Changeset 1836081 for doc


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Timestamp:
Mar 7, 2019, 3:17:13 PM (3 years ago)
Author:
Aaron Moss <a3moss@…>
Branches:
aaron-thesis, arm-eh, cleanup-dtors, jacob/cs343-translation, jenkins-sandbox, master, new-ast, new-ast-unique-expr
Children:
d438111
Parents:
247c8f3
Message:

thesis: second draft of ch.5

Location:
doc
Files:
3 edited

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  • doc/bibliography/pl.bib

    r247c8f3 r1836081  
    24992499    pages       = {325-361},
    25002500}
     2501
     2502@article{Tarjan75,
     2503 keywords = {union-find},
     2504 contributer = {a3moss@uwaterloo.ca},
     2505 author = {Tarjan, Robert Endre},
     2506 title = {Efficiency of a Good But Not Linear Set Union Algorithm},
     2507 journal = {J. ACM},
     2508 issue_date = {April 1975},
     2509 volume = {22},
     2510 number = {2},
     2511 month = apr,
     2512 year = {1975},
     2513 issn = {0004-5411},
     2514 pages = {215--225},
     2515 numpages = {11},
     2516 url = {http://doi.acm.org/10.1145/321879.321884},
     2517 doi = {10.1145/321879.321884},
     2518 acmid = {321884},
     2519 publisher = {ACM},
     2520 address = {New York, NY, USA},
     2521}
    25012522
    25022523@book{Eiffel,
  • doc/theses/aaron_moss_PhD/phd/macros.tex

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    1515\newcommand{\Csharp}{C\raisebox{-0.7ex}{\Large$^\sharp$}} % C# symbolic name
    1616
    17 \newcommand{\ie}{\textit{i.e.}}
    18 \newcommand{\eg}{\textit{e.g.}}
    19 \newcommand{\etc}{\textit{etc.}}
    20 \newcommand{\etal}{\textit{et~al.}}
     17\newcommand{\ie}{\textit{i.e.}\@}
     18\newcommand{\eg}{\textit{e.g.}\@}
     19\newcommand{\etc}{\textit{etc.}\@}
     20\newcommand{\etal}{\textit{et~al.}\@}
    2121
    2222\newcommand{\myset}[1]{\left\{#1\right\}}
  • doc/theses/aaron_moss_PhD/phd/type-environment.tex

    r247c8f3 r1836081  
    44One key data structure for expression resolution is the \emph{type environment}.
    55As discussed in Chapter~\ref{resolution-chap}, 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 Furthermore, expression resolution involves a search through many related possible solutions, so being able to re-use shared subsets of type environment data and to switch between environments quickly is desirable for performance.
    7 In this chapter I discuss a number of type environment data structure variants, including some novel variations on the union-find \cite{Galler64} data structure introduced in this thesis.
     6Furthermore, expression resolution involves a search through many related possible solutions, so the ability to re-use shared subsets of type-environment data and to switch between environments quickly is desirable for performance.
     7In this chapter, I discuss a number of type environment data-structure variants, including some novel variations on the union-find \cite{Galler64} data structure introduced in this thesis.
    88Chapter~\ref{expr-chap} contains empirical comparisons of the performance of these data structures when integrated into the resolution algorithm.
    99
     
    1111
    1212For purposes of this chapter, a \emph{type environment} $T$ is a set of \emph{type classes} $\myset{T_1, T_2, \cdots, T_{|T|}}$.
    13 Each type class $T_i$ contains a set of \emph{type variables} $\myset{v_{i,1}, v_{i,2}, \cdots, v_{i,|T_i|}}$; note that the sets of variables contained in two distinct classes in the same environment must be disjoint.
    14 Each individual type class $T_i$ may also be associated with a \emph{bound}, $b_i$; this bound contains the \emph{bound type} which 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).
     13Each type class $T_i$ contains a set of \emph{type variables} $\myset{v_{i,1}, v_{i,2}, \cdots, v_{i,|T_i|}}$.
     14Since 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 disjoint.
     15Each 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).
    1516
    1617\begin{table}
     
