Match example
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unify.tex
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unify.tex
@ -247,10 +247,192 @@ Afterwards \rulename{Prepare} can be used eventually leading to the erasure of t
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}
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\end{displaymath}
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Note that the \rulename{Prepare} rule is always applied together with the \rulename{Capture} and the \rulename{Reduce} rule:
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\rulename{Trim} removes unused wildcard declarations
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whereas \rulename{Clear} and \rulename{Exclude} remove wildcard placeholders or wildcards to
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\rulename{Trim} removes unused wildcard declarations.
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\rulename{Clear} and \rulename{Exclude} remove wildcard placeholders or wildcards to
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allow the constraint to be processed by \rulename{Prepare}.
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\unify{} keeps $\tv{a} \lessdot \type{T}$ constraints as long as possible.
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The \rulename{Match} rule reduces two $\tv{a} \lessdot \type{T}$ constraints to one.
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There has to be a common subtype $\type{C}$ of $\type{D}$ and $\type{D'}$ for this rule to work
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expressed as premises $\type{C} \ll \type{D}$ and $\type{C} \ll \type{D'}$.
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The $\ll$ relation is the reflexive and transitive closure of the \texttt{extends} relations:
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\begin{displaymath}
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\begin{array}[c]{c}
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\exptype{C}{\ol{X} \triangleleft \ol{N}} \triangleleft \exptype{D}{\ol{N}} \\
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\hline
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\vspace*{-0.4cm}\\
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\texttt{C} \ll \texttt{D}
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\end{array}
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\quad
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\begin{array}[c]{l}
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\\
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\hline
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\vspace*{-0.4cm}\\
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\texttt{C} \ll \texttt{C}
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\end{array}
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\quad
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\begin{array}[c]{l}
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\texttt{C} \ll \texttt{D}, \texttt{D} \ll \texttt{E} \\
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\hline
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\vspace*{-0.4cm}\\
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\texttt{C} \ll \texttt{E}
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\end{array}
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\end{displaymath}
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For example the constraints $\tv{a} \lessdot \type{String}$ and $\tv{a} \lessdot \type{Integer}$
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are unsolvable, because there exists no common subtype of $\type{String}$ and $\type{Integer}$.
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Whereas the constraints $\tv{a} \lessdotCC \exptype{List}{\type{Integer}}$ and
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$\tv{a} \lessdotCC \exptype{List}{\tv{b}}$ can be processed by \rulename{Match}:
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\begin{displaymath}
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\prftree[r]{\rulename{Tame}}{
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\prftree[r]{\rulename{Reduce}}{
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\prftree[r]{\rulename{Capture}}{
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\prftree[r]{\rulename{Match}}{
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\tv{a} \lessdotCC \exptype{List}{\type{Integer}}, \tv{a} \lessdotCC \exptype{List}{\tv{b}}
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}{
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\begin{array}{l}
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\tv{a} \lessdot \wctype{\wildcard{A}{\tv{u}}{\tv{l}}}{List}{\rwildcard{A}}, \\
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\wctype{\wildcard{A}{\tv{u}}{\tv{l}}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\type{Integer}}, \\
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\wctype{\wildcard{A}{\tv{u}}{\tv{l}}}{List}{\rwildcard{A}} \lessdotCC \exptype{List}{\tv{b}}
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\end{array}
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}}{
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\wildcard{A}{\tv{u}}{\tv{l}} \vdash
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\begin{array}[t]{l}
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\tv{a} \lessdot \wctype{\wildcard{A}{\tv{u}}{\tv{l}}}{List}{\rwildcard{A}}, \\
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\exptype{List}{\rwildcard{A}} \lessdot \exptype{List}{\type{Integer}}, \\
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\exptype{List}{\rwildcard{A}} \lessdot \exptype{List}{\tv{b}}
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\end{array}
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}}{
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\wildcard{A}{\tv{u}}{\tv{l}} \vdash \tv{a} \lessdot \wctype{\wildcard{A}{\tv{u}}{\tv{l}}}{List}{\rwildcard{A}},
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\rwildcard{A} \doteq \type{Integer}, \rwildcard{A} \doteq \tv{b}
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}}{
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\wildcard{A}{\tv{u}}{\tv{l}} \vdash \tv{a} \lessdot \wctype{\wildcard{A}{\tv{u}}{\tv{l}}}{List}{\rwildcard{A}},
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\tv{u} \doteq \type{Integer}, \tv{l} \doteq \type{Integer}, \tv{u} \doteq \tv{b}
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}
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\end{displaymath}
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After \rulename{Subst} and \rulename{Same} the remaining constraints are $\tv{b} \doteq \type{Integer}$ and
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$\tv{a} \lessdot \wctype{\wildcard{A}{\type{Integer}}{\type{Integer}}}{List}{\rwildcard{A}}$
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%which is equal to $\tv{a} \lessdot \exptype{List}{\type{Integer}}$ and additionally we have $\tv{b} \doteq \type{Integer}$.
