Cleanup, rephrase introduction. Fix and Cleanup type rules. Remove override
This commit is contained in:
Andreas Stadelmeier
parent
6c716c5138
commit
26678767c2
@ -126,7 +126,7 @@ $\begin{array}{rcl}
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\input{tRules}
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\input{tiRules}
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%\input{tiRules}
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\input{constraints}
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@ -1,14 +1,15 @@
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\section{Constraint generation}
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Our type inference algorithm is split into two parts.
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A constraint generation step \textbf{TYPE} and a \unify{} step.
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% Our type inference algorithm is split into two parts.
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% A constraint generation step \textbf{TYPE} and a \unify{} step.
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Method names are not unique.
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It is possible to define the same method in multiple classes.
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The \TYPE{} algorithm accounts for that by generating Or-Constraints.
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This can lead to multiple possible solutions.
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% Method names are not unique.
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% It is possible to define the same method in multiple classes.
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% The \TYPE{} algorithm accounts for that by generating Or-Constraints.
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% This can lead to multiple possible solutions.
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%\subsection{Well-Formedness}
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The \fjtype{} algorithm assumes capture conversions for every method parameter.
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%Why do we need a constraint generation step?
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%% The problem is NP-Hard
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@ -17,11 +17,21 @@ In Java this is done implicitly by a process called capture conversion \cite{Jav
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The type system in \cite{WildcardsNeedWitnessProtection} makes this process explicit by using \texttt{let} statements.
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Our type inference algorithm will accept an input program without let statements and add them where necessary.
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The input to the type inference algorithm is a Featherweight Java program without let statements (see figure \ref{fig:syntax}).
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The input to the type inference algorithm is a Featherweight Java program (example in figure \ref{fig:nested-list-example-typeless}) conforming to the syntax shown in figure \ref{fig:syntax}.
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Type inference adds \texttt{let} statements in a fashion similar to the Java capture conversion \cite{WildFJ}.
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We wrap every parameter of a method invocation in \texttt{let} statement unknowing if a capture conversion is necessary (see figure \ref{fig:nested-list-example-let}).
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The \fjtype{} algorithm calculates constraints based on this intermediate representation,
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which are then solved by the \unify{} algorithm
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resulting in a correctly typed program (see figure \ref{fig:nested-list-example-typed}).
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We figured the \texttt{let} statements to be obsolete for our use case.
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Once the type inference algorithm found a correct type solution
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Once the type inference algorithm found a correct type solution they can be inferred by the given type annotations.
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% 1. Constraint generation
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% 2. Insert typing
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% # Showing soundness
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% Every program in our calculus can be converted to a WildcardsNeedWitnessProtection program
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\begin{figure}[tp]
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\begin{subfigure}[t]{\linewidth}
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@ -48,6 +58,27 @@ class List<A> {
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List<A> add(A v) { ... }
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}
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class Example {
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m(l, la, lb){
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return let r2 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = {
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let r1 : (*@$\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}$@*) = l in {
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let p1 : (*@$\exptype{List}{\type{Integer}}$@*) = {
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let xa = la in xa.add(1)
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} in x1.add(p1)
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} in {
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let p2 = {
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let xb = lb in xb.add("str")
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} in x2.add(p2)
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};
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}
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}
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\end{lstlisting}
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\begin{lstlisting}[style=tfgj]
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class List<A> {
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List<A> add(A v) { ... }
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}
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class Example {
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m((*@$\exptype{List}{\wctype{\wildcard{X}{\type{Object}}{\bot}}{List}{\rwildcard{X}}}$@*) l, List<Integer> la, List<String> lb){
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return l
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@ -57,7 +88,7 @@ class Example {
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}
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\end{lstlisting}
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\caption{Featherweight Java Representation}
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\label{fig:nested-list-example-typed}
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\label{fig:nested-list-example-let}
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\end{subfigure}
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~
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\begin{subfigure}[t]{\linewidth}
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@ -198,8 +198,8 @@
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\end{tcolorbox}
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}
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\newcommand{\wcNtype}[2]{#1 .\ntype{#2}}
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\newcommand{\wctype}[3]{#1 .\exptype{#2}{#3}}
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\newcommand{\wcNtype}[2]{\exists #1 .\ntype{#2}}
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\newcommand{\wctype}[3]{\exists #1 .\exptype{#2}{#3}}
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\newcommand{\wtype}[1]{\mathit{#1}}
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\newcommand{\ntype}[1]{\mathtt{#1}}
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149
soundness.tex
149
soundness.tex
@ -420,3 +420,152 @@ $\sigma(\wildcardEnv) = \sigma([\type{T}/\tv{a}]\wildcardEnv)$
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%Proof by Lemma 5 \emph{Type substitution preserves subtyping} from \cite{WildcardsNeedWitnessProtection}.