    1920\centering
    2021\begin{tabular}{r@{\hskip 0.25em}ll}
    21         $find(T, v_{i,j})$ & $\rightarrow T_i | \bot$           & Locate class for variable             \\
     22        \hline
     23        $find(T, v_{i,j})$ & $\rightarrow T_i~|~\mathsf{fail}$          & Locate class for variable             \\
    2224        $report(T_i)$ & $\rightarrow \{ v_{i,j} \cdots \}$      & List variables for class              \\
    23         $bound(T_i)$ & $\rightarrow b_i | \bot$                         & Get bound for class                   \\
    24         $insert(v_{i,1})$ &                                                                     & New single-variable class             \\
     25        $bound(T_i)$ & $\rightarrow b_i~|~\mathsf{fail}$                                & Get bound for class                   \\
     26        $insert(T, v_{i,1})$ &                                                          & New single-variable class             \\
    2527        $add(T_i, v_{i,j})$ &                                                           & Add variable to class                 \\
    2628        $bind(T_i, b_i)$ &                                                                      & Set or update class bound             \\
    27         $unify(T, T_i, T_j)$ & $\rightarrow \top | \bot$        & Combine two type classes              \\
     29        \hline
     30        $unify(T, T_i, T_j)$ & $\rightarrow \mathsf{pass}~|~\mathsf{fail}$      & Combine two type classes              \\
    2831        $split(T, T_i)$ & $\rightarrow T'$                                      & Revert the last $unify$ operation on $T_i$            \\
    29         $combine(T, T')$ & $\rightarrow \top | \bot$            & Merge two environments                \\
     32        $combine(T, T')$ & $\rightarrow \mathsf{pass}~|~\mathsf{fail}$          & Merge two environments                \\
    3033        $save(T)$ & $\rightarrow H$                                                     & Get handle for current state  \\
    31         $backtrack(T, H)$ &                                                                     & Return to handle state
     34        $backtrack(T, H)$ &                                                                     & Return to handle state                \\
     35        \hline
    3236\end{tabular}
    3337\end{table}
    3438
    3539Given this basic structure, type environments in \CFACC{} need to support eleven basic operations, summarized in Table~\ref{env-op-table}.
    36 The first seven operations are straightforward queries and updates on these data structures:
    37 The lookup operation $find(T, v_{i,j})$ produces $T_i$, the type class in $T$ which contains variable $v_{i,j}$, or an invalid sentinel value for no such class.
     40The first six operations are straightforward queries and updates on these data structures:
     41The 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.
    3842The other two query operations act on type classes, where $report(T_i)$ produces the set $\myset{v_{i,1}, v_{i,2}, \cdots, v_{i,|T_i|}}$ of all type variables in a class $T_i$ and $bound(T_i)$ produces the bound $b_i$ of that class, or a sentinel indicating no bound is set.
    3943
    40 The update operation $insert(T, v_{i,1})$ creates a new type class $T_i$ in $T$ that contains only the variable $v_{i,1}$ and no bound; due to the disjointness property $v_{i,1}$ cannot belong to any other type class in $T$.
     44The update operation $insert(T, v_{i,1})$ creates a new type class $T_i$ in $T$ that contains only the variable $v_{i,1}$ and no bound; due to the disjointness property, $v_{i,1}$ cannot belong to any other type class in $T$.
    4145The $add(T_i, v_{i,j})$ operation adds a new type variable $v_{i,j}$ to class $T_i$; again, $v_{i,j}$ cannot exist elsewhere in $T$.
    4246$bind(T_i, b_i)$ mutates the bound for a type class, setting or updating the current bound.
    4347
    44 The $unify$ operation is the fundamental non-trivial operation a type environment data structure must support.
     48The $unify$ operation is the fundamental non-trivial operation a type-environment data-structure must support.
    4549$unify(T, T_i, T_j)$ 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.
    4650It 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 $unifyBound$ 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.
     51$unify$ depends on an internal $unifyBound$ operation, which may fail.
     52In \CFACC{}, $unifyBound(b_i, b_j) \rightarrow b'_i~|~\mathsf{fail}$ 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.
    4953If 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!.
     54For instance, unifying !R*! with !S*! for type variables !R! and !S! results in a call to $unify(T, find($!R!$), find($!S!$))$, while unifying !R*! with !int*! results in a call to $unifyBound$ on !int! and the bound type of the class containing !R!.
    5155As 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.
    5256For more information on \CFA{} unification, see \cite{Bilson03}.
    53 The inverse of $unify$ is $split(T, T_i)$, which produces a new environment $T'$ which is the same as $T$ except that $T_i$ has been replaced by two classes corresponding to the arguments to the previous call to $unify$ on $T_i$.
    54 If there has been no call to $unify$ on $T_i$ (\ie{} $T_i$ is a single-element class) $T_i$ is absent in $T'$.
    55 
    56 Given the nature of the expression resolution problem as backtracking search, caching and concurrency are both useful tools to decrease runtime.
     57The inverse of $unify$ is $split(T, T_i)$, which produces a new environment $T'$ that is the same as $T$ except that $T_i$ has been replaced by two classes corresponding to the arguments to the previous call to $unify$ on $T_i$.
     58If there is no prior call to $unify$ on $T_i$ (\ie{} $T_i$ is a single-element class) $T_i$ is absent in $T'$.
     59
     60Given the nature of the expression resolution problem as a backtracking search, caching and concurrency are both useful tools to decrease runtime.
    5761However, both of these approaches may produce multiple distinct descendants of the same initial type environment, which have possibly been mutated in incompatible ways.
    58 As such, to effectively employ either concurrency or caching, the type environment data structure must support an efficient method to check if two type environments are compatible and merge them if so.
    59 $combine(T,T')$ attempts to merge an environment $T'$ into another environment $T$, producing $\top$ if successful or leaving $T$ in an invalid state and producing $\bot$ otherwise.
    60 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.
     62As such, to effectively employ either caching or concurrency, the type environment data structure must support an efficient method to check if two type environments are compatible and merge them if so.
     63$combine(T,T')$ attempts to merge an environment $T'$ into another environment $T$, producing $\mathsf{pass}$ if successful or leaving $T$ in an invalid state and producing $\mathsf{fail}$ otherwise.
     64The invalid state of $T$ on failure is not important, given that a combination failure results in the resolution algorithm backtracking to a different environment.
    6165$combine$ 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 $T'$.
    6266Like $unify$, $combine$ can always find a mutually-consistent 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.
     