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\begin{figure}
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\begin{center}
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\leavevmode
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\fbox{
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\begin{tabular}[t]{l@{~}l}
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\rulename{Match}
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& $
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \, \set{
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\tv{a} \lessdot_1 \wctype{\Delta}{D}{\ol{T}}, \tv{a} \lessdot_2 \wctype{\Delta'}{D'}{\ol{T'}} }\\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash C \cup \, \left\{ \begin{array}[c]{l}
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\tv{a} \lessdot \wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}},
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\ol{\tv{l}} \lessdot \ol{\tv{u}}, \\
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\wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}}
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\lessdot_1 \wctype{\Delta}{D}{\ol{T}}, \\
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\wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}}
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\lessdot_2 \wctype{\Delta'}{D'}{\ol{T'}}
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\end{array}
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\right\}
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\end{array}
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&\begin{array}[c]{l}
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\text{fresh}\ \overline{\wildcard{A}{\tv{u}}{\tv{l}}} \\
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\type{C} \ll \type{D}\\
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\type{C} \ll \type{D'} % TODO: THe match rule has to pick the most general type for C
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\end{array}
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\end{array}
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$
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\\\\
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\ruleReduceWC{}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
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\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv
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\vdash C \cup \, \set{
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\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
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\ol{\wtv{a}} \lessdot [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{U}, [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{L} \lessdot \ol{\wtv{a}} }
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\ol{\wtv{a}} \ \text{fresh}\\
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%\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
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%\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
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%\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Capture}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \cup \overline{\wildcard{C}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{U}}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{L}}}
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\vdash C \cup \, \set{
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[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \type{T} }
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\ol{\rwildcard{C}} \ \text{fresh}\\
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%\text{fv}(\type{T}) \neq \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Adopt}
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& $
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \, \set{
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\tv{b} \lessdot_1 \tv{a},
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\tv{a} \lessdot_2 \type{N}, \tv{b} \lessdot_3 \type{N'}} \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash C \cup \, \set{
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\tv{b} \lessdot \type{N},
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\tv{b} \lessdot_1 \tv{a},
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\tv{a} \lessdot_2 \type{N} , \tv{b} \lessdot_3 \type{N'}
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}
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Adapt}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \, \set{ \wctype{\Delta}{C}{\ol{T}} \lessdot
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\wctype{\Delta'}{D'}{\ol{T'}} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash C \cup \, \set{ \wctype{\Delta}{D}{[\ol{\type{T}}/\ol{X}]\ol{S}} \lessdot \wctype{\Delta'}{D'}{\ol{T'}} }
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\end{array}
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& \begin{array}[c]{l}
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\type{C} \ll \type{D'} \\
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\texttt{class} \ \exptype{C}{\ol{X} \triangleleft \ol{N}} \triangleleft \exptype{D}{\ol{S}}
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\end{array}
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\end{array}
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$
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\end{tabular}}
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\end{center}
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\caption{Constraint reduce rules}\label{fig:reduce-rules}
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\end{figure}
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\subsection{Adding Wildcards to the mix}
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%\unify{} is able to create wildcard solutions even when the input set of constraints do not contain any wildcard variables.
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%Input constraints originating from a completely untyped input program do not contain any existential types.
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@ -590,10 +772,6 @@ Otherwise the generation rules \rulename{GenSigma} and \rulename{GenDelta} will
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\end{description}
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The $\wtv{a}$ type variables are flagged as wildcard type variables.
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These type variables can be substituted by a wildcard or a type with free wildcard variables.
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As long as a type variable is flagged as $\wtv{a}$ it can be used by the \rulename{Subst-WC} rule in step 1.
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With \texttt{C} being class names and \texttt{A} being wildcard names.
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The wildcard type $\wildcard{X}{U}{L}$ consist of an upper bound $\type{U}$, a lower bound $\type{L}$
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and a name $\mathtt{X}$.
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@ -617,36 +795,10 @@ This is used by the \rulename{Tame} rule.
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\item[$\tph{}$] returns all type placeholders inside a given type.
