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Add A-Normal Form transform

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Andreas Stadelmeier 2024-03-18 14:57:56 +01:00
parent e7b6786f08
commit e9f86ffda3
4 changed files with 141 additions and 56 deletions
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43
constraints.tex
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@ -64,12 +64,10 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
\type{T}, \type{U} &::= \tv{a} \mid \wtv{a} \mid \mv{X} \mid {\wcNtype{\Delta}{N}} && \text{types and type placeholders}\\ \type{T}, \type{U} &::= \tv{a} \mid \wtv{a} \mid \mv{X} \mid {\wcNtype{\Delta}{N}} && \text{types and type placeholders}\\
\type{N} &::= \exptype{C}{\il{T}} && \text{class type (with type variables)} \\ \type{N} &::= \exptype{C}{\il{T}} && \text{class type (with type variables)} \\
% Constraints % Constraints
\simpleCons &::= \type{T} \lessdot \type{U} && \text{subtype constraint}\\ \constraint &::= \type{T} \lessdot \type{U} \mid \type{T} \lessdotCC \type{U} && \text{Constraint}\\
\orCons{} &::= \set{\set{\overline{\simpleCons_1}}, \ldots, \set{\overline{\simpleCons_n}}} && \text{or-constraint}\\
\constraint &::= \simpleCons \mid \orCons && \text{constraint}\\
\consSet &::= \set{\constraints} && \text{constraint set}\\ \consSet &::= \set{\constraints} && \text{constraint set}\\
% Method assumptions: % Method assumptions:
\methodAssumption &::= \exptype{C}{\ol{X} \triangleleft \ol{N}}.\texttt{m} : \exptype{}{\ol{Y} \methodAssumption &::= \texttt{m} : \exptype{}{\ol{Y}
\triangleleft \ol{P}}\ \ol{\type{T}} \to \type{T} && \triangleleft \ol{P}}\ \ol{\type{T}} \to \type{T} &&
\text{method \text{method
type assumption}\\ type assumption}\\
@ -79,7 +77,7 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
& \text{method type environment} \\ & \text{method type environment} \\
\typeAssumptionsSymbol &::= ({\mtypeEnvironment} ; \overline{\localVarAssumption}) \typeAssumptionsSymbol &::= ({\mtypeEnvironment} ; \overline{\localVarAssumption})
\end{align*} \end{align*}
\caption{Syntax of constraints and type assumptions.} \caption{Syntax of constraints and type assumptions}
\label{fig:syntax-constraints} \label{fig:syntax-constraints}
\end{figure} \end{figure}
@ -141,6 +139,23 @@ For the example above a correct solution would be $\sigma(\tv{a}) = \wctype{\rwi
% <X, Y> Y m(Example<X> this, Example<Y> p){ ... } % <X, Y> Y m(Example<X> this, Example<Y> p){ ... }
% \end{verbatim} % \end{verbatim}
\begin{displaymath}
\begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2) = \\
& \begin{array}{ll}
\textbf{let}
& \tv{e} \ \text{fresh} \\
& \consSet_R = \typeExpr({\mtypeEnvironment}, \expr{e}_1, \tv{e})\\
& \constraint =
\set{
\tv{e} \lessdotCC \tv{x}
}\\
{\mathbf{in}} & {
\consSet_R \cup \set{\constraint}} \cup \typeExpr(\mtypeEnvironment \cup \set{\expr{x} : \tv{x}})
\end{array}
\end{array}
\end{displaymath}
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}
\typeExpr{} &({\mtypeEnvironment} , \texttt{e}.\texttt{f}, \tv{a}) = \\ \typeExpr{} &({\mtypeEnvironment} , \texttt{e}.\texttt{f}, \tv{a}) = \\
@ -168,6 +183,24 @@ The set of method assumptions returned by the \textit{mtypes} function is used t
There are two kinds of method calls. There are two kinds of method calls.
The ones to already typed methods and calls to untyped methods. The ones to already typed methods and calls to untyped methods.
\begin{displaymath}
\begin{array}{@{}l@{}l}
\typeExpr{}' & ({\mtypeEnvironment} , \expr{v}.\mathtt{m}(\overline{\expr{v}}), \tv{a}) = \\
& \begin{array}{ll}
\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{Y} \lessdot \ol{N} }\\
\mathbf{in} & (\consSet_R \cup \overline{\consSet} \cup \constraint, \type{T}) \\
& \mathbf{where}\ \begin{array}[t]{l}
\expr{v}, \ol{v} : \ol{S} \in \localVarAssumption \\
\texttt{m} : \generics{\ol{Y} \triangleleft \ol{N}}\overline{\type{T}} \to \type{T} \in {\mtypeEnvironment}
\end{array}
\end{array}
\end{array}
\end{displaymath}
\begin{displaymath} \begin{displaymath}
\begin{array}{@{}l@{}l} \begin{array}{@{}l@{}l}

33
introduction.tex
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@ -521,19 +521,34 @@ The \unify{} algorithm only sees the constraints with no information about the p
The main challenge was to find an algorithm which computes $\sigma(\wtv{a}) = \rwildcard{X}$ for example \ref{intro-example1} but not for example \ref{intro-example2}. The main challenge was to find an algorithm which computes $\sigma(\wtv{a}) = \rwildcard{X}$ for example \ref{intro-example1} but not for example \ref{intro-example2}.
