Commit c6b32498 authored by Kim Nguyễn's avatar Kim Nguyễn
Browse files

WIP Journal version :

- Make all negative vertical space conditional.
- Add mention about timing in the experimental part.
parent 21a3e35e
......@@ -16,7 +16,7 @@ rules%
particular connective (here, a type constructor, that is, either
$\to$, or $\times$, or $b$), while identity rules (e.g., axioms and
cuts) and structural rules (e.g., weakening and contraction) do
not.\vspace{-3.3mm}}
not.\svvspace{-3.3mm}}
such as \Rule{Subs} and \Rule{Inter}. We handle this presence
in the classic way: we define an algorithmic system that tracks the
miminum type of an expression; this system is obtained from the
......@@ -76,7 +76,7 @@ things get more difficult, since a function can be typed by, say, a
union of intersection of arrows and negations of types. Checking that
the function has a functional type is easy since it corresponds to
checking that it has a type subtype of $\Empty{\to}\Any$. Determining
its domain and the type of the application is more complicated and needs the operators $\dom{}$ and $\circ$ we informally described in Section~\ref{sec:ideas} where we also introduced the operator $\worra{}{}$. These three operators are used by our algorithm and formally defined as:\vspace{-0.5mm}
its domain and the type of the application is more complicated and needs the operators $\dom{}$ and $\circ$ we informally described in Section~\ref{sec:ideas} where we also introduced the operator $\worra{}{}$. These three operators are used by our algorithm and formally defined as:\svvspace{-0.5mm}
\begin{eqnarray}
\dom t & = & \max \{ u \alt t\leq u\to \Any\}
\\[-1mm]
......@@ -93,12 +93,12 @@ We need similar operators for projections since the type $t$
of $e$ in $\pi_i e$ may not be a single product type but, say, a union
of products: all we know is that $t$ must be a subtype of
$\pair\Any\Any$. So let $t$ be a type such that $t\leq\pair\Any\Any$,
then we define:\vspace{-0.7mm}
then we define:\svvspace{-0.7mm}
\begin{equation}
\begin{array}{lcrlcr}
\bpl t & = & \min \{ u \alt t\leq \pair u\Any\}\qquad&\qquad
\bpr t & = & \min \{ u \alt t\leq \pair \Any u\}
\end{array}\vspace{-0.7mm}
\end{array}\svvspace{-0.7mm}
\end{equation}
All the operators above but $\worra{}{}$ are already present in the
theory of semantic subtyping: the reader can find how to compute them
......@@ -111,10 +111,10 @@ furthermore $t\leq\Empty\to\Any$, then $t \simeq \bigvee_{i\in
I}\left(\bigwedge_{p\in P_i}(s_p\to t_p)\bigwedge_{n\in
N_i}\neg(s_n'\to t_n')\right)$ with $\bigwedge_{p\in P_i}(s_p\to
t_p)\bigwedge_{n\in N_i}\neg(s_n'\to t_n') \not\simeq \Empty$ for all
$i$ in $I$. For such a $t$ and any type $s$ then we have:\vspace{-1.0mm}
$i$ in $I$. For such a $t$ and any type $s$ then we have:\svvspace{-1.0mm}
%
\begin{equation}
\worra t s = \dom t \wedge\bigvee_{i\in I}\left(\bigwedge_{\{P\subseteq P_i\alt s\leq \bigvee_{p \in P} \neg t_p\}}\left(\bigvee_{p \in P} \neg s_p\right) \right)\vspace{-1.0mm}
\worra t s = \dom t \wedge\bigvee_{i\in I}\left(\bigwedge_{\{P\subseteq P_i\alt s\leq \bigvee_{p \in P} \neg t_p\}}\left(\bigvee_{p \in P} \neg s_p\right) \right)\svvspace{-1.0mm}
\end{equation}
The formula considers only the positive arrows of each summand that
forms $t$ and states that, for each summand, whenever you take a subset
......@@ -136,7 +136,7 @@ extends $\Gamma$ with hypotheses on the occurrences of $e$ that are
the most general that can be deduced by assuming that $e\,{\in}\,t$ succeeds. For that we need the notation $\tyof{e}{\Gamma}$ which denotes the type deduced for $e$ under the type environment $\Gamma$ in the algorithmic type system of Section~\ref{sec:algorules}.
That is, $\tyof{e}{\Gamma}=t$ if and only if $\Gamma\vdashA e:t$ is provable.
