) void f(T, int); //3, even less polymorphic
129forall(T) void f(T*, int); //4, least polymorphic
130\end{cfa}
131
132This argument is not entirely correct.
133Although it is true that both the sequence 1,2 and 1,3,4 are increasingly more constrained on the argument  types, the option 2 is not comparable to either of option 3 or 4; they actually describe independent constraints on the two arguments.
134In natural language, option 3 says that the second argument must have type \textbf{int}, while option 2 says that the two arguments must have the same type.
135These two constraints can independently be satisfied, therefore neither should be considered a better match when trying to resolve a call to f with argument types (int, int), and reporting such an expression as ambiguous is the most appropriate action.
136This is a limitation of using a numerical cost value as it cannot these logically complicated cases.
137
138\item Finally, the introduction of generic types means that it may require more type variables to describe a more specific type and that means simply counting the number of polymorphic type variables is no longer correct in general to order the function candidates as being more constrained.
139Suppose we have a generic pair type defined and writing a function that takes an arbitrary pair would require using two type variables
140\begin{cfa}
141forall (T,U) void f (pair(T,U)); // 1
142\end{cfa}
143and compare that with a function that takes any type at all:
144\begin{cfa}
145forall (T) void f (T); // 2
146\end{cfa}
147
148Passing a pair variable to f gives a cost of 1 poly, 2 variable for the pair overload, and a cost of 1 poly, 1 variable for the unconstrained overload.
149Clearly we would like the former to be chosen but the cost model cannot handle that correctly.
150
151\end{enumerate}
152
153These inconsistencies do not seem to be easily solvable and currently the \CFA codebase has to work around with these known defects.
154One potential path that could possibly be taken is a mix of the conversion cost and \CC-like partial ordering of specializations.
155Observe that in the \CFA cost tuple, the first three elements (unsafe, polymorphic and safe conversions) are related to the argument types, while the other elements (polymorphic variable and assertion counts) are properties of the function declarations independent of the arguments.
156This means it may be reasonable to have an ordering that compares the argument conversion costs first and uses the partial ordering of specializations as a tiebreaker.
157The algorithm used by \CC template specialization ordering can be applied for \CFA with some slight modifications.
158
159
160At the meantime, some other improvements have been made to the expression cost system.
161Most notably, the expression resolution algorithm now consistently uses the globally minimal cost interpretation, as discussed in a previous technical report.
162While implementing the change, there are also two detailed issues that need to be addressed for the new rules to fully work.
163
164The first one deals with an idiom commonly used in \CFA that would cause a lot of overloads to have equal costs.
165These kinds of expressions are so ubiquitous in \CFA code that we do not want them to be deemed ambiguous in the language.
166Many C library functions have multiple versions for different argument types, for example there are absolute value functions defined for basic arithmetic types with different names, since C does not support any kind of overloading:
167\begin{cfa}
168int abs (int);
169long labs (long);
170double fabs (double);
171float fabsf (float);
172\end{cfa}
173It is cumbersome for the programmers to remember all these different function names and select the correct ones, and even worse, if the incorrect version is picked, the program still compiles but with undesired conversions, which can sometimes even change the result, such as using the int version for floating point argument.
174In \CFA all of these functions are renamed to simply @abs@.
175This causes multiple overloads to have the same total cost when some conversion is needed.
176For example @long x = abs(42);@ could be either calling @long abs(long)@ with the argument 42 converted to @long@ or calling @int abs(int)@ and converting the result to @long@.
177In this example the choice could be arbitrary because both yields identical results.
178In some other cases, the choice can have an actual impact on the final result.
179While testing the effects of using the updated cost rule we found this piece of code in \CFA standard library:
180\begin{cfa}
181static inline __readyQ_avg_t __to_readyQ_avg(unsigned long long intsc) {
182        if(unlikely(0 == intsc)) return 0.0;
183        else return log2(intsc); // implicit conversion happens here
184} // __readyQ_avg_t is defined to be double
185\end{cfa}
186This is a helper function for performance logging that calculate the geometric mean of a counter value, and it does so by summing up the logarithm value of the counter.
187The function @log2@ is similarly overloaded in \CFA for both integer and floating point types, however in this case, the integer overload returns an integer, so the fractional part of logarithm is truncated.
188With the previous cost rules the integer version of @log2@ is selected, and when experimenting the updated cost rules this got picked up as an ambiguous expression at first.