\subsection{ANF transformation} \subsection{ANF transformation}
The input is transformed to A-normal form.
%TODO: describe ANF syntax (which is different then the one from the wiki: https://en.wikipedia.org/wiki/A-normal_form) %TODO: describe ANF syntax (which is different then the one from the wiki: https://en.wikipedia.org/wiki/A-normal_form)
The input is a \letfj{} program without \texttt{let} statements, which is then transformed to A-normal form \cite{aNormalForm}
before getting processed by \fjtype{}.
$ $
\begin{array}{lrcl} \begin{array}{lrcl|l}
\hline \hline
\text{Terms} & t & ::= & v \\ & & & \textbf{\letfj{}} & \textbf{A-Normal form} \\
& & \ \ | & \texttt{let}\ x = v \ \texttt{in}\ x.f\\ \text{Terms} & t & ::=
& & \ \ | & \texttt{let}\ x_1 = v \ \texttt{in}\ \texttt{let}\ \overline{x} = \overline{v} \ \texttt{in}\ x_1.\texttt{m}(\overline{x})\\ & \expr{x}
& & \ \ | & t \elvis{} t\\ & \expr{x}
\text{Var} & v & ::= & \expr{x} \\ \\
& & \ \ | & \texttt{new} \ \type{C}(\overline{t})\\ & & \ \ |
\text{Variable contexts} & \Gamma & ::= & \overline{x:\type{T}}\\ & \texttt{new} \ \type{C}(\overline{t})
& \texttt{let}\ \overline{x} = \overline{t} \ \texttt{in}\ \texttt{new} \ \type{C}(\overline{x})
\\
& & \ \ |
& t.f
& \texttt{let}\ x = t \ \texttt{in}\ x.f
\\
& & \ \ |
& t.\texttt{m}(\overline{t})
& \texttt{let}\ x_1 = t \ \texttt{in}\ \texttt{let}\ \overline{x} = \overline{v} \ \texttt{in}\ x_1.\texttt{m}(\overline{x})
\\
& & \ \ |
& t \elvis{} t
& t \elvis{} t\\
%
\hline \hline
\end{array} \end{array}
$ $

10
martin.bib
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@ -456,3 +456,13 @@ keywords = {Compilation, Java, generic classes, language design, language semant
year={1996} year={1996}
} }
@article{aNormalForm,
title={Reasoning about programs in continuation-passing style.},
author={Sabry, Amr and Felleisen, Matthias},
journal={ACM SIGPLAN Lisp Pointers},
number={1},
pages={288--298},
year={1992},
publisher={ACM New York, NY, USA}
}

111
tRules.tex
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@ -196,72 +196,98 @@ $\begin{array}{l}
$\begin{array}{l} $\begin{array}{l}
\typerule{T-Field}\\ \typerule{T-Field}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad \Delta | \Gamma \vdash \expr{v} : \type{T} \quad \quad
\Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad \Delta \vdash \type{T} <: \wcNtype{}{N} \quad \quad
\textit{fields}(\type{N}) = \ol{U\ f} \\ \textit{fields}(\type{N}) = \ol{U\ f} \\
\Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
\\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S} \Delta | \Gamma \vdash \expr{v}.\texttt{f}_i : \type{U}_i
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
% $\begin{array}{l}
% \typerule{T-Field}\\
% \begin{array}{@{}c}
% \Delta | \Gamma \vdash \texttt{e} : \type{T} \quad \quad
% \Delta \vdash \type{T} <: \wcNtype{\Delta'}{N} \quad \quad
% \textit{fields}(\type{N}) = \ol{U\ f} \\
% \Delta, \Delta' \vdash \type{U}_i <: \type{S} \quad \quad
% \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
% \Delta \vdash \type{S}, \wcNtype{\Delta'}{N} \ \ok
% \\
% \hline
% \vspace*{-0.3cm}\\
% \Delta | \Gamma \vdash \texttt{e}.\texttt{f}_i : \type{S}
% \end{array}
% \end{array}$
% \\[1em]
% $\begin{array}{l}
% \typerule{T-New}\\
% \begin{array}{@{}c}
% \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad
% \text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad
% \Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad
% \Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\
% \Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad
% \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
% \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
% \Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok
% \\
% \hline
% \vspace*{-0.3cm}\\
% \triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}}
% \end{array}
% \end{array}$
% \\[1em]
$\begin{array}{l} $\begin{array}{l}
\typerule{T-New}\\ \typerule{T-New}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad \Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} \ \ok \quad \quad