We start by defining the algorithm for each single occurrence, that is for the deduction of $\pvdash \Gamma e t \varpi:t'$. This is obtained by defining two mutually recursive functions $\constrf$ and $\env{}{}$:\vspace{-1.3mm}
We start by defining the algorithm for each single occurrence, that is for the deduction of $\pvdash \Gamma e t \varpi:t'$. This is obtained by defining two mutually recursive functions $\constrf$ and $\env{}{}$:\svvspace{-1.3mm}
\begin{eqnarray}
\constr\epsilon{\Gamma,e,t} & = & t\label{uno}\\[\sk]
\constr{\varpi.0}{\Gamma,e,t} & = & \neg(\arrow{\env {\Gamma,e,t}{(\varpi.1)}}{\neg \env {\Gamma,e,t} (\varpi)})\label{due}\\[\sk]
......@@ -146,7 +146,7 @@ We start by defining the algorithm for each single occurrence, that is for the d
\constr{\varpi.f}{\Gamma,e,t} & = & \pair{\env {\Gamma,e,t} (\varpi)}\Any\label{sei}\\[\sk]
\constr{\varpi.s}{\Gamma,e,t} & = & \pair\Any{\env {\Gamma,e,t} (\varpi)}\label{sette}\\[.8mm]
\env {\Gamma,e,t} (\varpi) & = & {\constr \varpi {\Gamma,e,t} \wedge \tyof {\occ e \varpi} \Gamma}\label{otto}
\end{eqnarray}\vspace{-5mm}\\
\end{eqnarray}\svvspace{-5mm}\\
All the functions above are defined if and only if the initial path
$\varpi$ is valid for $e$ (i.e., $\occ e{\varpi}$ is defined) and $e$
is well-typed (which implies that all $\tyof {\occ e{\varpi}} \Gamma$
......@@ -159,7 +159,7 @@ in the definition are defined)%
this is defined for all $\varpi$ since the first premisses of
\Rule{Case\Aa} states that $\Gamma\vdash e:t_0$ (and this is
possible only if we were able to deduce under the hypothesis
$\Gamma$ the type of every occurrence of $e$.)\vspace{-3mm}}
$\Gamma$ the type of every occurrence of $e$.)\svvspace{-3mm}}
\else
; the well foundness of the definition can be deduced by analysing the rule~\Rule{Case\Aa} of Section~\ref{sec:algorules}.
\fi
......@@ -223,7 +223,7 @@ $\Refinef$ yields for $x$ a type strictly more precise than the type deduced in
previous iteration.
The solution we adopt in practice is to bound the number of iterations to some number $n_o$. This is obtained by the following definition of $\Refinef$\vspace{-1mm}
The solution we adopt in practice is to bound the number of iterations to some number $n_o$. This is obtained by the following definition of $\Refinef$\svvspace{-1mm}
\[
\begin{array}{rcl}
\Refinef_{e,t} \eqdef (\RefineStep{e,t})^{n_o}\\[-2mm]
......@@ -234,7 +234,7 @@ The solution we adopt in practice is to bound the number of iterations to some
\Gamma(e') & \text{otherwise, if } e'\in\dom\Gamma\\
\text{undefined} & \text{otherwise}
\end{array}\right.
\end{array}\vspace{-1.5mm}
\end{array}\svvspace{-1.5mm}
\]
Note in particular that $\Refine{e,t}\Gamma$ extends $\Gamma$ with hypotheses on the expressions occurring in $e$, since
$\dom{\Refine{e,t}\Gamma} = \dom{\RefineStep {e,t}(\Gamma)} = \dom{\Gamma} \cup \{e' \alt \exists \varpi.\ \occ e \varpi \equiv e'\}$.
