Changeset eeb0767 for doc/theses/aaron_moss_PhD/phd/typeenvironment.tex
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 Oct 16, 2018, 7:22:39 PM (5 years ago)
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doc/theses/aaron_moss_PhD/phd/typeenvironment.tex
r7b61ce8 reeb0767 5 5 As discussed in Chapter~\ref{resolutionchap}, being able to efficiently determine which type variables are bound to which concrete types or whether two type environments are compatible is a core requirement of the resolution algorithm. 6 6 Furthermore, expression resolution involves a search through many related possible solutions, so being able to reuse shared subsets of type environment data and to switch between environments quickly is desirable for performance. 7 In this chapter I discuss and empirically compare a number of type environment data structure variants, including some novel variations on the unionfind\cit {} data structure introduced in this thesis.7 In this chapter I discuss and empirically compare a number of type environment data structure variants, including some novel variations on the unionfind\cite{Galler64} data structure introduced in this thesis. 8 8 9 \section{Definitions} 9 \section{Definitions} \label{envdefnsec} 10 10 11 11 For purposes of this chapter, a \emph{type environment} $T$ is a set of \emph{type classes} $\myset{T_1, T_2, \cdots, T_{T}}$. … … 45 45 $unify(T, T_i, T_j)$ merges a type class $T_j$ into another $T_i$, producing a failure result and leaving $T$ in an invalid state if this merge fails. 46 46 It is always possible to unify the type variables of both classes by simply taking the union of both sets; given the disjointness property, no checks for set containment are required, and the variable sets can simply be concatenated if supported by the underlying data structure. 47 $unify$ depends on an internal $unify _bound$ operation which may fail.48 In \CFACC{}, $unify _bound(b_i, b_j) \rightarrow b'_i\bot$ checks that the type classes contain the same sort of variable, takes the tighter of the two conversion permissions, and checks if the bound types can be unified.49 If the bound types cannot be unified (\eg{} !struct A! with !int*!), then $unify _bound$ fails, while other combinations of bound types may result in recursive calls.50 For instance, unifying !R*! with !S*! for type variables !R! and !S! will result in a call to $unify(T, find($!R!$), find($!S!$))$, while unifying !R*! with !int*! will result in a call to $unify _bound$ on !int! and the bound type of the class containing !R!.47 $unify$ depends on an internal $unifyBound$ operation which may fail. 48 In \CFACC{}, $unifyBound(b_i, b_j) \rightarrow b'_i\bot$ checks that the type classes contain the same sort of variable, takes the tighter of the two conversion permissions, and checks if the bound types can be unified. 49 If the bound types cannot be unified (\eg{} !struct A! with !int*!), then $unifyBound$ fails, while other combinations of bound types may result in recursive calls. 50 For instance, unifying !R*! with !S*! for type variables !R! and !S! will result in a call to $unify(T, find($!R!$), find($!S!$))$, while unifying !R*! with !int*! will result in a call to $unifyBound$ on !int! and the bound type of the class containing !R!. 51 51 As such, a call to $unify(T, T_i, T_j)$ may touch every type class in $T$, not just $T_i$ and $T_j$, collapsing the entirety of $T$ into a single type class in extreme cases. 52 52 … … 57 57 The invalid state of $T$ on failure is not important, given that a combination failure will result in the resolution algorithm backtracking to a different environment. 58 58 $combine$ proceeds by calls to $insert$, $add$, and $unify$ as needed, and can be roughly thought of as calling $unify$ on every pair of classes in $T$ that have variables $v'_{i,j}$ and $v'_{i,k}$ in the same class $T'_i$ in $T'$. 59 Like for $unify$, $combine$ can always find a mutuallyconsistent division of type variables into classes (in the extreme case, all type variables from $T$ and $T'$ in a single type class), but may fail due to inconsistent bounds on merged type classes.59 Like for $unify$, $combine$ can always find a mutuallyconsistent partition of type variables into classes (in the extreme case, all type variables from $T$ and $T'$ in a single type class), but may fail due to inconsistent bounds on merged type classes. 60 60 61 61 Finally, the backtracking access patterns of the compiler can be exploited to reduce memory usage or runtime through use of an appropriately designed data structure. 62 62 The set of mutations to a type environment across the execution of the resolution algorithm produce an implicit tree of related environments, and the backtracking search typically focuses only on one leaf of the tree at once, or at most a small number of closelyrelated nodes as arguments to $combine$. 63 63 As such, the ability to save and restore particular type environment states is useful, and supported by the $save(T) \rightarrow H$ and $backtrack(T, H)$ operations, which produce a handle for the current environment state and mutate an environment back to a previous state, respectively. 64 These operations can be naively implemented by a deep copy of $T$ into $H$ and vice versa, but have more efficient implementations in persistencyaware data structures. 