189I reported this issue to the author of library code and got the reply that the expectation was that \CFA would choose the floating point overload, by the return type overloading selection.
190This mistake went unnoticed since it is only inside a performance logging function and does not serve any critical purposes, and the only effect it has caused is that the performance data becomes inaccurate as the fractional parts got truncated before the sum.
191Investigating this example leads to the decision that matching the return type higher up in the expression tree is prioritized, in case the total expression cost is equal.
192
193Another change addresses the issue that C arithmetic expressions have unique meanings governed by the arithmetic promotion rules, however in \CFA they are all modelled as function calls for overload resolution purposes.
194The previous, partially greedy resolution rules will pick the locally optimal match and it matches the C rules naturally.
195Care needs to be taken to maintain the C semantics when switching to the total expression cost approach.
196
197
198This problem is already partially recognized, when Aaron Moss suggested overload resolution by total cost, in the form of handling cast expressions.
199To quote directly the example:
200
201If 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.
202This is demonstrated in the following example, adapted from the C standard library:
203\begin{cfa}
204unsigned long long x;
205(unsigned)(x >> 32);
206\end{cfa}
207In C semantics, this example is unambiguously upcasting 32 to @unsigned long long@, performing the shift, then downcasting the result to @unsigned@, at cost (1, 0, 3, 1, 0, 0, 0).
208If 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).
209However, this break from C semantics is not backwards compatible, 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.
210
211However, a cast expression is not necessary to have such inconsistency to C semantics.
212With any implicit argument-parameter type conversion in function calls we can replicate this issue without an explicit cast.
213For example
214\begin{cfa}
215unsigned long long x;
216void f (unsigned);
217f (x >> 32);
218\end{cfa}
219This has the same effect as using an explicit cast to coerce the type of expression @x >> 32@ to @unsigned@.
220This shows that fundamentally the problem is not coming from the cast expressions, but modelling the C built-in operators as overloaded functions.
221A different rule is enforced in selecting the built-in function candidates to fix this problem.
222If an expression has any legal interpretations as a C built-in operation, only the lowest cost one is kept, regardless of the result types.
223
224
225\section{Type Unification}
226
227Type unification is the algorithm that assigns values to each (free) type parameters such that the types of the provided arguments and function parameters match.
228
229
230\CFA does not attempt to do any type \textit{inference}: it has no anonymous functions (i.e.
231lambdas, commonly found in functional programming and also used in \CC and Java), and the variable types must all be explicitly defined (no auto typing).
232This makes the unification problem more tractable in \CFA as the argument types at each call site are usually all specified.
233There is a single exception case, which happens when the function return type contains a free type variable that does not occur in any of the argument types, and subsequently passed into the parent expression.
234A top level expression whose type still contains an unbounded type variable is considered ill-formed as such expression is inherently ambiguous.
235
236The unification algorithm in \CFA is originally presented in Richard Bilson's thesis and it has remained as the basis of the algorithm currently in use.
237Some additions have been made in order to accommodate for the newly added type features to the language.
238To summarize, the \CFA type unification has two minor variants: an \textit{exact} mode and an \textit{inexact} mode.
239The inexact mode is applied at top level argument-parameter matching, and attempts to find an assignment to the type variables such that the argument types can be converted to parameter types with minimal cost as defined in the previous section.
240The exact mode is required since the type matching algorithm operates recursively and the inner types often have to match exactly, for example there is no common type for the pointer types \textbf{int*} and \textbf{long*} while there is for \textbf{int} and \textbf{long}.
241With the introduction of generic record types, the parameters must match exactly as well; currently there are no covariance or contravariance supported for the generics.
242
243One simplification was made to the \CFA language that makes modelling the type system easier: polymorphic function pointer types are no longer allowed in declarations.
244The polymorphic function declarations themselves are still treated as function pointer types internally, however the change means that formal parameter types can no longer be polymorphic.
245Previously it is possible to write function prototypes such as
246\begin{cfa}
247void f( forall( T | { T -?( T ); } ) T (*p)( T, T ) );
248\end{cfa}
249Notably, the unification algorithm implemented in the \CFA compiler has never managed to trace the assertion parameters on the formal types at all, and the problem of determining if two assertion sets are compatible may very likely be undecidable in general, given the ability of synthesizing more complicated types by the nesting of generics.