    6569The 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 closely-related nodes as arguments to $combine$.
    6670As 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.
    67 These operations can be naively implemented by a deep copy of $T$ into $H$ and vice versa, but have more efficient implementations in persistency-aware data structures.
    68 
    69 \section{Approaches}
     71These operations can be naively implemented by a deep copy of $T$ into $H$ and vice versa, but have more efficient implementations in persistency-aware data structures such as the persistent union-find introduced in Section~\ref{env-persistent-union-find}.
     72
     73\section{Approaches} \label{env-approaches-sec}
    7074
    7175\subsection{Na\"{\i}ve} \label{naive-env-sec}
     
    7377The type environment data structure used in Bilson's~\cite{Bilson03} original implementation of \CFACC{} is a straightforward translation of the definitions in Section~\ref{env-defn-sec} to \CC{} code; a !TypeEnvironment! contains a list of !EqvClass! type equivalence classes, each of which contains the type bound information and a tree-based sorted set of type variables.
    7478This approach has the benefit of being easy to understand and not imposing life-cycle or inheritance constraints on its use, but, as can be seen in Table~\ref{env-bounds-table}, does not support many of the desired operations with any particular efficiency.
    75 Some variations on this structure may improve performance somewhat; for instance, replacing the !EqvClass! variable storage with a hash-based 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.
     79Some variations on this structure may improve performance somewhat; for instance, replacing the !EqvClass! variable storage with a hash-based 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.
    7680These improvements do not change the fundamental issues with this data structure, however.
    7781
    7882\subsection{Incremental Inheritance} \label{inc-env-sec}
    7983
    80 One more invasive modification to this data structure which I investigated is to support swifter combinations of closely-related 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.
     84One more invasive modification to this data structure that I investigated is to support swifter combinations of closely-related 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 that have been modified with respect to the parent.
    8185This approach provides constant-time 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.
    82 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 re-using storage and reducing computation in many cases.
    83 
    84 For this environment I also employed a lazily-generated index of type variables to their containing class, which could be in either the current environment or an ancestor.
    85 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.
     86Since 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 re-using storage and reducing computation in many cases.
     87
     88For this environment, I also employed a lazily-generated index of type variables to their containing class, which could be in either the current environment or an ancestor.
     89Any mutation of a type class in an ancestor environment causes that class to be copied into the current environment before mutation, as well as added to the index, ensuring all local changes to the type environment are listed in its index.
    8690However, not adding type variables to the index until lookup or mutation preserves the constant-time environment copy operation in the common case in which the copy is not mutated from its parent during its life-cycle.
    8791
    88 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.
     92This 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 difference, but is correct as long as the ``null parent'' base-case is properly handled.
    8993The life-cycle 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.
    90 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 reference-counted or otherwise automatically garbage-collected heap storage.
     94These issues can be solved by ``flattening'' parent nodes into their children before the parent's scope ends, but given the tree structure of the inheritance graph it is more straightforward to store the parent nodes in reference-counted or otherwise automatically garbage-collected heap storage.
    9195
    9296\subsection{Union-Find} \label{env-union-find-approach}
    9397
    94 Given the nature of the classes of type variables as disjoint sets, another natural approach to implementing a type environment is the union-find disjoint set data structure~\cite{Galler64}.