\text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f} \quad \quad \Delta \vdash \overline{\type{S}} <: \overline{\type{U}} \\
\Delta | \Gamma \vdash \overline{t : \type{S}} \quad \quad \Delta | \Gamma \vdash \overline{\expr{v} : \type{S}} \quad \quad
\Delta \vdash \overline{\type{S}} <: \overline{\wcNtype{\Delta}{N}} \\ \text{fields}(\exptype{C}{\ol{T}}) = \overline{\type{U}\ f}
\Delta, \overline{\Delta} \vdash \overline{\type{N}} <: \overline{\type{U}} \quad \quad
\Delta, \overline{\Delta} \vdash \exptype{C}{\ol{T}} <: \type{T} \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} \quad \quad
\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{t}) : \exptype{C}{\ol{T}} \triangle | \Gamma \vdash \texttt{new}\ \type{C}(\ol{v}) : \exptype{C}{\ol{T}}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \hfill
$\begin{array}{l} % $\begin{array}{l}
\typerule{T-Call}\\ % \typerule{T-Call}\\
\begin{array}{@{}c} % \begin{array}{@{}c}
\Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad % \Delta, \Delta', \overline{\Delta} \vdash \ol{\type{N}} <: [\ol{S}/\ol{X}]\ol{U} \quad \quad
\generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad % \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'} % \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
\\ % \\
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad % \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \quad \quad
\Delta | \Gamma \vdash \texttt{e}, \ol{e} : \ol{T} \quad \quad % \Delta | \Gamma \vdash \texttt{e}, \ol{e} : \ol{T} \quad \quad
\Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}} % \Delta \vdash \ol{T} <: \overline{\wcNtype{\Delta}{N}}
\\ % \\
\Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad % \Delta \vdash \type{T}, \overline{\wcNtype{\Delta}{N}} \ \ok \quad \quad
\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad % \Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} <: \type{T} \quad \quad
\text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad % \text{dom}(\Delta') \subseteq \text{fv}(\type{N}) \quad \quad
\overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})} % \overline{\text{dom}(\Delta) \subseteq \text{fv}(\type{N})}
\\ % \\
\hline % \hline
\vspace*{-0.3cm}\\ % \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T} % \Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T}
\end{array} % \end{array}
\end{array}$ % \end{array}$
\\[1em] % \\[1em]
$\begin{array}{l} $\begin{array}{l}
\typerule{T-Call}\\ \typerule{T-Call}\\
\begin{array}{@{}c} \begin{array}{@{}c}
\generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad \generics{\ol{X \triangleleft U'}} \ol{U} \to \type{U} \in \Pi(\texttt{m}) \quad \quad
\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'} \Delta \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}
\\ \\
\Delta \vdash \ol{S} \ \ok \quad \quad \Delta \vdash \ol{S} \ \ok \quad \quad
\Delta | \Gamma \vdash \texttt{e}, \ol{e} : \ol{T} \quad \quad \Delta | \Gamma \vdash \expr{v}, \ol{v} : \ol{T} \quad \quad
\Delta \vdash \ol{T} <: [\ol{S}/\ol{X}]\ol{U} \Delta \vdash \ol{T} <: [\ol{S}/\ol{X}]\ol{U}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{e}.\texttt{m}(\ol{e}) : \type{T} \Delta | \Gamma \vdash \expr{v}.\texttt{m}(\ol{v}) : \type{T}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
@ -278,7 +304,7 @@ $\begin{array}{l}
\\ \\
\hline \hline
\vspace*{-0.3cm}\\ \vspace*{-0.3cm}\\
\Delta | \Gamma \vdash \texttt{let} \expr{x} : \wcNtype{\Delta'}{N} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2 : \type{T} \Delta | \Gamma \vdash \texttt{let}\ \expr{x} = \expr{e}_1 \ \texttt{in} \ \expr{e}_2 : \type{T}
\end{array} \end{array}
\end{array}$ \end{array}$
\\[1em] \\[1em]
@ -368,4 +394,5 @@ $\begin{array}{l}
\text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f} \text{fields}(\exptype{C}{\ol{T}}) = \ol{U\ g}, [\ol{T}/\ol{X}]\ol{S\ f}
\end{array} \end{array}
\end{array}$ \end{array}$
\caption{Field access}
\end{figure} \end{figure}