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Same as Subst
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\end{description}
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\subsection{Converting to Wild FJ}
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Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
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They have been merged with let statements and simplified.
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The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
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are necessary for the soundness proof.
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%TODO: Show that well-formed implies witnessed!
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We change the type rules to require the well-formedness instead of the witnessed property.
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See figure \ref{fig:well-formedness}.
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Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
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\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
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With \texttt{witnessed} being the stronger one.
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We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
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$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
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Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
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Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
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The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
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The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
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Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
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$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
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A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
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$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
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$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
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and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
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\begin{figure}[tp]
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$\begin{array}{l}
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\typerule{T-Var}\\
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\begin{array}{@{}c}
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x : \type{T} \in \Gamma
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash x : \type{T}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-New}\\
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\begin{array}{@{}c}
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\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad
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\text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
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\Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
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\Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
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\Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad
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\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
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\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
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\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Field}\\
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\begin{array}{@{}c}
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\Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad
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\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
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\textit{fields}(\type{N}) = \ol{U\ f} \\
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\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
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\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
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\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Call}\\
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\begin{array}{@{}c}
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\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
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\textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad
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\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
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\\
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\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
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\Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad
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\Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad
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\Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
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\Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
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\\
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\Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
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\Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
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\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
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\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}
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{\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Elvis}\\
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\begin{array}{@{}c}
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\Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad
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\Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad
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\Delta \vdash \type{T}_1 <: \type{T} \quad \quad
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\Delta \vdash \type{T}_2 <: \type{T}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Method}\\
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\begin{array}{@{}c}
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\text{dom}(\Delta)=\ol{X} \quad \quad
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\Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad
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\Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad
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\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\
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\Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad
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\Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad
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\text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C}
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\end{array}
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\end{array}$
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\\[1em]
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$\begin{array}{l}
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\typerule{T-Class}\\
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\begin{array}{@{}c}
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\Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad
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\Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad
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\Delta \vdash \type{N} \ \ok \quad \quad
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\Delta \vdash \ol{M} \ \ok \texttt{ in C}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok
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\end{array}
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\end{array}$
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\caption{T-Call and T-Field} \label{fig:tletexpr}
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\end{figure}
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215
tRules.tex
215
tRules.tex
@ -1,4 +1,15 @@
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\section{Syntax}
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\section{Syntax and Typing}
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The input syntax for our algorithm is shown in figure \ref{fig:syntax}
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and the respective type rules in figure \ref{fig:expressionTyping} and \ref{fig:typing}.
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The type system in \cite{WildcardsNeedWitnessProtection} allows a method to \textit{override} an existing method declaration in one of its super classes,
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but only by a method with the exact same type.
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The type system presented here does not allow the \textit{overriding} of methods.
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Our type inference algorithm consumes the input classes in succession and could only do a type check instead of type inference
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on overriding methods, because their type is already determined.
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Allowing overriding therefore has no implication on our type inference algorithm.
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\begin{figure}
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$
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\begin{array}{lrcl}
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@ -21,13 +32,12 @@ $
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\caption{Input Syntax}\label{fig:syntax}
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\end{figure}
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Each class type has a set of wildcard types $\overline{\Delta}$ attached to it.
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The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$,
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which can be used inside the type parameters $\ol{T}$.
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\section{Type rules}
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% Each class type has a set of wildcard types $\overline{\Delta}$ attached to it.
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% The type $\wctype{\overline{\Delta}}{C}{\ol{T}}$ defines a set of wildcards $\overline{\Delta}$,
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% which can be used inside the type parameters $\ol{T}$.
