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Start Implication Rules

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Andreas Stadelmeier 2024-06-10 09:03:46 +02:00
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aspUnify.tex
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@ -19,6 +19,11 @@
%\renewcommand\UrlFont{\color{blue}\rmfamily}
%\urlstyle{rm}
%
\include{prolog}
\usepackage{mathpartir}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
%
\title{Global Type Inference for Java using SAT Solvers}
@ -70,4 +75,78 @@ Java has adopted more and more type inference features over time.
Currently Java only has local type inference.
We want to bring type inference for Java to the next level.
\section{Unify}
We have to formulate the unification algorithm with implication rules.
Those can be directly translated to ASP.
\begin{mathpar}
\inferrule[Subst-L]{
\tv{a} \doteq \type{N} \\
\tv{a} \lessdot \type{T}
}{
\type{N} \lessdot \type{T}
}
\and
\inferrule[Subst-R]{
\tv{a} \doteq \type{N} \\
\type{T} \lessdot \tv{a}
}{
\type{T} \lessdot \type{N}
}
\and
\inferrule[Fail]{
\type{T} \lessdot \type{N}\\
\type{T} \nless : \type{N}
}{
\emptyset
}
\and
\inferrule[Fail]{
\exptype{C}{\ldots} \doteq \exptype{D}{\ldots}\\
\type{C} \neq \type{D}
}{
\emptyset
}
\and
\inferrule[Fail]{
\tv{a} \lessdot \type{T}_1 \\
\tv{a} \lessdot \type{T}_2 \\
\type{T}_1 \nless : \type{T}_2 \\
\type{T}_2 \nless : \type{T}_1
}{
\emptyset
}
\and
\inferrule[Swap]{
\type{T}_1 \doteq \type{T}_2
}{
\type{T}_2 \doteq \type{T}_1
}
\and
\inferrule[Adopt]{
\tv{a} \lessdot \type{T}_1 \\
\tv{a} \lessdot \type{T}_2 \\
\type{T}_1 <: \type{T}_2
}{
\type{T}_1 \lessdot \type{T}_2
}
\and
\inferrule[Subst-Param]{
\tv{a} \doteq \type{N} \\
\tv{a} = \type{T}_i \\
\exptype{C}{\type{T}_1 \ldots \type{T}_n} <: \type{T} \\
}{
\type{T}_i \doteq \type{N} \\
}
\end{mathpar}
% Subst
% a =. N, a <. T, N <: T
% --------------
% N <. T
% how to proof completeness and termination?
% TODO: how to proof termination?
\section{Completeness}
To proof completeness we have to show that every type can be replaced by a placeholder in a correct constraint set.
\end{document}