......@@ -275,7 +275,7 @@ We now have all the definitions we need for our typing algorithm%
{ }
{ \Gamma \vdashA x: \Gamma(x) }
{ x\in\dom\Gamma}
\vspace{-2mm}\\
\svvspace{-2mm}\\
\Infer[Env\Aa]
{ \Gamma\setminus\{e\} \vdashA e : t }
{ \Gamma \vdashA e: \Gamma(e) \wedge t }
......@@ -285,7 +285,7 @@ We now have all the definitions we need for our typing algorithm%
{ }
{\Gamma\vdashA c:\basic{c}}
{c\not\in\dom\Gamma}
\vspace{-2mm}\\
\svvspace{-2mm}\\
\ifsubmission\else
\end{mathpar}
\begin{mathpar}
......@@ -296,7 +296,7 @@ We now have all the definitions we need for our typing algorithm%
\Gamma\vdashA\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:\textstyle\wedge_{i\in I} {\arrow {s_i} {t_i}}
}
{\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e\not\in\dom\Gamma}
\vspace{-2mm}\\
\svvspace{-2mm}\\
\Infer[App\Aa]
{
\Gamma \vdashA e_1: t_1\\
......@@ -306,7 +306,7 @@ We now have all the definitions we need for our typing algorithm%
}
{ \Gamma \vdashA {e_1}{e_2}: t_1 \circ t_2 }
{ {e_1}{e_2}\not\in\dom\Gamma}
\vspace{-2mm}\\
\svvspace{-2mm}\\
\Infer[Case\Aa]
{\Gamma\vdashA e:t_0\\
%\makebox{$\begin{array}{l}
......@@ -326,7 +326,7 @@ We now have all the definitions we need for our typing algorithm%
{\Gamma\vdashA \tcase {e} t {e_1}{e_2}: t_1\vee t_2}
%{\ite {e} t {e_1}{e_2}\not\in\dom\Gamma}
{ \tcase {e} {t\!} {\!e_1\!}{\!e_2}\not\in\dom\Gamma}
\vspace{-2mm} \\
\svvspace{-2mm} \\
\Infer[Proj\Aa]
{\Gamma \vdashA e:t\and \!\!t\leq\pair{\Any\!}{\!\Any}}
{\Gamma \vdashA \pi_i e:\bpi_{\mathbf{i}}(t)}
......@@ -429,11 +429,11 @@ rule \Rule{Path} and $(ii)$ the use of nested \Rule{PAppL} that yields
a precision that the algorithm loses by using type schemes in defining of \constrf{} (case~\eqref{due} is the critical
one). Completeness is recovered by $(i)$ limiting the depth of the
derivations and $(ii)$ forbidding nested negated arrows on the
left-hand side of negated arrows.\vspace{-.7mm}
left-hand side of negated arrows.\svvspace{-.7mm}
\begin{definition}[Rank-0 negation]
A derivation of $\Gamma \vdash e:t$ is \emph{rank-0 negated} if
\Rule{Abs--} never occurs in the derivation of a left premise of a
\Rule{PAppL} rule.\vspace{-.7mm}
\Rule{PAppL} rule.\svvspace{-.7mm}
\end{definition}
\noindent The use of this terminology is borrowed from the ranking of higher-order
types, since, intuitively, it corresponds to typing a language in
......
......@@ -150,7 +150,7 @@ $(\Int\to\textsf{Empty})\land(\neg\Int\to{}2)$\newline
}
\caption{Types inferred by the implementation}
\ifsubmission%
\vspace{-10mm}
\svvspace{-10mm}
\fi%
\label{tab:implem}
\end{table}
......@@ -64,11 +64,11 @@ field $\ell = t \vee \Undef$%
For what concerns \emph{expressions}, we cannot use CDuce record expressions
as they are, but we must adapt them to our analysis. In particular, we
consider records that are built starting from the empty record expression \erecord{} by adding, updating, or removing fields:\vspace{-0.75mm}
consider records that are built starting from the empty record expression \erecord{} by adding, updating, or removing fields:\svvspace{-0.75mm}
\[
\begin{array}{lrcl}
\textbf{Expr} & e & ::= & \erecord {} ~\alt~ \recupd e \ell e ~\alt~ \recdel e \ell ~\alt~ e.\ell
\end{array}\vspace{-.75mm}
\end{array}\svvspace{-.75mm}
\]
in particular $\recdel e \ell$ deletes the field $\ell$ from $e$, $\recupd e \ell e'$ adds the field $\ell=e'$ to the record $e$ (deleting any existing $\ell$ field), while $e.\ell$ is field selection with the reduction:
\(\erecord{...,\ell=e,...}.\ell\ \reduces\ e\).
......@@ -77,7 +77,7 @@ To define record type subtyping and record expression type inference we need thr
%
Then two record types $t_1$ and $t_2$ are in subtyping relation, $t_1 \leq t_2$, if and only if for all $\ell \in \Labels$ we have $\proj \ell {t_1} \leq \proj \ell {t_2}$. In particular $\orecord{\!\!}$ is the largest record type.
Expressions are then typed by the following rules (already in algorithmic form).\vspace{-.1mm}
Expressions are then typed by the following rules (already in algorithmic form).\svvspace{-.1mm}
\begin{mathpar}
\Infer[Record]
{~}
......@@ -90,7 +90,7 @@ Expressions are then typed by the following rules (already in algorithmic form).
{\recupd{e_1\!\!}{\!\!\ell}{e_2} \not\in\dom\Gamma}
%\end{mathpar}
%\begin{mathpar}
\vspace{-1.9mm}
\svvspace{-1.9mm}
\\
\Infer[Delete]
{\Gamma \vdash e:t\and t\leq\orecord {\!\!}}
......@@ -100,7 +100,7 @@ Expressions are then typed by the following rules (already in algorithmic form).