250Eventually, the reason of not allowing such constructs is that they mostly do not provide useful type features for actual programming tasks.
251A subroutine of a program operates on the arguments provided at the call site together with (if any) local and global variables, and even though the subroutine can be polymorphic, the types will be supported at each call site.
252On each invocation the types to be operate on can be determined from the arguments provided, and therefore there should not be a need to pass a polymorphic function pointer, which can take any type in principle.
253
254For example, consider a polymorphic function that takes one argument of type \textbf{T} and another pointer to a subroutine
255\begin{cfa}
256forall (T) void f (T x, forall (U) void (*g) (U));
257\end{cfa}
258Making \textbf{g} polymorphic in this context would almost certainly be unnecessary, since it can only be called inside the body of \textbf{f} and the types of the argument would have been known anyways, although it can potentially depend on \textbf{T}.
259Moreover, requesting a function parameter to be able to potentially work on any input type at all would always impose too much constraint on the arguments, as it only needs to make each calls inside the body of \textbf{f} valid.
260
261Rewriting the prototype to
262\begin{cfa}
263forall (T) void f (T x, void (*g) (T));
264\end{cfa}
265will be sufficient (or potentially, some compound type synthesized from \textbf{T}), in which case \textbf{g} is no longer a polymorphic type on itself.
266The "monomorphization" conversion is readily supported in \CFA, either by explicitly assigning a polymorphic function name to a compatible function pointer type, or implicitly done in deducing assertion parameters (which will be discussed in the next section).
267Such technique can be directly applied to argument passing, which is essentially just assignment to function parameter variables.
268There might be some edge cases where the supplied subroutine \textbf{g} is called on arguments of different types inside the body of \textbf{f} and so declared as polymorphic, but such use case is rare and the benefit of allowing such constructs seems to be minimal in practice.
269
270
271The result of this change is that the unification algorithm no longer needs to distinguish "open" and "closed" type variables, as the latter is not allowed to exist.
272The only type variables that need to be handled are those introduced by the \textbf{forall} clause from the function prototype.
273The subtype relationship between function types is now also rendered redundant since none of the function parameter or return types can be polymorphic, and no basic types or non-polymorphic function types are subtypes of any other type.
274Therefore the goal of (exact) type unification now simply becomes finding a substitution that produces identical types.
275The assertion set need to be resolved is also always just the declarations on the function prototype, which also simplifies the assertion satisfaction algorithm by a bit, as will be discussed further in the next section.
276
277The type unification results are stored in a type environment data structure, which represents all the type variables currently in scope as equivalent classes, together with their bound types and some other extra information, such as whether the bound type is allowed to be opaque (i.e.
278a forward declaration without definition in scope), and whether the bounds are allowed to be widened.
279In the more general approach commonly used in functional languages, the unification variables are given a lower bound and an upper bound to account for covariance and contravariance of types.
280\CFA currently does not implement any variance with its generic types, and does not allow polymorphic function types, therefore no explicit upper bound is needed and one simple binding value for each equivalence class suffices.
281However, since type conversions are allowed in \CFA, the type environment needs to keep track on which type variables are allowed conversions.
282This behavior is notably different from \CC template argument deduction which enforces an exact match everywhere unless the template argument types are explicitly given.
283For example, a polymorphic maximum function in \CFA can be called with arguments of different arithmetic types and the result follows the usual arithmetic conversion rules, while such expression is not allowed by \CC:
284\begin{cfa}
285forall (T | {int ?<? (T, T); }) T max (T, T);
286
287max (42, 3.14); // OK, T=double; requires explicit type annotation in C++ such as max<double>(42, 3.14);
288\end{cfa}
289The current \CFA documentation does not include a formal set of rules for type unification.
290In practice, the algorithm implemented in the \CFA translator can be summarized as follows, given a function signature forall$(T_1,..., T_n) f(p_1, ..., p_m)$ and argument types $(a_1, ..., a_m)$, the unification algorithm performs the following steps: \footnote{This assumes argument tuples are already expanded to the individual components.}
291\begin{enumerate}
292\item The type environment is initialized as the union of all type environments of the arguments, plus $(T_1,...,T_n)$ as free variables.
293The inclusion of argument environments serves the purpose of resolving polymorphic return types that needs to be deduced.