     98Given the nature of the classes of type variables as disjoint sets, another natural approach to implementing a type environment is the union-find disjoint-set data-structure~\cite{Galler64}.
    9599Union-find 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.
    96100The union-find 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.
    97101$find$ 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.
    98102$union$ involves making the representative of one set a child of the representative of the other, generally employing a rank- or size-based heuristic to ensure that the tree remains somewhat balanced.
    99 If both path compression and a balancing heuristic are employed, both $union$ and $find$ run in amortized $O(\alpha(n))$ worst-case time; this bound by the inverse Ackermann function is a small constant for all practical values of $n$.
     103If both path compression and a balancing heuristic are employed, both $union$ and $find$ run in amortized $O(\alpha(n))$ worst-case time; this inverse Ackermann bound is a small constant for all practical values of $n$ \cite{Tarjan75}.
    100104
    101105The union-find $find$ and $union$ operations have obvious applicability to the $find$ and $unify$ type environment operations in Table~\ref{env-op-table}, but the union-find data structure must be augmented to fully implement the type environment operations.
    102 In particular, the type class bound cannot be easily included in the union-find data structure, as the requirement to make it the class representative breaks the balancing properties of $union$, and requires too-close integration of the type environment $unifyBound$ internal operation.
    103 This issue can be solved by including a side map from class representatives to the type class bound.
     106In particular, the type-class bound cannot be easily included in the union-find data structure, as the requirement to make it the class representative breaks the balancing properties of $union$, and requires too-close integration of the type environment $unifyBound$ internal operation.
     107This issue can be solved by including a side map from class representatives to the type-class bound.
    104108If 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 union-find data structure without a linear search for class representatives through all elements.
    105109
     
    108112Another type environment operation not supported directly by the union-find data structure is $report$, which lists the type variables in a given class, and similarly $split$, which reverts a $unify$ operation.
    109113Since the union-find data structure stores only links from children to parents and not vice-versa, 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.
    110 The situation is even worse for the $split$ operation, which would require extra information to maintain the order that each child was added to its parent node.
    111 Unfortunately, the literature \cite{Tarjan84,Galil91,Patwary10} on union-find 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.
     114The situation is even worse for the $split$ operation, which requires extra information to maintain the order that each child is added to its parent node.
     115Unfortunately, the literature \cite{Tarjan84,Galil91,Patwary10} on union-find 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.
    112116
    113117The core idea of this ``union-find with classes'' data structure and algorithm is to keep the members of each class stored in a circularly-linked list.
     
    133137
    134138\begin{theorem} \label{env-reverse-thm}
    135 The !next! pointer of a class representative in the union-find with classes algorithm without path compression points to a leaf from the most-recently-added subtree.
     139The !next! pointer of a class representative in the union-find with classes algorithm, without path compression, points to a leaf from the most-recently-added subtree.
    136140\end{theorem}
    137141
     
    139143        By induction on the height of the tree. \\
    140144        \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 most-recently-added (and only) leaf in the tree. \\
    141         \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 most-recently-added subtree $T'$ has a smaller height than $T$, thus by the inductive hypothesis before the most-recent $union$ 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.
     145        \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 most-recently-added subtree $T'$ has a smaller height than $T$, thus by the inductive hypothesis before the most-recent $union$ 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.
    142146\end{proof}
    143147
     