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\begin{figure}[tp]
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\begin{center}
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$\begin{array}{l}
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\typerule{S-Refl}\\
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\begin{array}{@{}c}
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@ -74,7 +84,7 @@ $\begin{array}{l}
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \wctype{\Delta'}{C}{\ol{T}} <: \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{S}}
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\Delta \vdash \wcNtype{\Delta'}{\type{N}} <: \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}}
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\end{array}
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\end{array}$
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$\begin{array}{l}
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@ -82,15 +92,16 @@ $\begin{array}{l}
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\begin{array}{@{}c}
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\Delta', \Delta \vdash [\ol{T}/\ol{\type{X}}]\ol{L} <: \ol{T} \quad \quad
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\Delta', \Delta \vdash \ol{T} <: [\ol{T}/\ol{\type{X}}]\ol{U} \\
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\text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad \quad
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\text{fv}(\ol{T}) \subseteq \text{dom}(\Delta, \Delta') \quad
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\text{dom}(\Delta') \cap \text{fv}(\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}) = \emptyset
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\\
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\hline
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\vspace*{-0.3cm}\\
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\Delta \vdash \wctype{\Delta'}{C}{[\ol{T}/\ol{\type{X}}]\ol{S}} <:
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\wctype{\ol{\wildcard{X}{U}{L}}}{C}{\ol{S}}
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\Delta \vdash \wcNtype{\Delta'}{[\ol{T}/\ol{X}]\type{N}} <:
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\wcNtype{\ol{\wildcard{X}{U}{L}}}{N}
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\end{array}
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\end{array}$
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\end{center}
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\caption{Subtyping}\label{fig:subtyping}
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\end{figure}
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@ -141,69 +152,32 @@ $\begin{array}{l}
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\end{center}
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\caption{Well-formedness}\label{fig:well-formedness}
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\end{figure}
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%TODO: Proof that well-formed (ok) implies that a type is witnessed
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We change the type rules to require the well-formedness instead of the witnessed property.
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See figure \ref{fig:well-formedness}.
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Our well-formedness criteria is more restrictive than the ones used for \wildFJ{}.
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\cite{WildcardsNeedWitnessProtection} works with the two different judgements $\ok$ and \texttt{witnessed}.
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With \texttt{witnessed} being the stronger one.
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We rephrased the $\ok$ judgement to include \texttt{witnessed} aswell.
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$\type{T} \ \ok$ in this paper means $\type{T} \ \ok$ and $\type{T} \ \texttt{witnessed}$ in the sense of the \wildFJ{} type rules.
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Java's type system is complex enough as it is. Simplification, when possible, is always appreciated.
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Our $\ok$ rule may not be able to express every corner of the Java type system, but is sufficient to show soundness regarding type inference for a Java core calculus.
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The \rulename{WF-Class} rule requires upper and lower bounds to be direct subtypes of each other $\type{L} <: \type{U}$.
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The type rules from \cite{WildcardsNeedWitnessProtection} use a witness type instead.
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Stating that a type is well formed (\texttt{witnessed}) if there exists atleast one simple type $\type{N}$ as a possible subtype:
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$\Delta \vdash \type{N} <: \wctype{\Delta'}{C}{\ol{T}}$.
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A witness type is easy to find by replacing every wildcard $\ol{W}$ in $\wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$ by its upper bound:
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$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} <: \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}}$.
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$\Delta \vdash \exptype{C}{[\ol{U}/\ol{W}]\ol{T}} \ \texttt{witnessed}$ is given due to $\Delta \vdash \ol{T}, \ol{L}, \ol{U} \ \ok$
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and $\Delta \vdash \wctype{\ol{\wildcard{W}{U}{L}}}{C}{\ol{T}} \ \ok$.
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%Not necessary!
