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Remove comments (cleanup). Add Clear and Exclude rules. Change Unify Soundness premise

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This commit is contained in:
Andreas Stadelmeier 2024-03-25 19:12:35 +01:00
parent f2002ea833
commit e40693a7de
3 changed files with 59 additions and 229 deletions
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6
constraints.tex
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@ -144,11 +144,11 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
\typeExpr{} &({\mtypeEnvironment} , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2) = \\
& \begin{array}{ll}
\textbf{let}
& \tv{e} \ \text{fresh} \\
& \tv{e}, \tv{x} \ \text{fresh} \\
& \consSet_R = \typeExpr({\mtypeEnvironment}, \expr{e}_1, \tv{e})\\
& \constraint =
\set{
\tv{e} \lessdotCC \tv{x}
\tv{e} \lessdot \tv{x}
}\\
{\mathbf{in}} & {
\consSet_R \cup \set{\constraint}} \cup \typeExpr(\mtypeEnvironment \cup \set{\expr{x} : \tv{x}})
@ -190,7 +190,7 @@ The ones to already typed methods and calls to untyped methods.
\textbf{let}
& \tv{r}, \ol{\tv{r}} \text{ fresh} \\
& \constraint = [\overline{\wtv{b}}/\ol{Y}]\set{
\ol{S} \lessdot \ol{T}, \type{T} \lessdot \tv{a},
\ol{S} \lessdotCC \ol{T}, \type{T} \lessdot \tv{a},
\ol{Y} \lessdot \ol{N} }\\
\mathbf{in} & (\consSet_R \cup \overline{\consSet} \cup \constraint, \type{T}) \\
& \mathbf{where}\ \begin{array}[t]{l}

19
soundness.tex
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@ -329,18 +329,17 @@ If there is a solution for a constraint set $C$, then there is also a solution f
% does this really work. We have to show that the algorithm never gets stuck as long as there is a solution
% maybe there are substitutions with types containing free variables that make additional solutions possible. Check!
\begin{lemma}\label{lemma:unifySoundness}
The \unify{} algorithm only produces correct output.
\begin{lemma}{\unify{} Soundness:}\label{lemma:unifySoundness}
\unify{}'s type solutions are correct respective to the subtyping rules defined in figure \ref{fig:subtyping}.
\begin{description}
\item[If] $(\sigma, \Delta) = \unify{}( \Delta', \, \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdotCC \type{T'} } )$ %\cup \overline{ \type{S} \doteq \type{S'} })$
%\item[and] $fv(\overline{ \type{S} }) = \emptyset$, $fv(\overline{ \type{T} }) = \emptyset$
\item[Then] there exists a $\sigma'$ with:
$\sigma \subseteq \sigma'$ and
$\Delta, \Delta' \vdash \overline{\sigma'(\type{S}) <: \sigma'(\type{T})}$
and $\Delta, \Delta', \overline{\Delta} \vdash \overline{\type{N} <: \sigma'(\type{T'})}$ where $\overline{\sigma(\type{S'}) = \wcNtype{\Delta}{N}}$,
otherwise $\Delta, \Delta' \vdash \overline{\sigma'(\type{S'}) <: \sigma'(\type{T'})}$
% and $\sigma(\type{T'}) = \sigma(\type{T'})$.
\item[Then] there exists a substitution $\sigma'$ and a set of types $\overline{\wcNtype{\Delta}{N}}$ with:
\begin{itemize}
\item $\sigma \subseteq \sigma'$
\item $\Delta, \Delta' \vdash \overline{\sigma'(\type{S}) <: \sigma'(\type{T})}$
\item $\Delta, \Delta' \vdash \overline{\sigma'(\type{S'}) <: \wcNtype{\Delta}{N}}$
\item $\Delta, \Delta', \overline{\Delta} \vdash \overline{\type{N} <: \sigma'(\type{T'})}$
\end{itemize}
\end{description}
\end{lemma}

263
unify.tex
View File

@ -259,11 +259,8 @@ $
\end{figure}
The capture constraints are preserved when applying the \rulename{Upper} rule.
