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 Apr 20, 2019, 7:28:50 PM (4 years ago)
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doc/theses/aaron_moss_PhD/phd/typeenvironment.tex
rc1f3d1a8 r2834e99 13 13 Each type class $T_i$ contains a set of \emph{type variables} $\myset{v_{i,1}, v_{i,2}, \cdots, v_{i,T_i}}$. 14 14 Since the type classes represent an equivalence relation over the type variables the sets of variables contained in two distinct classes in the same environment must be \emph{disjoint}. 15 Each individual type class $T_i$ may also be associated with a \emph{bound}, $b_i$; this bound contains the \emph{bound type} that the variables in the type class are replaced with, but also includes other information in \CFACC{}, including whether type conversions are permissible on the bound type and what sort of type variables are contained in the class (data types, function types, or variadic tuples). 15 Each individual type class $T_i$ may also be associated with a \emph{bound}, $b_i$; this bound contains the \emph{bound type} that the variables in the type class are replaced with, but also includes other information in \CFACC{}, including whether type conversions are permissible on the bound type and what sort of type variables are contained in the class (data types, function types, or variadic tuples). 16 17 The following example demonstrates the use of a type environment for unification: 18 19 \begin{cfa} 20 forall(otype F) F f(F, F); 21 forall(otype G) G g(G); 22 23 f( g(10), g(20) ); 24 \end{cfa} 25 26 Expression resolution starts from an empty type environment; from this empty environment, the calls to !g! can be independently resolved. 27 These resolutions result in two new type environments, $T = \{ \myset{\mathsf{G}_1} \rightarrow$ !int!$\}$ and $T' = \{ \myset{\mathsf{G}_2} \rightarrow$ !int!$\}$; the calls to !g! have generated distinct type variables !G!$_1$ and !G!$_2$, each bound to !int! by unification with their argument type. 28 To complete resolution of the call to !f!, both environments must be combined; resolving the first argument to !f! produces a new type environment $T'' = \{ \myset{\mathsf{G}_1, \mathsf{F}_1} \rightarrow$ !int!$\}$: the new type variable !F!$_1$ has been introduced and unified with !G!$_1$ (the return type of !g(10)!), and consequently bound to !int!. 29 To resolve the second argument to !f!, $T''$ must be checked for compatibility with $T'$; since !F!$_1$ unifies with !G!$_2$, their type classes must be merged. 30 Since both !F!$_1$ and !G!$_2$ are bound to !int!, this merge succeeds, producing the final environment $T'' = \{ \myset{\mathsf{G}_1, \mathsf{F}_1, \mathsf{G}_2} \rightarrow$ !int!$\}$. 16 31 17 32 \begin{table} … … 37 52 \end{table} 38 53 39 Given this basic structure, type environments in \CFACC{} need to support eleven basic operations, summarized in Table~\ref{envoptable}.54 Type environments in \CFACC{} need to support eleven basic operations, summarized in Table~\ref{envoptable}. 40 55 The first six operations are straightforward queries and updates on these data structures: 41 56 The lookup operation $find(T, v_{i,j})$ produces $T_i$, the type class in $T$ that contains variable $v_{i,j}$, or an invalid sentinel value for no such class. … … 75 90 \subsection{Na\"{\i}ve} \label{naiveenvsec} 76 91 77 The type environment data structure used in Bilson's~\cite{Bilson03} original implementation of \CFACC{} is a s traightforwardtranslation of the definitions in Section~\ref{envdefnsec} to \CC{} code; a !TypeEnvironment! contains a list of !EqvClass! type equivalence classes, each of which contains the type bound information and a treebased sorted set of type variables.92 The type environment data structure used in Bilson's~\cite{Bilson03} original implementation of \CFACC{} is a simple translation of the definitions in Section~\ref{envdefnsec} to \CC{} code; a !TypeEnvironment! contains a list of !EqvClass! type equivalence classes, each of which contains the type bound information and a treebased sorted set of type variables. 78 93 This approach has the benefit of being easy to understand and not imposing lifecycle or inheritance constraints on its use, but, as can be seen in Table~\ref{envboundstable}, does not support many of the desired operations with any particular efficiency. 79 94 Some variations on this structure may improve performance somewhat; for instance, replacing the !EqvClass! variable storage with a hashbased set reduces search and update times from $O(\log n)$ to amortized $O(1)$, while adding an index for the type variables in the entire environment removes the need to check each type class individually to maintain the disjointness property. … … 244 259 The na\"{\i}ve $combine$ operation must traverse each of the classes of one environment, merging in any class of the other environment that shares a type variable. 245 260 Since there are at most $n$ classes to unify, the unification cost is $O(nm + nu(n))$, while traversal and $find$ costs to locate classes to merge total $O(n^2m)$, for an overall cost of $O(n^2m + nu(n))$. 246 The incremental $combine$ operation works similarly, but only needs to consider classes modified in either environment with respect to the common ancestor of both environments, allowing the $n$ cost terms to be substituted for $p$, for an overall cost of $O(p^ m + pu(n))$.261 The incremental $combine$ operation works similarly, but only needs to consider classes modified in either environment with respect to the common ancestor of both environments, allowing the $n$ cost terms to be substituted for $p$, for an overall cost of $O(p^2m + pu(n))$. 247 262 Neither variant supports the $split$ operation to undo a $unify$. 248 263
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