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% Additionally we do not allow nested wildcards. %TODO: Unify cannot create them (or does it?) What is when the input contains them
|
||||
|
||||
% The \unify{} algorithm is not capable of generating types like $\wctype{\rwildcard{X}, \wildcard{Y}{\bot}{\rwildcard{X}}}{Pair}{\rwildcard{X}, \rwildcard{Y}}$.
|
||||
% %But they could be created in intermidiate types (\ol{S}) in method calls by capture conversion:
|
||||
% class WList<A> extends List<List<? extends A>>
|
||||
% m(List<X> l) % TODO
|
||||
% m(X.Wlist<X>) % S = Y^X.List<Y>
|
||||
|
||||
% X.WList<X> <. List<x?>
|
||||
% X.List<Y^X.List<Y>> <. List<x?>
|
||||
% x =. Y^X.List<Y>
|
||||
|
||||
% %TODO: do we need \Delta' \Delta \vdash T, L ,U ok ? or is \Delta \vdashh ... sufficient?
|
||||
% %<A> m(List<? extends A> l)
|
||||
% %This is important because it means different things maybe for the proof of our OK being more restrictive than "witnessed"
|
||||
|
||||
% % where can nested wildcards occur?
|
||||
% WBox<X> extends Box<Box<? extends X>>
|
||||
% WBox<?> => Y.WBox<Y> <: Y.Box<X^Y.Box<X>>
|
||||
|
||||
% Everything else regarding subtyping stays the same as in \cite{WildcardsNeedWitnessProtection}.
|
||||
% \begin{lemma}
|
||||
% If $\Delta \vdash $
|
||||
% \end{lemma}
|
||||
|
||||
Figure \ref{fig:tletexpr} shows type rules for fields and method calls.
|
||||
They have been merged with let statements and simplified.
|
||||
|
||||
The let statements and the type $\wcNtype{\Delta'}{N}$, which the type inference algorithm can freely choose,
|
||||
are necessary for the soundness proof.
|
||||
|
||||
\begin{figure}[tp]
|
||||
\begin{center}
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Var}\\
|
||||
\begin{array}{@{}c}
|
||||
x : \type{T} \in \Gamma
|
||||
\texttt{x} : \type{T} \in \Gamma
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash x : \type{T}
|
||||
\triangle | \Gamma \vdash \texttt{x} : \type{T}
|
||||
\end{array}
|
||||
\end{array}$ \hfill
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Field}\\
|
||||
\begin{array}{@{}c}
|
||||
\Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad
|
||||
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
|
||||
\textit{fields}(\type{N}) = \ol{U\ f} \\
|
||||
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
|
||||
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
|
||||
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
@ -221,23 +195,7 @@ $\begin{array}{l}
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash \letstmt{\ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}{\texttt{new} \ \exptype{C}{\ol{T}}(\overline{t})} : \type{T}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Field}\\
|
||||
\begin{array}{@{}c}
|
||||
\Delta | \Gamma \vdash \texttt{t} : \type{T} \quad \quad
|
||||
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
|
||||
\textit{fields}(\type{N}) = \ol{U\ f} \\
|
||||
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
|
||||
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
|
||||
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash \texttt{let}\ x : \wcNtype{\Delta'}{N} = \texttt{t} \ \texttt{in}\ x.\texttt{f}_i : \type{S}
|
||||
\triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
@ -245,72 +203,109 @@ $\begin{array}{l}
|
||||
\typerule{T-Call}\\
|
||||
\begin{array}{@{}c}
|
||||
\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
|
||||
\textit{mtype}(\texttt{m}, \type{N}) = \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \quad \quad
|
||||
\generics{\ol{X \triangleleft U'}} \type{N} \to \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