%because \texttt{let} statements like
%let x : X = v in
%can be transformed to
%let x : U = v in
%The capture constraints are preserved when applying the \rulename{Upper} rule.
% This is legal: a T <c S constraint indicates a let-statement can be inserted. Therefore there must exist a type T' with T <. T' <c S
\begin{figure}
\begin{center}
@ -439,6 +436,11 @@ The capture constraints are preserved when applying the \rulename{Upper} rule.
\wildcardEnv \vdash C \cup \set{\type{N} \doteq \rwildcard{A}}\\
\hline
\wildcardEnv \vdash C \cup \set{\rwildcard{A} \doteq \type{N}}
\end{array} \quad
\begin{array}[c]{l}
\wildcardEnv \vdash C \cup \set{\ntv{a} \doteq \rwildcard{A}}\\
\hline
\wildcardEnv \vdash C \cup \set{\rwildcard{A} \doteq \ntv{a}}
\end{array}$
\end{tabular}}
\end{center}
@ -583,6 +585,46 @@ Their upper and lower bounds are fresh type variables.
\end{array}
$
\\\\
\rulename{Clear}
&
$
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \cup \set{\wildcard{A}{\type{U}}{\type{L}}} \vdash
C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
\hline
\vspace*{-0.4cm}\\
\subst{\type{U}}{\rwildcard{A}}\wildcardEnv \vdash
[\type{U}/\rwildcard{A}]C \cup \, [\type{U}/\rwildcard{A}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T}, \type{U} \doteq \type{L} } \\
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\Delta \neq \emptyset\\
\rwildcard{A} \in \text{fv}(\type{T})
\end{array}
\end{array}
$
\\\\
\rulename{Exclude}
&
$
\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash
C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
\hline
\vspace*{-0.4cm}\\
\subst{\tv{a}}{\wtv{a}}\wildcardEnv \vdash
[\tv{a}/\wtv{a}]C \cup \, [\tv{a}/\wtv{a}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T}, \type{U} \doteq \type{L} } \\
\end{array}
%\quad \ol{Y} = \textit{fresh}(\ol{X})
\quad \begin{array}[c]{l}
\Delta \neq \emptyset\\
\wtv{a} \in \text{fv}(\type{T}), \tv{a} \ \text{fresh}
\end{array}
\end{array}
$
\\\\
\rulename{Adopt}
& $
\begin{array}[c]{@{}ll}
@ -1002,137 +1044,6 @@ This builds a search tree over multiple possible solutions.
\label{fig:step2-rules}
\end{figure}
% \begin{figure}
% \begin{center}
% \fbox{
% \begin{tabular}[t]{l@{~}l}
% \rulename{Same}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \vdash
% C \cup \type{G} \lessdot \tv{a}\\
% \hline
% \wildcardEnv \vdash
% C \cup \set{
% \tv{a} \doteq \type{G}
% }
% \end{array} \quad \begin{array}[c]{l}
% \text{fv}(\type{G}) \in \Delta_{in}
% \end{array}
% $
% \\\\
% \rulename{\generalizeRule}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
% \hline
% \wildcardEnv \vdash C \cup \set{\wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a},
% \tv{a} \doteq \wctype{\overline{\wildcard{X}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{X}}},
% %\overline{\tv{l} \lessdot \tv{u}}, % not needed, due to subst and reduce rule which are used afterwards
% \overline{\tv{u} \lessdot \type{S}}
% }
% \end{array} \quad \begin{array}[c]{l}
% \texttt{class} \ \exptype{C}{\ol{X \triangleleft \type{S}}} \triangleleft \exptype{D}{\ol{N}} \\
% \text{fresh}\ \overline{\wildcard{X}{\tv{u}}{\tv{l}}}
% \end{array}
% $
% \\\\
% \rulename{Subst-X}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
% C \cup \rwildcard{X} \lessdot \tv{a}\\
% \hline
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
% C \cup \set{
% \tv{a} \doteq \rwildcard{X}
% }
% \end{array} \quad \begin{array}[c]{l}
% \rwildcard{X} \in \Delta_{in}
% \end{array}
% $
% \\\\
% \rulename{Gen-X}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
% C \cup \rwildcard{X} \lessdot \tv{a}\\