    147151\label{env-persistent-union-find}
    148152
    149 Given the backtracking nature of the resolution algorithm discussed in Section~\ref{env-defn-sec}, the abilities to quickly switch between related versions of a type environment and to de-duplicate shared data between environments are both assets to performance.
     153Given the backtracking nature of the resolution algorithm discussed in Section~\ref{env-defn-sec}, the abilities to quickly switch between related versions of a type environment and to de-duplicate shared data among environments are both assets to performance.
    150154Conchon and Filli\^{a}tre~\cite{Conchon07} present a persistent union-find data structure based on the persistent array of Baker~\cite{Baker78,Baker91}.
    151155
    152 In Baker's persistent array, an array reference contains either a pointer to the array or a pointer to an \emph{edit node}; these edit nodes contain an array index, the value in that index, and another array reference pointing either to the array or a different edit node.
    153 In this manner, a tree of edits is formed, rooted at the actual array.
     156In Baker's persistent array, an \emph{array reference} contains either a pointer to the array or a pointer to an \emph{edit node}; these edit nodes contain an array index, the value in that index, and another array reference pointing either to the array or a different edit node.
     157By construction, these array references always point to a node more like the actual array, forming a tree of edits rooted at the actual array.
    154158Reads from the actual array at the root can be performed in constant time, as with a non-persistent array.
    155159The persistent array can be mutated in constant time by directly modifying the underlying array, then replacing its array reference with an edit node containing the mutated index, the previous value at that index, and a reference to the mutated array. If the current array reference is not the root, mutation consists simply of constructing a new edit node encoding the change and referring to the current array reference. 
    156 The mutation algorithm at the root is in some sense a special case of the key operation on persistent arrays, $reroot$.
    157 
     160
     161The mutation algorithm at the root is a special case of the key operation on persistent arrays, $reroot$.
    158162A rerooting operation takes any array reference and makes it the root node of the array.
    159 This is accomplished by tracing the path from some edit node to the root node of the array (always the underlying array), recursively applying the edits to the underlying array and replacing each edit node's successor with the inverse edit.
     163This operation is accomplished by tracing the path from some edit node to actual array at the root node, recursively applying the edits to the underlying array and replacing each edit node's successor with the inverse edit.
    160164In this way, any previous state of the persistent array can be restored in time proportional to the number of edits to the current state of the array.
    161165While $reroot$ does maintain the same value mapping in every version of the persistent array, the internal mutations it performs break thread-safety, and thus it must be used behind a lock in a concurrent context.
     
    164168
    165169While Conchon and Filli\^{a}tre~\cite{Conchon07} implement their persistent union-find data structure over a universe of integer elements in the fixed range $[1,N]$, the type environment problem needs more flexibility.
    166 In particular, an arbitrary number of type variables must be added to the environment.
    167 As such, a persistent hash table is a more suitable structure than a persistent array, providing the same expected asymptotic time bounds while allowing a dynamic number of elements.
    168 Besides replacing the underlying array with a hash table, the other major change in this approach is to replace the two types of array references, !Array! and !Edit!, with four node types, !Table!,  !Edit!, !Add!, and !Remove!, where !Add! adds a new key-value pair, !Remove! removes a key, and !Edit! mutates an existing key-value pair.
    169 In this variant of \CFACC{}, this persistent hash table is used as the side map discussed in Section~\ref{env-union-find-approach} for class bounds.
    170 The actual union-find data structure is slightly modified from this approach, with a !Base! node containing the root union-find data structure, !Add! nodes adding new elements, !AddTo! nodes defining the union of two type classes, and !Remove! and !RemoveFrom! nodes as inverses of the previous two elements, for purposes of maintaining the edit list; Figure~\ref{persistent-union-find-fig} demonstrates the structure of a simple example.
     170In particular, an arbitrary number of type variables may be added to the environment.
     171As such, a persistent hash table is a more suitable structure than a persistent array, providing the same expected asymptotic time bounds, while allowing a dynamic number of elements.
     172Besides replacing the underlying array with a hash table, the other major change in this approach is to replace the two types of array references, !Array! and !Edit!, with four node types, !Table!,  !Edit!, !Add!, and !Remove!, where !Add! adds a new key-value pair, !Remove! removes a key-value pair, and !Edit! mutates an existing key-value pair.
     173In this variant of \CFACC{}, this persistent hash-table is used as the side map discussed in Section~\ref{env-union-find-approach} for class bounds.
     174The actual union-find data structure is slightly modified from this approach, with a !Base! node containing the root union-find data structure, !Add! nodes adding new elements, !AddTo! nodes defining the union of two type classes, and !Remove! and !RemoveFrom! nodes as inverses of the previous two elements, for purposes of maintaining the edit list.
     175Figure~\ref{persistent-union-find-fig} demonstrates the structure of a simple example.
    171176Making !AddTo! and !RemoveFrom! single nodes provides semantic information missing from the raw array updates in Conchon and Filli\^{a}tre's data structure.
    172177!RemoveFrom! is implemented using the ``leaf of last union'' approach discussed in Section~\ref{env-union-find-classes-approach}; this does, however, preclude the use of path-compression algorithms in this persistent union-find data structure.
     