|
||||
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
|
||||
\\
|
||||
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
|
||||
\Delta | \Gamma \vdash \texttt{t}_r : \type{T}_r \quad \quad
|
||||
\Delta | \Gamma \vdash \ol{t} : \ol{T} \quad \quad
|
||||
\Delta | \Gamma \vdash \texttt{e} : \type{T}_r \quad \quad
|
||||
\Delta | \Gamma \vdash \ol{e} : \ol{T} \quad \quad
|
||||
\Delta \vdash \type{T}_r <: \wcNtype{\Delta'}{N} \quad \quad
|
||||
\Delta \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
|
||||
\Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}}
|
||||
\\
|
||||
\Delta \vdash \type{T}, \wcNtype{\Delta'}{N}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
|
||||
\Delta, \Delta', \Delta'' \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
|
||||
\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
|
||||
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
|
||||
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash \letstmt{x : \wcNtype{\Delta'}{N} = t_r, \ol{x} : \ol{\wcNtype{\Delta}{N}} = \ol{t}}
|
||||
{\texttt{x}.\generics{\ol{S}}\texttt{m}(\ol{x})} : \type{T}
|
||||
\Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Elvis}\\
|
||||
\begin{array}{@{}c}
|
||||
\Delta | \Gamma \vdash t_1 : \type{T}_1 \quad \quad
|
||||
\Delta | \Gamma \vdash t_2 : \type{T}_2 \quad \quad
|
||||
\Delta \vdash \type{T}_1 <: \type{T} \quad \quad
|
||||
\Delta \vdash \type{T}_2 <: \type{T}
|
||||
\triangle | \Gamma \vdash \texttt{t} : \type{T}_1 \quad \quad
|
||||
\triangle | \Gamma \vdash \texttt{t}_2 : \type{T}_2 \quad \quad
|
||||
\triangle \vdash \type{T}_1 <: \type{T} \quad \quad
|
||||
\triangle \vdash \type{T}_2 <: \type{T} \quad \quad
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta | \Gamma \vdash t_1 \elvis{} t_2 : \type{T}
|
||||
\triangle | \Gamma \vdash \texttt{t}_1 \ \texttt{?:} \ \texttt{t}_2 : \type{T}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\end{center}
|
||||
\caption{Expression Typing}\label{fig:expressionTyping}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\begin{center}
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Method}\\
|
||||
\begin{array}{@{}c}
|
||||
\exptype{C}{\ol{X}} \to \ol{T} \to \type{T} \in \mathtt{\Pi}(\texttt{m})\quad \quad
|
||||
%\text{dom}(\triangle) = \ol{X} \quad \quad
|
||||
\triangle' = \overline{\type{Y} : \bot .. \type{P}} \quad \quad
|
||||
\triangle, \triangle' \vdash \ol{U}, \type{T}, \ol{T} \ \ok \\
|
||||
%\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \ \{ \ldots \} \\
|
||||
\mathtt{\Pi} | \triangle, \triangle' | \ol{x : T}, \texttt{this} : \exptype{C}{\ol{X}} \vdash \texttt{e} : \type{S} \quad \quad
|
||||
\triangle \vdash \type{S} <: \type{T}
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\triangle \vdash \texttt{m}(\ol{x}) = \texttt{e} \ \ok \text{ in C with } \generics{\ol{Y \triangleleft P}}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Method}\\
|
||||
\typerule{T-Class}\\
|
||||
\begin{array}{@{}c}
|
||||
\text{dom}(\Delta)=\ol{X} \quad \quad
|
||||
\Delta' = \overline{\type{Y} : \bot .. \type{U}} \quad \quad
|
||||
\Delta, \Delta' \vdash \ol{U}, \type{T}, \ol{T}\ \ok \quad \quad
|
||||
\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ldots} \\
|
||||
\Delta, \Delta' | \overline{x:\type{T}}, \texttt{this} : \exptype{C}{\ol{X}} \vdash t:\type{S} \quad \quad
|
||||
\Delta, \Delta' \vdash \type{S} <: \type{T} \quad \quad
|
||||
\text{override}(\texttt{m}, \type{N}, \generics{\ol{Y \triangleleft U}}\ol{T} \to \type{T})
|
||||