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\textit{Example:} $\tph(\wctype{\wildcard{X}{\tv{a}}{\bot}}{Pair}{\wtv{b},\rwildcard{X}}) = \set{\tv{a}, \wtv{b}}$
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\item [$\ll$ relation:]
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The $\ll$ relation is the reflexive and transitive closure of the \texttt{extends} relations:
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\begin{displaymath}
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\begin{array}[c]{c}
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\exptype{C}{\ol{X} \triangleleft \ol{N}} \triangleleft \exptype{D}{\ol{N}} \\
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\hline
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\vspace*{-0.4cm}\\
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\texttt{C} \ll \texttt{D}
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\end{array}
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\quad
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\begin{array}[c]{l}
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\\
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\hline
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\vspace*{-0.4cm}\\
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\texttt{C} \ll \texttt{C}
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\end{array}
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\quad
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\begin{array}[c]{l}
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\texttt{C} \ll \texttt{D}, \texttt{D} \ll \texttt{E} \\
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\hline
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\vspace*{-0.4cm}\\
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\texttt{C} \ll \texttt{E}
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\end{array}
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\end{displaymath}
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The algorithm uses it to determine if two types are possible subtypes of one another.
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This is needed in the \rulename{adapt} and \rulename{match} rules.
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%\textbf{Wildcard renaming}\\
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\item[Wildcard renaming:]
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The \rulename{reduce} rule separates wildcards from their environment.
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The \rulename{Reduce} rule separates wildcards from their environment.
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At this point each wildcard gets a new and unique name.
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To only rename the respective wildcards the reduce rule renames wildcards up to alpha conversion:
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($[\ol{C}/\ol{B}]$ in the \rulename{reduce} rule)
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@ -727,6 +879,19 @@ which are used for the upper and lower bounds.
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% if there are a <. List<x?> constraints remaining in the end, then this can be a sign of a irregular input constraint set.
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\unify{} must not replace normal type placeholders with free variables
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except variables initially passed by $\Delta_{in}$.
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The \rulename{Subst} rule checks if a type $\type{T}$ contains any
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free variables or wildcard placeholders before replacing a normal type placeholder with it.
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This ensures that free variables are never substituted for normal type placeholders.
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\rulename{Subst-WC} does not need to do that and can freely replace wildcard placehodlers with types despite their free variables.
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We do not keep replacements for wildcard placeholders and they also will not show up in the final type solution.
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If the \rulename{Subst} rule is not applicable then either the \rulename{Normalize}
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or \rulename{Contract} transformation has to be used to remove wildcard placeholders and
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wildcards.
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\begin{figure}
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\begin{center}
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\leavevmode
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@ -779,6 +944,9 @@ C \cup [\type{U}/\type{A}]\set{\ntv{a} \doteq \type{T}, \type{L} \doteq \type{U}
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\caption{Substitution rules}\label{fig:subst-rules}
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\end{figure}
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% all possible variations have to be converted
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There are n different rules to deal with $\type{N} \lessdot \type{N}$ constraints.
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Prepare, Capture, Reduce, Trim, Clear, Exclude, Adapt
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@ -916,124 +1084,6 @@ Prepare, Capture, Reduce, Trim, Clear, Exclude, Adapt
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\caption{Rules for normal placeholders}\label{fig:reduce-rules}
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\end{figure}
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\begin{figure}
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\begin{center}
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\leavevmode
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\fbox{
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\begin{tabular}[t]{l@{~}l}
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\rulename{Match}
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& $
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \, \set{
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\tv{a} \lessdot_1 \wctype{\Delta}{D}{\ol{T}}, \tv{a} \lessdot_2 \wctype{\Delta'}{D'}{\ol{T'}} }\\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash C \cup \, \left\{ \begin{array}[c]{l}