\Infer[Proj]
{\Gamma \vdash e:t\and t\leq\orecord{\ell = \Any}}
{\Gamma \vdash e.\ell:\proj \ell {t}}
{e.\ell\not\in\dom\Gamma}\vspace{-2mm}
{e.\ell\not\in\dom\Gamma}\svvspace{-2mm}
\end{mathpar}
To extend occurrence typing to records we add the following values to paths: $\varpi\in\{\ldots,a_\ell,u_\ell^1,u_\ell^2,r_\ell\}^*$, with
\(e.\ell\downarrow a_\ell.\varpi =\occ{e}\varpi\),
......@@ -108,7 +108,7 @@ To extend occurrence typing to records we add the following values to paths: $\v
\(\recupd{e_1}\ell{e_2}\downarrow u_\ell^i.\varpi = \occ{e_i}\varpi\)
and add the following rules for the new paths:
\begin{mathpar}\vspace{-8.7mm}\\
\begin{mathpar}\svvspace{-8.7mm}\\
\Infer[PSel]
{ \pvdash \Gamma e t \varpi:t'}
{ \pvdash \Gamma e t {\varpi.a_\ell}:\orecord {\ell:t'} }
......@@ -118,7 +118,7 @@ and add the following rules for the new paths:
{ \pvdash \Gamma e t \varpi:t'}
{ \pvdash \Gamma e t \varpi.r_\ell: (\recdel {t'} \ell) + \crecord{\ell \eqq \Any}}
{ }
\vspace{-3mm}\\
\svvspace{-3mm}\\
\Infer[PUpd1]
{ \pvdash \Gamma e t \varpi:t'}
{ \pvdash \Gamma e t \varpi.u_\ell^1: (\recdel {t'} \ell) + \crecord{\ell \eqq \Any}}
......
......@@ -15,7 +15,7 @@ In a sense, occurrence typing is a
discipline designed to push forward the frontiers beyond which gradual
typing is needed, thus reducing the amount of runtime checks needed. For
instance, the JavaScript code of~\eqref{foo} and~\eqref{foo2} in the introduction can also be
typed by using gradual typing:%\vspace{-.2mm}
typed by using gradual typing:%\svvspace{-.2mm}
\begin{alltt}\color{darkblue}\morecompact
function foo(x\textcolor{darkred}{ : \pmb{\dyn}}) \{
return (typeof(x) === "number")? x+1 : x.trim(); \refstepcounter{equation} \mbox{\color{black}\rm(\theequation)}\label{foo3}
......@@ -105,7 +105,7 @@ other words, if a function expects an argument of type $\tau$ but can be
typed under the hypothesis that the argument has type $\tauUp$, then no casts
are needed, since every cast that succeeds will be a subtype of
$\tauUp$. Taking advantage of this property, we modify the rule for
functions as: \vspace{-2mm}
functions as: \svvspace{-2mm}
%
%\begin{mathpar}
% \Infer[Abs]
......@@ -128,7 +128,7 @@ functions as: \vspace{-2mm}
}
{
\Gamma\vdash\lambda x:\sigma'.e:\textstyle\bigwedge_{(\sigma,\tau) \in T}\sigma\to \tau
}\vspace{-2mm}
}\svvspace{-2mm}
\]
The main idea behind this rule is the same as before: we first collect all the
information we can into $\psi$ by analyzing the body of the function. We then
......@@ -149,7 +149,7 @@ $\code{number} \land \dyn\simeq \code{number}$ is
not an option, since it would force us to choose between having
the gradual guarantee or having, say,
$\code{number} \land \code{string}$ be more precise than
$\code{number} \land \dyn$.\vspace{-2mm}}
$\code{number} \land \dyn$.\svvspace{-2mm}}
Hence the need for the second part of
Rule~\textsc{[AbsInf+]}: the maximal interpretation of $\code{number} \land \dyn$
is $\code{number}$, and it is clear that, if $\code x$ is given type \code{number},
......
......@@ -238,7 +238,8 @@ that $e_1e_2$ has type $t$ succeeds or fails. Let us start with refining the typ
which the test succeeds. Intuitively, we want to remove from $t_2$ all
the values for which the application will surely return a result not
in $t$, thus making the test fail. Consider $t_1$ and let $s$ be the
largest subtype of $\dom{t_1}$ such that\vspace{-1.29mm}
largest subtype of $\dom{t_1}$ such that%
\svvspace{-1.29mm}
\begin{equation}\label{eq1}
t_1\circ s\leq \neg t
\end{equation}
......@@ -293,9 +294,10 @@ Ergo $t_1\setminus (t_2^+\to
Let us see all this on our example \eqref{exptre}, in particular, by showing how this technique deduces that the type of $x_1$ in the positive branch is (a subtype of) $\Int{\vee}\String\to\Int$.
