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 Jan 22, 2019, 11:23:43 PM (5 years ago)
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doc/theses/aaron_moss_PhD/phd/resolutionheuristics.tex
rf2c5726 r104a13e 3 3 4 4 % consider using "satisfaction" throughout when talking about assertions 5 % "valid" instead of "feasible" interpretations 5 6 6 7 The main task of the \CFACC{} typechecker is \emph{expression resolution}, determining which declarations the identifiers in each expression correspond to. … … 147 148 If 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. 148 149 149 \subsection{ Analysis} \label{resnanalysissec}150 \subsection{Worstcase Analysis} \label{resnanalysissec} 150 151 151 152 Expression resolution has a number of components which contribute to its runtime, including argumentparameter type unification, recursive traversal of the expression tree, and satisfaction of type assertions. 152 153 153 154 If the bound type for a type variable can be looked up or mutated in constant time (as asserted in Table~\ref{envboundstable}), then the runtime of the unification algorithm to match an argument to a parameter is proportional to the complexity of the types being unified. 154 In C, complexity of type representation is bounded by the mostcomplex type explicitly written in a declaration ; in \CFA{}, however, polymorphism can generate morecomplex types:155 In C, complexity of type representation is bounded by the mostcomplex type explicitly written in a declaration, effectively a small constant; in \CFA{}, however, polymorphism can generate morecomplex types: 155 156 156 157 \begin{cfa} … … 161 162 162 163 To 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!. 163 According to this argument, then, the cost of a single argumentparameter unification is !O(d)!, where !d! is the depth of the expression tree, and the cost of argumentparameter unification for a single candidate for a given function call expression is !O(pd)!, where !p! is the number of parameters. 164 164 Accordingly, the cost of a single argumentparameter unification is $O(d)$, where !d! is the depth of the expression tree, and the cost of argumentparameter unification for a single candidate for a given function call expression is $O(pd)$, where $p$ is the number of parameters. 165 166 Considering the recursive traversal of the expression tree, polymorphism again greatly expands the worstcase runtime. 167 Letting $i$ 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. 168 Ambiguous minimalcost 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. 169 One safe pruning operation during expression resolution is to discard all subexpression interpretations with greaterthanminimal cost for their return type, as such interpretations will never beat the minimalcost interpretation with their return type for the overall optimal solution. 170 As such, with no polymorphism each declaration will generate no more than one minimalcost interpretation with its return type, so the number of possible subexpression interpretations is $O(i)$ (note that in C, which lacks overloading, $i \leq 1$). 171 With 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. 172 Calculated recursively, the bound on the total number of candidate interpretations is $O(i^{p^d})$, each with a distinct type. 173 174 Given 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 $O( i^{p^d} \cdot pd )$. 175 Since there are $O(p^d)$ subexpressions in a single toplevel expression, the total worstcase cost of argumentparameter matching with the overloading and polymorphism features of \CFA{} is $O( i^{p^d} \cdot pd \cdot p^d )$. 176 177 This already high bound does not yet account for the cost of assertion resolution, though. 165 178 % continue from here 179 180 % need to discuss that n == p^d, that that i^p^d bound is **super** rare, and will usually be closer to $i$ 181 182 % missing a +1 somewhere in these bounds; also, pd is O(n) () 166 183 167 184 \subsection{Related Work}
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