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\tv{a} \lessdot \wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}},
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\ol{\tv{l}} \lessdot \ol{\tv{u}}, \\
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\wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}}
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\lessdot_1 \wctype{\Delta}{D}{\ol{T}}, \\
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\wctype{\overline{\wildcard{A}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{A}}}
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\lessdot_2 \wctype{\Delta'}{D'}{\ol{T'}}
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\end{array}
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\right\}
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\end{array}
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&\begin{array}[c]{l}
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\text{fresh}\ \overline{\wildcard{A}{\tv{u}}{\tv{l}}} \\
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\type{C} \ll \type{D}\\
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\type{C} \ll \type{D'} % TODO: THe match rule has to pick the most general type for C
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\end{array}
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\end{array}
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$
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\\\\
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\ruleReduceWC{}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
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\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv
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\vdash C \cup \, \set{
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\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
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\ol{\wtv{a}} \lessdot [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{U}, [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{L} \lessdot \ol{\wtv{a}} }
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\ol{\wtv{a}} \ \text{fresh}\\
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%\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
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%\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
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%\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Capture}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \cup \overline{\wildcard{C}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{U}}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{L}}}
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\vdash C \cup \, \set{
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[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \type{T} }
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\ol{\rwildcard{C}} \ \text{fresh}\\
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%\text{fv}(\type{T}) \neq \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Adopt}
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& $
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
|
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\wildcardEnv \vdash C \cup \, \set{
|
||||
\tv{b} \lessdot_1 \tv{a},
|
||||
\tv{a} \lessdot_2 \type{N}, \tv{b} \lessdot_3 \type{N'}} \\
|
||||
\hline
|
||||
\vspace*{-0.4cm}\\
|
||||
\wildcardEnv \vdash C \cup \, \set{
|
||||
\tv{b} \lessdot \type{N},
|
||||
\tv{b} \lessdot_1 \tv{a},
|
||||
\tv{a} \lessdot_2 \type{N} , \tv{b} \lessdot_3 \type{N'}
|
||||
}
|
||||
\end{array}
|
||||
\end{array}
|
||||
$
|
||||
\\\\
|
||||
\rulename{Adapt}
|
||||
&
|
||||
$
|
||||
\begin{array}[c]{@{}ll}
|
||||
\begin{array}[c]{l}
|
||||
\wildcardEnv \vdash C \cup \, \set{ \wctype{\Delta}{C}{\ol{T}} \lessdot
|
||||
\wctype{\Delta'}{D'}{\ol{T'}} } \\
|
||||
\hline
|
||||
\vspace*{-0.4cm}\\
|
||||
\wildcardEnv \vdash C \cup \, \set{ \wctype{\Delta}{D}{[\ol{\type{T}}/\ol{X}]\ol{S}} \lessdot \wctype{\Delta'}{D'}{\ol{T'}} }
|
||||
|
||||
\end{array}
|
||||
& \begin{array}[c]{l}
|
||||
\type{C} \ll \type{D'} \\
|
||||
\texttt{class} \ \exptype{C}{\ol{X} \triangleleft \ol{N}} \triangleleft \exptype{D}{\ol{S}}
|
||||
\end{array}
|
||||
\end{array}
|
||||
$
|
||||
\end{tabular}}
|
||||
\end{center}
|
||||
\caption{Constraint reduce rules}\label{fig:reduce-rules}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
If we find an illicit constraint assigning a type containing free variables to a type placeholder not flagged as a wildcard placeholder the algorithm fails.
|
||||
|
||||
@ -1366,20 +1416,15 @@ $\begin{array}{c}
|
||||
\end{NiceTabular}
|
||||
|
||||
\subsection{Capture Conversion during Unification}
|
||||
The \unify{} algorithm applies a capture conversion when needed.
|
||||
A constraint of the form $\wcNtype{\Delta'}{N} \lessdot \type{T}$,
|
||||
where $\text{fv}(\type{T}) \neq \emptyset$ is not solvable without capture conversion.
|
||||
\unify{} converts those constraints to $\type{N} \lessdot \type{T}$.
|
||||
This is only possible for subtype constraints which originated from a method call.
|
||||
% The \unify{} algorithm applies a capture conversion when needed.
|
||||
% A constraint of the form $\wcNtype{\Delta'}{N} \lessdot \type{T}$,
|
||||
% where $\text{fv}(\type{T}) \neq \emptyset$ is not solvable without capture conversion.
|
||||
% \unify{} converts those constraints to $\type{N} \lessdot \type{T}$.
|
||||
% This is only possible for subtype constraints which originated from a method call.
|
||||
|
||||
Capture conversion only works with constraints containing free variables.
|
||||
It also introduces fresh free variables into the constraint set.
|
||||
Both have to be regulated.
|
||||
It is not allowed to substitute free type variables freely.
|
||||
The algorithm introduces a new type of variables: $\wtv{a}$.
|
||||
\unify{} treats those as free type variables.
|
||||
This makes it possible to replace a $\wtv{a}$ with a captured wildcard variable
|
||||
without having to worry about introducing free type variables at unwanted places.
|
||||
|
||||
The challenge for a type inference algorithm is to apply capture conversion during type inference.
|
||||
Given a program
|
||||
|
||||
Loading…
Reference in New Issue
Block a user