Take the static type of $x_1$, that is $(\Int{\vee}\String\to\Int)\vee(\Bool{\vee}\String\to\Bool)$ and intersect it with
$(t_2^+\to \neg t)$, that is, $\String\to\neg\Int$. Since intersection distributes over unions we
obtain\vspace{-1mm}
obtain
\svvspace{-1mm}
%
\[((\Int{\vee}\String{\to}\Int)\wedge\neg(\String{\to}\neg\Int))\vee((\Bool{\vee}\String{\to}\Bool)\wedge\neg(\String{\to}\neg\Int))\vspace{-1mm}\]
\[((\Int{\vee}\String{\to}\Int)\wedge\neg(\String{\to}\neg\Int))\vee((\Bool{\vee}\String{\to}\Bool)\wedge\neg(\String{\to}\neg\Int))\svvspace{-1mm}\]
%
and since
$(\Bool{\vee}\String{\to}\Bool)\wedge\neg(\String{\to}\neg\Int)$ is empty
......@@ -314,9 +316,18 @@ This is essentially what we formalize in Section~\ref{sec:language}, in the type
In the previous section we outlined the main ideas of our approach to occurrence typing. However, devil is in the details. So the formalization we give in Section~\ref{sec:language} is not so smooth as we just outlined: we must introduce several auxiliary definitions to handle some corner cases. This section presents by tiny examples the main technical difficulties we had to overcome and the definitions we introduced to handle them. As such it provides a kind of road-map for the technicalities of Section~\ref{sec:language}.
\paragraph{Typing occurrences} As it should be clear by now, not only variables but also generic expressions are given different types in the ``then'' and ``else'' branches of type tests. For instance, in \eqref{two} the expression $x_1x_2$ has type \Int{} in the positive branch and type \Bool{} in the negative one. In this specific case it is possible to deduce these typings from the refined types of the variables (in particular, thanks to the fact that $x_2$ has type \Int{} the positive branch and \Bool{} in the negative one), but this is not possible in general. For instance, consider $x_1:\Int\to(\Int\vee\Bool)$, $x_2:\Int$, and the expression\vspace{-1mm}
\paragraph{Typing occurrences} As it should be clear by now, not only variables
but also generic expressions are given different types in the ``then'' and
``else'' branches of type tests. For instance, in \eqref{two} the expression
$x_1x_2$ has type \Int{} in the positive branch and type \Bool{} in the negative
one. In this specific case it is possible to deduce these typings from the
refined types of the variables (in particular, thanks to the fact that $x_2$ has
type \Int{} the positive branch and \Bool{} in the negative one), but this is
not possible in general. For instance, consider $x_1:\Int\to(\Int\vee\Bool)$,
$x_2:\Int$, and the expression
\svvspace{-1mm}
\begin{equation}\label{twobis}
\ifty{x_1x_2}{\Int}{...x_1x_2...}{...x_1x_2...}\vspace{-1mm}
\ifty{x_1x_2}{\Int}{...x_1x_2...}{...x_1x_2...}\svvspace{-1mm}
\end{equation}
It is not possible to specialize the type of the variables in the
branches. Nevertheless, we want to be able to deduce that $x_1x_2$ has
......@@ -347,9 +358,9 @@ satisfies progress and type preservation. The latter property is
challenging because, as explained just above, our type
assumptions are not only about variables but also about
expressions. Two corner cases are particularly difficult. The first is
shown by the following example\vspace{-.9mm}
shown by the following example\svvspace{-.9mm}
\begin{equation}\label{bistwo}
\ifty{e(42)}{\Bool}{e}{...}\vspace{-.9mm}
\ifty{e(42)}{\Bool}{e}{...} \svvspace{-.9mm}
\end{equation}
If $e$ is an expression of type $\Int\to t$, then, as discussed before,
the positive branch will have type $(\Int\to t)\setminus(\Int\to\neg
......@@ -396,9 +407,9 @@ an approximation thereof.
Finally, a nested check may help refining
the type assumptions on some outer expressions. For instance, when typing
the positive branch $e$ of\vspace{-.9mm}
the positive branch $e$ of\svvspace{-.9mm}
\begin{equation}\label{pair}
\ifty{(x,y)}{(\pair{(\Int\vee\Bool)}\Int)}{e}{...}\vspace{-.9mm}
\ifty{(x,y)}{(\pair{(\Int\vee\Bool)}\Int)}{e}{...}\svvspace{-.9mm}
\end{equation}
we can assume that the expression $(x,y)$ is of type
$\pair{(\Int\vee\Bool)}\Int$ and put it in the type environment. But
......