64 These operations can be naively implemented by a deep copy of $T$ into $H$ and vice versa, but have more efficient implementations in persistencyaware data structures. 65 66 \section{Approaches} 67 68 \subsection{Na\"{\i}ve} 69 70 The type environment data structure used in Bilson's\cite{Bilson03} original implementation of \CFACC{} is a straightforward 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. 71 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. 72 Some variations on this structure may improve performance somewhat; for instance, replacing the !EqvClass! variable storage with a hashbased set would reduce search and update times from $O(log n)$ to amortized $O(1)$, while adding an index for the type variables in the entire environment would remove the need to check each type class individually to maintain the disjointness property. 73 These improvements do not change the fundamental issues with this data structure, however. 74 75 \subsection{Incremental Inheritance} 76 77 One more invasive modification to this data structure which I investigated is to support swifter combinations of closelyrelated environments in the backtracking tree by storing a reference to a \emph{parent} environment within each environment, and having that environment only store type classes which have been modified with respect to the parent. 78 This approach provides constanttime copying of environments, as a new environment simply consists of an empty list of type classes and a reference to its (logically identical) parent; since many type environments are no different than their parent, this speeds backtracking in this common case. 79 Since all mutations made to a child environment are by definition compatible with the parent environment, two descendants of a common ancestor environment can be combined by iteratively combining the changes made in one environment then that environment's parent until the common ancestor is reached, again reusing storage and reducing computation in many cases. 80 81 For this environment I also employed a lazilygenerated index of type variables to their containing class, which could be in either the current environment or an ancestor. 82 Any mutation of a type class in an ancestor environment would cause that class to be copied into the current environment before mutation, as well as added to the index, ensuring that all local changes to the type environment are listed in its index. 83 However, not adding type variables to the index until lookup or mutation preserves the constanttime environment copy operation in the common case in which the copy is not mutated from its parent during its lifecycle. 84 85 This approach imposes some performance penalty on $combine$ if related environments are not properly linked together, as the entire environment needs to be combined rather than just the diff, but is correct as long as the ``null parent'' base case is properly handled. 86 The lifecycle issues are somewhat more complex, as many environments may descend from a common parent, and all of these need their parent to stay alive for purposes of lookup. 87 These issues can be solved by ``flattening'' parent nodes into their children before the parents leave scope, but given the tree structure of the inheritance graph it is more straightforward to store the parent nodes in referencecounted or otherwise automatically garbagecollected heap storage. 88 89 \subsection{UnionFind} 90 91 Given the nature of the classes of type variables as disjoint sets, another natural approach to implementing a type environment is the unionfind disjoint set data structure\cite{Galler64}. 92 Unionfind efficiently implements two operations over a partition of a collection of elements into disjoint sets; $find(x)$ locates the \emph{representative} of $x$, the element which canonically names its set, while $union(r, s)$ merges two sets represented by $r$ and $s$, respectively. 93 The unionfind data structure is based on providing each element with a reference to its parent element, such that the root of a tree of elements is the representative of the set of elements contained in the tree. 94 $find$ is then implemented by a search up to the parent, generally combined with a \emph{path compression} step that links nodes more directly to their ancestors to speed up subsequent searches. 95 $union$ involves making the representative of one set a child of the representative of the other, generally employing a rank or sizebased heuristic to ensure that the tree remains somewhat balanced. 96 If both path compression and a balancing heuristic are employed, both $union$ and $find$ run in amortized $O(\alpha(n))$ worstcase time; this bound by the inverse Ackermann function is a small constant for all practical values of $n$. 97 98 The unionfind $find$ and $union$ operations have obvious applicability to the $find$ and $unify$ type environment operations in Table~\ref{envoptable}, but the unionfind data structure must be augmented to fully implement the type environment operations. 99 In particular, the type class bound cannot be easily included in the unionfind data structure, as the requirement to make it the class representative breaks the balancing properties of $union$, and requires tooclose integration of the type environment $unifyBound$ internal operation. 