\mathtt{\Pi}' = \mathtt{\Pi} \cup \set{ \texttt{m} \mapsto (\exptype{C}{\ol{X}} \to \ol{T}_\texttt{m} \to \type{T}_\texttt{m}) \mid \texttt{m} \in \ol{M}} \\
|
||||
\mathtt{\Pi}'' = \mathtt{\Pi} \cup \set{ \texttt{m} \mapsto
|
||||
\generics{\ol{X \triangleleft \type{N}}, \ol{Y \triangleleft P}}(\exptype{C}{\ol{X}} \to \ol{T}_\texttt{m} \to \type{T}_\texttt{m}) \mid \texttt{m} \in \ol{M} } \\
|
||||
\triangle = \overline{\type{X} : \bot .. \type{U}} \quad \quad
|
||||
\triangle \vdash \ol{U}, \ol{T}, \type{N} \ \ok \quad \quad
|
||||
\triangle \vdash \ol{M} \ \ok \text{ in C with} \ \generics{\ol{Y \triangleleft P}}
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\Delta \vdash \generics{\ol{Y \triangleleft U}} \type{T} \ \texttt{m}(\overline{\type{T} \ x}) = t \ \ok \ \texttt{in C}
|
||||
\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \{ \ol{T\ f}; \ol{M} \} : \mathtt{\Pi}''
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\\[1em]
|
||||
$\begin{array}{l}
|
||||
\typerule{T-Class}\\
|
||||
\typerule{T-Program}\\
|
||||
\begin{array}{@{}c}
|
||||
\Delta = \overline{\type{X} : \bot .. \type{U}} \quad \quad
|
||||
\Delta \vdash \ol{U}, \ol{T} \ \ok \quad \quad
|
||||
\Delta \vdash \type{N} \ \ok \quad \quad
|
||||
\Delta \vdash \ol{M} \ \ok \texttt{ in C}
|
||||
\emptyset \vdash \texttt{L}_1 : \mathtt{\Pi}_1 \quad \quad
|
||||
\mathtt{\Pi}_1 \vdash \texttt{L}_2 : \mathtt{\Pi}_1 \quad \quad
|
||||
\ldots \quad \quad
|
||||
\mathtt{\Pi}_{n-1} \vdash \texttt{L}_n : \mathtt{\Pi}_n \quad \quad
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\texttt{class}\ \exptype{C}{\ol{X \triangleleft U}} \triangleleft \type{N} \set{\overline{\type{T}\ f}; \ol{M} } \ \ok
|
||||
\vdash \ol{L} : \mathtt{\Pi}_n
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\caption{T-Call and T-Field} \label{fig:tletexpr}
|
||||
\end{center}
|
||||
\caption{Class and Method Typing rules}\label{fig:typing}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\begin{figure}
|
||||
$\text{fields}(\exptype{Object}{}) = \emptyset$
|
||||
\quad \quad
|
||||
$\begin{array}{l}
|
||||
\typerule{F-Class}\\
|
||||
\begin{array}{@{}c}
|
||||
\texttt{class}\ \exptype{C}{\ol{X \triangleleft \_ }} \triangleleft \type{N} \set{\ol{S\ f}; \ol{M}} \quad \quad
|
||||
\text{fields}([\ol{T}/\ol{X}]\type{N}) = \ol{U\ g}
|
||||
\\
|
||||
\hline
|
||||
\vspace*{-0.3cm}\\
|
||||
\text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f}
|
||||
\end{array}
|
||||
\end{array}$
|
||||
\end{figure}
|
||||
19
tiRules.tex
19
tiRules.tex
@ -1,24 +1,5 @@
|
||||
\section{Typeless FJ}
|
||||
|
||||
\subsection{Syntax}
|
||||
We use the TameFJ syntax for the most part (\ref{fig:syntax-tamingfj}).
|
||||
|
||||
\begin{figure}[tp]
|
||||
\begin{align*}
|
||||
\mv N &::= \exptype{C}{\ol{W}}\\
|
||||
\mv T &::= \mv X \mid \wctype{\ol{W}}{C}{\ol{W}}\\
|
||||
\mv W &::= \mv T \mid \wildcard{Z}{T}{T}\\
|
||||
\mv L &::= \mathtt{class} \ \exptype{C}{\ol{X} \triangleleft \ol{T}} \triangleleft \ \mv N\ \{ \ol{T} \ \ol{f}; \,\mv K \, \ol{M} \} \\
|
||||
\mv K &::= \mv C(\ol{f})\ \{\mathtt{super}(\ol{f}); \ \mathtt{this}.\ol{f}=\ol{f};\} \\
|
||||
\mv M &::= \mathtt{m}(\ol{x})\ = \mv e \\
|
||||
\mv e &::= \mv x \mid \mv e.\mv f \mid
|
||||
\mv e.\mathtt{m}(\ol{e}) \mid \mathtt{new}\ \mathtt{C}(\ol{e})
|
||||
\mid \mv e \elvis{} \mv e
|
||||
\end{align*}
|
||||
\caption{Syntax of TameFJ + Type Inference}
|
||||
\label{fig:syntax-tamingfj}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Type System}
|
||||
|
||||
The type system is similar to the ones used in \emph{Wildcards need Witness Protection}.
|
||||
|
||||
Loading…
Reference in New Issue
Block a user