% \hline
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
% C \cup \set{
% \type{U} \lessdot \tv{a}
% }
% \end{array}
% $
% \\\\
% \rulename{Super}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
% \hline
% \wildcardEnv \vdash C \cup \set{ \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{N}} \lessdot \tv{a} }
% %\set{\wctype{\ol{\wtype{W}}}{D}{[\ol{X}/\ol{Y}]\ol{Z}} \lessdot \tv{a}}
% \end{array} \quad
% \begin{array}{l}
% \texttt{class} \ \exptype{C}{\ol{X}} \triangleleft \exptype{D}{\ol{N}} \\
% \ol{X} \notin \wildcardEnv \cup \Delta,\, \Delta' = \Delta \cap \text{fv}([\ol{T}/\ol{X}]\ol{N})
% \end{array}
% $
% \\\\
% \rulename{Settle}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
% \tv{a} \lessdot \tv{b}}
% \\
% \hline
% \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \tv{b}, \tv{b} \lessdot \type{N} }
% \end{array}
% $
% \\\\
% \rulename{Raise}
% & $
% \begin{array}[c]{l}
% \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
% \tv{a} \lessdot \tv{b}}
% \\
% \hline
% \wildcardEnv \vdash C \cup \set{\tv{a} \lessdot \type{N}, \type{N} \lessdot \tv{b} }
% \end{array}
% $
% \\\\
% \end{tabular}}
% \end{center}
% \caption{Step 2 branching: Multiple rules can be applied to the same constraint}
% \label{fig:step2-rules}
% \end{figure}
%For every $\type{T} \lessdot \tv{a}$ constraint, the unify algorithm has to consider every possible supertype of $\type{T}$.
%For every $\type{N}$ with $\type{T} \leq \type{N}$: ($\texttt{class} \ \exptype{T}{\ol{Y} \triangleleft \ol{N}} \triangleleft \type{N}$)
%There are two different ways of handling a $\type{T} \lessdot \tv{a}$ constraint:
%TODO: why the \generalizeRule is basically the Same rule for regular type placeholders
%where is the mistake in the old unify algorithm?
%when working with equality the problems arise! Free variables should not escape their scope
% Replacing regular type placeholders causes problems related to method calls and capture conversion.
% <X> List<X> same(List<X> a, List<X> b){}
% This program has no correct type. the same method requires
% \begin{lstlisting}
% List<?> f;
% List<?> problem(){
% return same(problem(), problem()) ?: f;
% }
% \end{lstlisting}
% \begin{constraints}
% r <. List<x?>
% r <. List<x?>
% X.List<X> <. r
% \end{constraints}
% %TODO
\unify{} generates wildcard types for constraints of the form $\type{N} \lessdot \tv{a}$ with the \rulename{\generalizeRule} rule.
Otherwise only the wildcards already defined in the input constraints will be included in the result.
\rulename{\generalizeRule} attempts to give $\tv{a}$ a more general type by replacing only the type parameters with fresh wildcards.
@ -1145,15 +1056,6 @@ need to be handled in a similiar fashion.
The type variable $\tv{b}$ could either be a sub or a supertype of the type $\type{N}$.
We have to consider both possibilities.
\\[1em]
% The specification of the \unify{} algorithm has only two rules generating $\doteq$-Constraints
% , \rulename{Reduce} and \rulename{Ground}.
% $\doteq$-Constraints and the accompaning substitutions are dangerous respective to the soundness of the algorithm.
% For the soundness proof of the \unify{} algorithm we have to show every generation of equals constraints
% and the subsequent application of the \rulename{subst} rule is correct.
% We try to use them as sparsely as possible to simplify the soundness proof.
% You can notice this at \rulename{Equals} or \rulename{Force}:
% Instead of setting $\type{U} \doteq \type{L}$, we say
% $\type{U} \lessdot \type{L}, \type{L} \lessdot \type{U}$.