100 This issue can be solved by including a side map from class representatives to the type class bound. 101 If placeholder values are inserted in this map for type classes without bounds than this also has the useful property that the key set of the map provides an easily obtainable list of all the class representatives, a list which cannot be derived from the unionfind data structure without a linear search for class representatives through all elements. 102 103 \subsection{UnionFind with Classes} 104 105 Another type environment operation not supported directly by the unionfind data structure is $report$, which lists the type variables in a given class, and similarly $remove$, which removes a class and all its type variables from the environment. 106 Since the unionfind data structure stores only links from children to parents and not viceversa, there is no way to reconstruct a class from one of its elements without a linear search over the entire data structure, with $find$ called on each element to check its membership in the class. 107 The situation is even worse for the $remove$ operation, where a na\"{\i}ve removal of every element belonging to a specific class would likely remove some parents before their children, requiring either extra bookkeeping or passes through the data structure to remove the leaf elements of the class first. 108 Unfortunately, the literature\cite{Tarjan84,Galil91,Patwary10} on unionfind does not present a way to keep references to children without breaking the asymptotic time bounds of the algorithm; I have discovered a method to do so which, despite its simplicity, seems to be novel. 109 110 The core idea of this ``unionfind with classes'' data structure and algorithm is to keep the members of each class stored in a circularlylinked list. 111 Rem's algorithm\cite{Dijkstra76}, dating from the 1970s, also includes a circularlylinked list data structure. 112 \TODO{Check this from source} However, Rem's algorithm has an entirely flat class hierarchy, where all elements were direct children of the representative, giving constanttime $find$ at the cost of lineartime $union$ operations. 113 In my version, the list data structure does not affect the layout of the unionfind tree, maintaining the same asymptotic bounds as unionfind. 114 In more detail, each element is given a !next! pointer to another element in the same class; this !next! pointer initially points to the element itself. 115 When two classes are unified, the !next! pointers of the representatives of those classes are swapped, splicing the two circularlylinked lists together. 116 Importantly, though this approach requires an extra pointer per element, it does maintain the linear space bound of unionfind, and because it only requires updating the two root nodes in $union$ it does not asymptotically increase runtime either. 117 The basic approach is compatible with all pathcompression techniques, and allows the members of any class to be retrieved in time linear in the size of the class simply by following the !next! pointers from any element. 118 119 If the pathcompression optimization is abandoned, unionfind with classes also encodes a reversible history of all the $union$ operations applied to a given class. 120 Theorem~\ref{envreversethm} demonstrates that the !next! pointer of the representative of a class always points to a leaf from the lastadded subtree. 121 This property is sufficient to reverse the mostrecent $union$ operation by finding the ancestor of that leaf that is an immediate child of the representative, breaking its parent link, and swapping the !next! pointers back\footnote{Unionbysize may be a more appropriate approach than unionbyrank in this instance, as adding two known sizes is a reversible operation, but the rank increment operation cannot be reliably reversed.}. 122 Once the $union$ operation has been reversed, Theorem~\ref{envreversethm} still holds for the reduced class, and the process can be repeated recursively until the entire set is split into its component elements. 123 124 \begin{theorem} \label{envreversethm} 125 The !next! pointer of a class representative in the unionfind with classes algorithm without path compression points to a leaf from the mostrecentlyadded subtree. 126 \end{theorem} 127 128 \begin{proof} 129 By induction on the height of the tree. \\ 130 \emph{Base case:} A height 1 tree by definition includes only a single item. In such a case, the representative's !next! pointer points to itself by construction, and the representative is the mostrecentlyadded (and only) leaf in the tree. \\ 131 \emph{Inductive case:} By construction, a tree $T$ of height greater than 1 has children of the root (representative) node that were representative nodes of classes merged by $union$. By definition, the mostrecentlyadded subtree $T'$ has a smaller height than $T$, thus by the inductive hypothesis before the mostrecent $union$ operation the !next! pointer of the root of $T'$ pointed to one of the leaf nodes of $T'$; by construction the !next! pointer of the root of $T$ points to this leaf after the $union$ operation. 132 \end{proof} 65 133 66 134 % Future work: design multithreaded version of C&F persistent map  core idea is some sort of threadboundary edit node
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