294\item Initially, all type variables
295\end{enumerate}
296
297
298\section{Satisfaction of Assertions}
299
300The assertion satisfaction problem greatly increases the complexity of \CFA expression resolution.
301Past experiments have shown that the majority of time is spent in resolving the assertions for those expressions that takes the longest time to resolve.
302Even though a few heuristics-based optimizations are introduced to the compiler now, this remains to be the most costly part of compiling a \CFA program.
303The major difficulty of resolving assertions is that the problem can become recursive, since the expression used to satisfy an outstanding assertion can have its own assertions, and in theory this can go on indefinitely.
304Detecting infinite recursion cases in general is not algorithmically possible and it is not attempted in the compiler.
305Instead, a fixed maximum depth of recursive assertions is imposed.
306This approach is also taken by \CC compilers as template argument deduction is also similarly undecidable in general.
307
308
309In previous versions of \CFA this number was set at 4; as the compiler becomes more optimized and capable of handling more complex expressions in a reasonable amount of time, I have increased the limit to 8 and in most occasions it does not lead to trouble.
310Very rarely there will be a case where the infinite recursion produces an exponentially growing assertion set, causing minutes of time wasted before the limit is reached.
311Fortunately it is very hard to run into this situation with realistic \CFA code, and the ones that were found all have some clear characteristics, which can be prevented by some clever tricks.
312In fact, some of the performance optimizations come from analyzing these problematic cases.
313One example of such will be presented later in this section.
314
315While the assertion satisfaction problem in isolation looks like just another expression to resolve, the recursive nature makes some techniques applied to expression resolution without assertions no longer possible.
316The most significant impact is that the type unification has a side effect, namely editing the type environment (equivalence classes and bindings), which means that if one expression has multiple associated assertions, they are not independent as the changes to the type environment must be compatible for all the assertions to be resolved.
317Particularly, if one assertion parameter can be resolved in multiple different ways, all of the results need to be checked to make sure the change to type variable bindings are compatible with other assertions to be resolved.
318A naive algorithm that simply picks any pending assertion to resolve and continue in a depth-first search could be very inefficient and especially prone of falling into an infinite loop, while in many cases it can be avoided by examining other assertions that can provide insight on the desired type binding: if one assertion parameter can only be matched by a unique option, we can then update the type bindings confidently without the need of backtracking.
319
320
321The algorithm currently used in \CFA compiler is designed by Aaron Moss through a simplified prototype experiment that captures most of \CFA type system features and ported back to the actual language.
322It can be described as a mix of breadth- and depth-first search in a staged approach.
323
324
325To resolve a set of assertions, the algorithm first attempts to resolve each assertion item individually.
326There are three possible outcomes on resolving each assertion:
327\begin{enumerate}
328\item If no matches are found, the algorithm terminates with a failure.
329\item If exactly one match is found, the type environment is updated immediately, and used in resolving any remaining assertions.
330\item If multiple matches are found, the assertion candidates with their updated type environments are stored in a list that will be checked for compatibility at the end.
331\end{enumerate}
332When all assertion items are resolved successfully, the algorithm attempts to combine the ambiguously resolved assertions to produce mutually compatible assignments.
333If any new assertions are introduced by the selected candidates, the algorithm is applied recursively, until there are none pending resolution or the recursion limit is reached which results in a failure.
334
335It has been discovered in practice that the efficiency of such algorithm can sometimes be very sensitive to the order of resolving assertions.
336Suppose an unbound type variable @T@ appears in two assertions, one can be uniquely resolved and allow the type @T@ to be inferred immediately, and another has many ways to be resolved, each results in @T@ being bound to a different concrete type.
337If the first assertion is examined first by the algorithm, the deducted type can then be utilized in resolving the second assertion and eliminate many incorrect options without producing the list of candidates pending further checks.
338In practice, this have a significant impact when an unbound type @T@ is declared to satisfy the basic "object assertions"\footnote{The term is borrowed from object-oriented languages although \CFA is not object-oriented in principle.} of having a default constructor, destructor, and copy assignment operations.
339Since they are defined for every type currently in scope, there are often hundreds or even thousands of matches to these functions with an unspecified operand type, and in most of the cases the value of @T@ can be deduced by resolving another assertion first, which then allows specific object lifetime functions to be looked up since they are sorted internally by the operand type, and greatly reduces the number of wasted resolution attempts.