\noindent
\textbf{Step 3}
@ -1164,23 +1066,6 @@ If $C$ does not contain any wildcard variables the algorithm proceeds with step
\begin{center}
\fbox{
\begin{tabular}[t]{l@{~}l}
% \rulename{Remove} % kann beim Subst step gemacht werden und ist nur nötig wenn ein a =. T constraint mit wtv(T) > 1 entsteht
% & $
% \begin{array}[c]{@{}ll}
% \begin{array}[c]{l}
% \wildcardEnv \vdash C \\
% % \cup \, \set{ \wtv{a} \lessdot \type{T} }\\
% \hline
% \vspace*{-0.4cm}\\
% \subst{\tv{a}}{\wtv{a}}\wildcardEnv \vdash [\tv{a}/\wtv{a}]C
% \end{array}
% &\begin{array}[c]{l}
% \wtv{a} \in C \\
% \tv{a} \ \text{fresh}
% \end{array}
% \end{array}
% $
% \\\\
\rulename{Remove-Cons} & $
\begin{array}[c]{l}
\wildcardEnv \vdash C \cup \set{\type{S} \lessdotCC \type{T} } \\
@ -1192,23 +1077,6 @@ If $C$ does not contain any wildcard variables the algorithm proceeds with step
\end{tabular}}
\end{center}
% \textbf{Step 4:}
% If there are constraints of the form $(\tv{a} \lessdot \tv{b})$ remaining in the constraint set then
% apply the \rulename{Sub-Elim} rule and start over with step 1.
% Otherwise continue to step 5.
% \begin{center}
% \fbox{
% \begin{tabular}[t]{l@{~}l}
% \rulename{SubElim}
% & $\begin{array}[c]{l}
% \wildcardEnv \vdash C \cup \set{\tv{a} \lessdot \tv{b}}\\
% \hline
% [\tv{a}/\tv{b}]\wildcardEnv \vdash [\tv{a}/\tv{b}]C \cup \set{ \tv{b} \doteq \tv{a} }
% \end{array}
% $
% \end{tabular}}
% \end{center}
\noindent
\textbf{Step 4:}
We apply the rules in figure \ref{fig:cleanup-rules} exhaustively and proceed with step 6.
@ -1242,43 +1110,6 @@ C \cup \set{ \tv{a} \doteq \bot }
%\text{length}( \overline{\tv{a} \lessdot \type{T}} ) > 1
\end{array}
\end{array}$
% \\\\ % The force rule is unnecessary because every type placeholder has an upper bound Object (a <. Object) The match rule eliminates those wildcards
% \rulename{Force} &$
% \begin{array}[c]{@{}ll}
% \begin{array}[c]{l}
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
% \vdash C \cup \, \set{
% \tv{a} \lessdot \type{N} } \\
% \hline
% \vspace*{-0.4cm}\\
% \wildcardEnv
% \vdash
% C \cup \, \set{ \tv{a} \lessdot [\type{U}/\rwildcard{X}]\type{N},
% \type{U} \doteq \type{L} }
% \end{array}
% &\begin{array}[c]{l}
% \type{X} \in \text{fv}(\type{N}) \\
% %\Delta' = \Delta \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
% \end{array}
% \end{array}$
% \\\\
% \rulename{FlatOut} &$
% \begin{array}[c]{@{}ll}
% \begin{array}[c]{l}
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
% \vdash C \cup \, \set{
% \tv{a} \lessdot \wcNtype{\Delta}{N} } \\
% \hline
% \vspace*{-0.4cm}\\
% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}}
% \vdash
% C
% \end{array}
% &\begin{array}[c]{l}
% \type{X} \in \text{fv}(\wcNtype{\Delta}{N}) \\
% \tv{a} \notin C , \, \tv{a} \notin \wildcardEnv, \tv{a} \notin \sigma
% \end{array}
% \end{array}$
\end{tabular}}
\end{center}
\caption{Cleanup rules}\label{fig:cleanup-rules}