    181186This added semantic information on $union$ operations in the persistent union-find edit tree exposes a new option for combining type environments.
    182187If the type environments are part of the same edit tree, one environment $T'$ can be combined with another $T$ by only testing the edits on the path from $T'$ to $T$ in both the persistent union-find data structure describing the classes and the persistent hash table containing the class bounds.
    183 This is generally more efficient than testing the compatibility of all type classes in $T'$, as only those that are actually different than those in $T$ must be considered.
    184 This improved performance comes at the cost of some flexibility, however, as the edit-tree link must be maintained between any two environments to be combined under this algorithm.
     188This approach is generally more efficient than testing the compatibility of all type classes in $T'$, as only those that are actually different than those in $T$ must be considered.
     189However, the improved performance comes at the cost of some flexibility, as the edit-tree link must be maintained between any two environments to be combined under this algorithm.
    185190
    186191The procedure for $combine(T, T')$ based on edit paths is as follows:
    187192The shared edit trees for classes and bindings are rerooted at $T$, and the path from $T'$ to $T$ is followed to create a list of actual edits.
    188193By tracking the state of each element, redundant changes such as an !Edit! followed by an !Edit! can be reduced to their form in $T'$ by dropping the later (more like $T$) !Edit! for the same key; !Add! and !Remove! cancel similarly.
    189 This procedure is repeated for both the class edit tree and the binding edit tree.
    190 When the list of net changes to the environment has been produced, the additive changes are applied to $T$.
    191 For example, if a type class exists in $T'$ but not $T$, the corresponding !Add! edit will be applied to $T$, but in the reverse situation the !Remove! edit will not be applied to $T$, as the intention is to produce a new environment representing the union of the two sets of type classes; similarly, !AddTo! edits are applied to unify type-classes in $T$ that are united in $T'$, but !RemoveFrom! edits that split type classes are not.
    192 The new environment, $T''$ can always be constructed with a consistent partitioning of type variables; in the extreme case, all variables from both $T$ and $T'$ will be united in a single type class in $T''$.
    193 Where $combine$ can fail is in unifying the bound types; if any class in $T'$ has a class bound which does not unify with the merged class in $T''$ than $combine$ fails.
    194 
    195 \section{Analysis}
    196 
    197 In this section I present asymptotic analyses of the various approaches to a type environment data structure discussed in the previous section.
     194This procedure is repeated for both the class edit-tree and the binding edit-tree.
     195When the list of net changes to the environment is produced, the additive changes are applied to $T$.
     196For example, if a type class exists in $T'$ but not $T$, the corresponding !Add! edit is applied to $T$, but in the reverse situation the !Remove! edit is not applied to $T$, as the intention is to produce a new environment representing the union of the two sets of type classes; similarly, !AddTo! edits are applied to unify type-classes in $T$ that are united in $T'$, but !RemoveFrom! edits that split type classes are not.
     197A new environment, $T''$, can always be constructed with a consistent partitioning of type variables; in the extreme case, all variables from both $T$ and $T'$ are united in a single type class in $T''$.
     198$combine$ can fail to unify the bound types; if any class in $T'$ has a class bound that does not unify with the merged class in $T''$, then $combine$ fails.
     199
     200\section{Analysis} \label{env-analysis-sec}
     201
     202In this section, I present asymptotic analyses of the various approaches to the type environment data structure discussed in the previous section.
    198203My results are summarized in Table~\ref{env-bounds-table}; in all cases, $n$ is the number of type classes, $m$ is the maximum size of a type class, and $p$ the number of edits between two environments or one environment and the empty environment.
    199 $u(n)$ captures the recursive cost of class unification; I have kept this separate so that the $O(n)$ number of recursive class unifications can be separated from the direct cost of each recursive step.
     204$u(n)$ captures the recursive cost of class unification, which is kept separate so that the $O(n)$ number of recursive class unifications can be distinguished from the direct cost of each recursive step.
    200205
    201206\begin{table}
     