......@@ -7,7 +7,7 @@ In this section we formalize the ideas we outlined in the introduction. We start
\subsection{Types}
\begin{definition}[Types]\label{def:types}
%\iflongversion%%%%%%%
The set of types \types{} is formed by the terms $t$ coinductively produced by the grammar:\vspace{-1.45mm}
The set of types \types{} is formed by the terms $t$ coinductively produced by the grammar:\svvspace{-1.45mm}
\[
\begin{array}{lrcl}
\textbf{Types} & t & ::= & b\alt t\to t\alt t\times t\alt t\vee t \alt \neg t \alt \Empty
......@@ -17,10 +17,10 @@ and that satisfy the following conditions
\begin{itemize}[nosep]
\item (regularity) every term has a finite number of different sub-terms;
\item (contractivity) every infinite branch of a term contains an infinite number of occurrences of the
arrow or product type constructors.\vspace{-1mm}
arrow or product type constructors.\svvspace{-1mm}
\end{itemize}
\iffalse%%%%%%%%%%%%%%%%%%%%%%%%%%
A type $t\in\types{}$ is a term coinductively produced by the grammar:\vspace{-1.45mm}
A type $t\in\types{}$ is a term coinductively produced by the grammar:\svvspace{-1.45mm}
\[
\begin{array}{lrcl}
\textbf{Types} & t & ::= & b\alt t\to t\alt t\times t\alt t\vee t \alt \neg t \alt \Empty
......@@ -28,7 +28,7 @@ A type $t\in\types{}$ is a term coinductively produced by the grammar:\vspace{-1
\]
that satisfies the following conditions: $(1)$\emph{Regularity}: the
term has a finite number of different sub-terms; $(2)$ \emph{Contractivity}: every infinite branch of the term contains an infinite number of occurrences of the
arrow or product type constructors.\vspace{-1mm}
arrow or product type constructors.\svvspace{-1mm}
\fi%%%%%%%%%%%%%%%%%%%
\end{definition}
We use the following abbreviations: $
......@@ -79,7 +79,7 @@ union of the values of the two types). We use $\simeq$ to denote the
symmetric closure of $\leq$: thus $s\simeq t$ (read, $s$ is equivalent to $t$) means that $s$ and $t$ denote the same set of values and, as such, they are semantically the same type.
\subsection{Syntax}\label{sec:syntax}
The expressions $e$ and values $v$ of our language are inductively generated by the following grammars:\vspace{-1mm}
The expressions $e$ and values $v$ of our language are inductively generated by the following grammars:\svvspace{-1mm}
\begin{equation}\label{expressions}
\begin{array}{lrclr}
\textbf{Expr} &e &::=& c\alt x\alt ee\alt\lambda^{\wedge_{i\in I}s_i\to t_i} x.e\alt \pi_j e\alt(e,e)\alt\tcase{e}{t}{e}{e}\\[.3mm]
......@@ -110,7 +110,7 @@ values. We write $v\in t$ if the most specific type of $v$ is a subtype of $t$ (
\subsection{Dynamic semantics}\label{sec:opsem}
The dynamic semantics is defined as a classic left-to-right call-by-value reduction for a $\lambda$-calculus with pairs, enriched with specific rules for type-cases. We have the following notions of reduction:\vspace{-1.2mm}
The dynamic semantics is defined as a classic left-to-right call-by-value reduction for a $\lambda$-calculus with pairs, enriched with specific rules for type-cases. We have the following notions of reduction:\svvspace{-1.2mm}
\[
\begin{array}{rcll}
(\lambda^{\wedge_{i\in I}s_i\to t_i} x.e)\,v &\reduces& e\subst x v\\[-.4mm]
......@@ -137,7 +137,7 @@ standard, for what concerns the type system we will have to introduce several
unconventional features that we anticipated in
Section~\ref{sec:challenges} and are at the core of our work. Let
us start with the standard part, that is the typing of the functional
core and the use of subtyping, given by the following typing rules:\vspace{-1mm}
core and the use of subtyping, given by the following typing rules:\svvspace{-1mm}
\begin{mathpar}
\Infer[Const]
{ }
......@@ -182,7 +182,7 @@ core and the use of subtyping, given by the following typing rules:\vspace{-1mm}
{ \Gamma \vdash e:t\\t\leq t' }
{ \Gamma \vdash e: t' }
{ }
\qquad\vspace{-3mm}
\qquad\svvspace{-3mm}
\end{mathpar}
These rules are quite standard and do not need any particular explanation besides those already given in Section~\ref{sec:syntax}. Just notice subtyping is embedded in the system by the classic \Rule{Subs} subsumption rule. Next we focus on the unconventional aspects of our system, from the simplest to the hardest.