340
341Currently this issue also causes the capability of the assertion resolution algorithm to be limited.
342Assertion matching is implemented to be more restricted than expression resolution in general, in that the parameter types must match exactly, rather than just merely callable.
343If one function declaration includes an assertion of @void f(T)@ and only a @f(long)@ is currently in scope, trying to resolve the assertion with @T=int@ would not work.
344Loosening the assertion matching requirement causes many assertion variables to have multiple matches and makes the delayed combination step too costly.
345
346Given all the issues caused by assertion resolution potentially creating new type variable bindings, a natural progression is to put some restrictions on free type variables such that all the type variables will be bound when the expression reaches assertion resolution stage.
347A type variable introduced by the @forall@ clause of function declaration can appear in parameter types, return types and assertion variables.
348If it appears in the parameter types, it will be bound when matching the arguments to parameters at the call site.
349If it only appears in the return type, it can be eventually figured out by the context in principle.
350The current implementation in \CFA compiler does not do enough return type deduction as it performs eager assertion resolution, and the return type information cannot be known in general before the parent expression is resolved, unless the expression is in an initialization context, in which the type of variable to be initialized is certainly known.
351By delaying the assertion resolution until the return type becomes known, this problem can be circumvented.
352The truly problematic case occurs if a type variable does not appear in either of the parameter or return types and only appears in some assertion variables.
353Such case is very rare and it is not evident that forcing every type variable to appear at least once in parameter or return types limits the expressiveness of \CFA type system to a significant extent.
354In the next chapter I will discuss about a proposal of including type declarations in traits rather than having all type variables appear in the trait parameter list, which could be helpful for providing equivalent functionality of having an unbound type parameter in assertion variables, and also addressing some of the variable cost issue discussed in section 4.1.
355
356
357\subsection*{Caching Assertion Resolution Results}
358
359In Aaron Moss's prototype design and experiments, a potential optimization of caching the result of already resolved assertions is discussed.
360Based on the experiment results, this approach can improve the performance of expression resolution in general, and sometimes allow hard instances of assertion resolution problems to be solved that are otherwise infeasible, for example when the resolution would encounter infinite loops.
361
362The problem that makes this approach tricky to be implemented correctly is that the resolution algorithm has side effects, namely modifying the type bindings in the environment.
363If we were to cache those results that cause the type bindings to be modified, it would be necessary to store the changes to type bindings too, and in case where multiple candidates can be used to satisfy one assertion parameter, all of them needs to be cached including those that are not eventually selected, since the side effect can produce different results depending on the context.
364
365In the original design of \CFA that includes unrestricted polymorphic formal parameters that can have assertions on themselves, the problem is even more complicated as new declarations can be introduced in scope during expression resolution.
366Here is one such example taken from Bilson:
367\begin{cfa}
368void f( forall( T | { T -?( T ); } ) T (*p)( T, T ) );
369forall( U, V | { U -?( U ); V -?( V ); } ) U g( U, V ) );
370f( g );
371\end{cfa}
372The inner assertion parameter on the \textit{closed} type variable @T@ is used to satisfy the assertions on @U@ and @V@ in this example.
373
374However, as per the previous discussions on this topic, polymorphic function pointer types have been removed from \CFA, since correctly implementing assertion matching is not possible in general.
375Without closed parameters (and therefore no have-set for assertions) the set of declarations in scope remains unchanged while resolving any expression.
376The current \CFA implementation also does not attempt to widen any already bound type parameters to satisfy an assertion.
377Note that such restriction does mean that certain kinds of expressions cannot be resolved, for example:
378\begin{cfa}
379forall (T | {void f(T);}) void g(T);
380void f (long);
381g(42);
382\end{cfa}
383The call @g(42)@ is rejected since no attempt is made to widen the parameter type @T@ from @int@ to @long@.
384Such problem could be mitigated if we allow inexact matches of assertions, but cannot be eliminated completely, if @T@ is matched in a parameterized type, including pointers and references:
385\begin{cfa}
386forall (T | {void f(T*);}) void g(T);
387void f (long *);
388g(42);
389\end{cfa}
390Here the only way to resolve the call @g(42)@ is to allow assertion resolution to widen the parameter @T@, since even with inexact matching, @int*@ cannot be converted to @long*@.
391
392
393\section{Compiler Implementation Considerations}
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