    205210\begin{tabular}{rllll}
    206211        \hline
    207                                 & \textbf{Na\"{\i}ve}   & \textbf{Incremental}  & \textbf{Union-Find}           & \textbf{Persistent U-F}               \\
    208         \hline
    209         $find$          & $O(n)$                                & $O(p)$                                & $O(\alpha(m))$                        & $O(\log m)$                                   \\
    210         $report$        & $O(m)$                                & $O(m)$                                & $O(n\alpha(m))$                               & $O(m)$                                                \\
    211         $bound$         & $O(1)$                                & $O(1)$                                & $O(1)$                                        & $O(1)$                                                \\
    212         $insert$        & $O(1)$                                & $O(1)$                                & $O(1)$                                        & $O(1)$                                                \\
    213         $add$           & $O(1)$                                & $O(m)$                                & $O(1)$                                        & $O(1)$                                                \\
    214         $bind$          & $O(1)$                                & $O(1)$                                & $O(1)$                                        & $O(1)$                                                \\
    215         $unify$         & $O(m + u(n))$                 & $O(m + u(n))$                 & $O(1 + u(n))$         & $O(1 + u(n))$                 \\
    216         $split$         & ---                                   & ---                                   & ---                                           & $O(\log m)$                                   \\
    217         $combine$       & $O(n^2m + nu(n))$     & $O(p^2m + pu(n))$     & $O(n\alpha(m) + nu(n))$       & $O(p \log m + pu(n))$         \\
    218         $save$          & $O(nm)$                               & $O(1)$                                & $O(nm)$                                       & $O(1)$                                                \\
    219         $backtrack$     & $O(nm)$                               & $O(pm)$                               & $O(nm)$                                       & $O(p)$                                                \\
     212                                & \textbf{Na\"{\i}ve} & \textbf{Incremental} & \textbf{Union-Find} & \textbf{Persistent U-F} \\
     213        \hline
     214        $find$          & $O(n)$                        & $O(p)$                        & $O(\alpha(m))$        & $O(\log m)$           \\
     215        $report$        & $O(m)$                        & $O(m)$                        & $O(nm\alpha(m))$      & $O(m)$                        \\
     216        $bound$         & $O(1)$                        & $O(1)$                        & $O(1)$                        & $O(1)$                        \\
     217        $insert$        & $O(1)$                        & $O(1)$                        & $O(1)$                        & $O(1)$                        \\
     218        $add$           & $O(1)$                        & $O(m)$                        & $O(1)$                        & $O(1)$                        \\
     219        $bind$          & $O(1)$                        & $O(1)$                        & $O(1)$                        & $O(1)$                        \\
     220        $unify$         & $O(m + u(n))$         & $O(m + u(n))$         & $O(1 + u(n))$         & $O(1 + u(n))$         \\
     221        $split$         & ---                           & ---                           & ---                           & $O(\log m)$           \\
     222        $combine$       & $O(n^2m $                     & $O(p^2m $                     & $O(nm\alpha(m) $      & $O(p \log m $         \\
     223                                & $~~~+ nu(n))$         & $~~~+ pu(n))$         & $~~~+ nu(n))$         & $~~~+ pu(n))$         \\
     224        $save$          & $O(nm)$                       & $O(1)$                        & $O(nm)$                       & $O(1)$                        \\
     225        $backtrack$     & $O(nm)$                       & $O(pm)$                       & $O(nm)$                       & $O(p)$                        \\
    220226        \hline
    221227\end{tabular}
     
    225231\label{naive-incremental-analysis}
    226232
    227 The na\"{\i}ve type environment data structure does not have an overall index for type variables, but does have an index for each type class, assumed to be hash-based here.
     233The na\"{\i}ve type environment data structure does not have an environment-wide index for type variables, but does have an index for each type class, assumed to be hash-based here.
    228234As a result, every class's index must be consulted for a $find$ operation, at an overall cost of $O(n)$.
    229235The incremental variant builds an overall hash-based index as it is queried, but may need to recursively check its parent environments if its local index does not contain a type variable, and may have as many parents as times it has been modified, for cost $O(p)$.
     