......@@ -190,7 +190,7 @@ The first unconventional aspect is that, as explained in
Section~\ref{sec:challenges}, our type assumptions are about
expressions. Therefore, in our rules the type environments, ranged over
by $\Gamma$, map \emph{expressions}---rather than just variables---into
types. This explains why the classic typing rule for variables is replaced by a more general \Rule{Env} rule defined below:\vspace{-1mm}
types. This explains why the classic typing rule for variables is replaced by a more general \Rule{Env} rule defined below:\svvspace{-1mm}
\begin{mathpar}
\Infer[Env]
{ }
......@@ -200,7 +200,7 @@ types. This explains why the classic typing rule for variables is replaced by a
\Infer[Inter]
{ \Gamma \vdash e:t_1\\\Gamma \vdash e:t_2 }
{ \Gamma \vdash e: t_1 \wedge t_2 }
{ }\vspace{-3mm}
{ }\svvspace{-3mm}
\end{mathpar}
The \Rule{Env} rule is coupled with the standard intersection introduction rule \Rule{Inter}
which allows us to deduce for a complex expression the intersection of
......@@ -209,12 +209,12 @@ environment $\Gamma$ with the static type deduced for the same
expression by using the other typing rules. This same intersection
rule is also used to infer the second unconventional aspect of our
system, that is, the fact that $\lambda$-abstractions can have negated
arrow types, as long as these negated types do not make the type deduced for the function empty:\vspace{-.5mm}
arrow types, as long as these negated types do not make the type deduced for the function empty:\svvspace{-.5mm}
\begin{mathpar}
\Infer[Abs-]
{\Gamma \vdash \lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:t}
{ \Gamma \vdash\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:\neg(t_1\to t_2) }
{ ((\wedge_{i\in I}\arrow {s_i} {t_i})\wedge\neg(t_1\to t_2))\not\simeq\Empty }\vspace{-1.2mm}
{ ((\wedge_{i\in I}\arrow {s_i} {t_i})\wedge\neg(t_1\to t_2))\not\simeq\Empty }\svvspace{-1.2mm}
\end{mathpar}
%\beppe{I have doubt: is this safe or should we play it safer and
% deduce $t\wedge\neg(t_1\to t_2)$? In other terms is is possible to
......@@ -266,7 +266,7 @@ expression whose type environment contains an empty assumption:
\Infer[Efq]
{ }
{ \Gamma, (e:\Empty) \vdash e': t }
{ }\vspace{-3mm}
{ }\svvspace{-3mm}
\end{mathpar}
Once more, this kind of deduction was already present in the system
by~\citet{Frisch2008} to type full fledged overloaded functions,
......@@ -278,7 +278,7 @@ property of subject reduction.
Finally, there remains one last rule in our type system, the one that
implements occurrence typing, that is, the rule for the
type-case:\vspace{-1mm}
type-case:\svvspace{-1mm}
\begin{mathpar}
\Infer[Case]
{\Gamma\vdash e:t_0\\
......@@ -287,7 +287,7 @@ type-case:\vspace{-1mm}
%t_0\not\leq t \Rightarrow
\Gamma \evdash e {\neg t} \Gamma_2 \\ \Gamma_2 \vdash e_2:t'}
{\Gamma\vdash \tcase {e} t {e_1}{e_2}: t'}
{ }\vspace{-3mm}
{ }\svvspace{-3mm}
\end{mathpar}
The rule \Rule{Case} checks whether the expression $e$, whose type is
being tested, is well-typed and then performs the occurrence typing
......@@ -312,7 +312,7 @@ root of the tree).
Let $e$ be an expression and $\varpi\in\{0,1,l,r,f,s\}^*$ a
\emph{path}; we denote $\occ e\varpi$ the occurrence of $e$ reached by
the path $\varpi$, that is (for $i=0,1$, and undefined otherwise)\vspace{-.4mm}
the path $\varpi$, that is (for $i=0,1$, and undefined otherwise)\svvspace{-.4mm}
%% \[
%% \begin{array}{l}
%% \begin{array}{r@{\downarrow}l@{\quad=\quad}l}
......@@ -339,7 +339,7 @@ application, $l$ and $r$ for the $l$eft and $r$ight expressions forming a pair,
and $f$ and $s$ for the argument of a $f$irst or of a $s$econd projection. Note also that we do not consider occurrences
under $\lambda$'s (since their type is frozen in their annotations) and type-cases (since they reset the analysis).