    238244The 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.
    239245Since 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 $O(n^2m + nu(n))$.
    240 The incremental $combine$ operation works similarly, but only needs to consider classes modified in either environment since the common ancestor of both environments, allowing the $n$ cost terms to be substituted for $p$, for an overall cost of $O(p^m + pu(n))$.
     246The 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 $O(p^m + pu(n))$.
    241247Neither variant supports the $split$ operation to undo a $unify$.
    242248
     
    248254
    249255The union-find data structure is designed to support $find$ efficiently, and thus for any variant, the cost is simply the distance up the tree to the representative element.
    250 For basic union-find, this is amortized to the inverse Ackermann function, $O(\alpha(m))$, essentially a small constant, though the loss of path compression in persistent union-find raises this cost to the still cheap $O(\log m)$.
    251 Basic union-find is not designed to support the $report$ operation, however, so it must be simulated by checking the representative of every type variable, at cost $O(n\alpha(m))$.
     256For basic union-find, this is amortized to the inverse Ackermann function, $O(\alpha(m))$, essentially a small constant, though the loss of path compression in persistent union-find raises this cost to $O(\log m)$.
     257Basic union-find is not designed to support the $report$ operation, however, so it must be simulated by checking the representative of every type variable, at cost $O(nm\alpha(m))$.
    252258Persistent union-find, on the other hand, uses the ``with classes'' extension to union-find described in Section~\ref{env-union-find-classes-approach} to run $report$ in $O(m)$ time.
    253259
     
    256262Since $add$ simply involves attaching a new child to the class root, it is also a $O(1)$ operation for all union-find environment variants.
    257263
    258 Union-find is also designed to support $unify$ in constant time, and as such, given class representatives, the variable-merging most of $unify$ for both variants is $O(1)$ to make one representative the child of the other, plus the $O(u(n))$ term for $unifyBound$.
     264Union-find is also designed to support $unify$ in constant time, and as such, given class representatives, the variable-merging cost of $unify$ for both variants is $O(1)$ to make one representative the child of the other, plus the $O(u(n))$ term for $unifyBound$.
    259265Basic union-find does not support $split$, but persistent union-find can accomplish it using the mechanism described in Section~\ref{env-union-find-classes-approach} in $O(\log m)$, the cost of traversing up to the root of a class from a leaf without path compression.
    260266$combine$ on the basic union-find data structure works similarly to the data structures discussed above in Section~\ref{naive-incremental-analysis}, with a $O(nu(n))$ term for the $O(n)$ underlying $unify$ operations, and a $O(n\alpha(m))$ term to find the classes which need unification by checking the class representatives of each corresponding type variable in both environments for equality.
     
    268274\section{Conclusion \& Future Work}
    269275
    270 This chapter has discussed the type environment data structure, as well as some more sophisticated approaches to optimize its performance for workloads encountered in the expression resolution problem, and asymptotic analysis of each approach.
     276This chapter presents the type environment data structure in Section~\ref{env-defn-sec}, as well as some more sophisticated approaches to optimize performance for workloads encountered in the expression resolution problem in Section~\ref{env-approaches-sec}, and asymptotic analysis of each approach in Section~\ref{env-analysis-sec}.
    271277Chapter~\ref{expr-chap} provides experimental performance results for a representative set of these approaches.
    272278One contribution of this thesis is the union-find with classes data structure for efficient retrieval of union-find class members, along with a related algorithm for reversing the history of $union$ operations in this data structure.
     
    275281
    276282This persistent union-find data structure is efficient, but not thread-safe; as suggested in Section~\ref{resn-conclusion-sec}, it may be valuable to parallelize the \CFA{} expression resolver.
    277 However, allowing multiple threads concurrent access to the persistent data structure is likely to result in a sort of ``reroot thrashing'', as different threads reroot the data structure to their own versions of interest.
    278 It is possible this could be mitigated by partitioning the data structure into separate subtrees for each thread, with each subtree having its own root node, and the boundaries between them implemented with a lock-equipped !ThreadBoundary! edit node.
     283However, allowing multiple threads concurrent access to the persistent data structure is likely to result in ``reroot thrashing'', as different threads reroot the data structure to their own versions of interest.
     284This contention could be mitigated by partitioning the data structure into separate subtrees for each thread, with each subtree having its own root node, and the boundaries among them implemented with a lock-equipped !ThreadBoundary! edit node.
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