%
The judgments $\Gamma \evdash e t \Gamma'$ are then deduced by the following two rules:\vspace{-1mm} \begin{mathpar}
The judgments $\Gamma \evdash e t \Gamma'$ are then deduced by the following two rules:\svvspace{-1mm} \begin{mathpar}
% \Infer[Base]
% { \Gamma \vdash e':t' }
% { \Gamma \cvdash p e t e':t' }
......@@ -357,7 +357,7 @@ The judgments $\Gamma \evdash e t \Gamma'$ are then deduced by the following tw
\Infer[Path]
{ \pvdash {\Gamma'} e t \varpi:t' \\ \Gamma \evdash e t \Gamma' }
{ \Gamma \evdash e t \Gamma',(\occ e \varpi:t') }
{ }\vspace{-1.5mm}
{ }\svvspace{-1.5mm}
\end{mathpar}
These rules describe how to produce by occurrence typing the type
environments while checking that an expression $e$ has type $t$. They state that $(i)$ we can
......@@ -390,7 +390,7 @@ of rules.
{ \Gamma \vdash \occ e \varpi:t' }
{ \pvdash \Gamma e t \varpi:t' }
{ }
\vspace{-1.2mm}\\
\svvspace{-1.2mm}\\
\Infer[PEps]
{ }
{ \pvdash \Gamma e t \epsilon:t }
......@@ -400,7 +400,7 @@ of rules.
{ \pvdash \Gamma e t \varpi.0:\arrow{t_1}{t_2} \\ \pvdash \Gamma e t \varpi:t_2'}
{ \pvdash \Gamma e t \varpi.1:\neg t_1 }
{ t_2\land t_2' \simeq \Empty }
\end{mathpar}\begin{mathpar}\vspace{-2mm}
\end{mathpar}\begin{mathpar}\svvspace{-2mm}
\Infer[PAppL]
{ \pvdash \Gamma e t \varpi.1:t_1 \\ \pvdash \Gamma e t \varpi:t_2 }
{ \pvdash \Gamma e t \varpi.0:\neg (\arrow {t_1} {\neg t_2}) }
......@@ -410,7 +410,7 @@ of rules.
{ \pvdash \Gamma e t \varpi:\pair{t_1}{t_2} }
{ \pvdash \Gamma e t \varpi.l:t_1 }
{ }
\vspace{-1.2mm}\\
\svvspace{-1.2mm}\\
\Infer[PPairR]
{ \pvdash \Gamma e t \varpi:\pair{t_1}{t_2} }
{ \pvdash \Gamma e t \varpi.r:t_2 }
......@@ -424,7 +424,7 @@ of rules.
\Infer[PSnd]
{ \pvdash \Gamma e t \varpi:t' }
{ \pvdash \Gamma e t \varpi.s:\pair \Any {t'} }
{ }\vspace{-0.9mm}
{ }\svvspace{-0.9mm}
\end{mathpar}
These rules implement the analysis described in
Section~\ref{sec:ideas} for functions and extend it to products. Let
......@@ -453,13 +453,13 @@ $\pair{t'}\Any$ (respectively, $\pair\Any{t'}$).
This concludes the presentation of all the rules of our type system
(they are summarized for the reader's convenience in Appendix~\ref{sec:declarative}), which satisfies
the property of safety, deduced, as customary, from the properties
of progress and subject reduction (\emph{cf.} Appendix~\ref{app:soundness}).\vspace{-.5mm}
of progress and subject reduction (\emph{cf.} Appendix~\ref{app:soundness}).\svvspace{-.5mm}
\begin{theorem}[type safety]
For every expression $e$ such that $\varnothing\vdash e:t$ either $e$
diverges or there
exists a value $v$ of type $t$ such that $e\reduces^* v$.
\end{theorem}
\vspace{-2.1mm}
\svvspace{-2.1mm}
......
......@@ -42,9 +42,11 @@
\withcommentsfalse
\newif\ifsubmission
\submissiontrue
%\submissionfalse
\newif\iflongversion
\longversiontrue
\newcommand{\svvspace}[1]{
\iflongversion\else\vspace{#1}\fi
}
%\longversionfalse
\usepackage{setup}
......
......@@ -50,12 +50,11 @@ these annotations.
_**Reviewer A:** What were the running times for the examples in Tab
1_
All examples run in 1 millisecond or less, save example 9 that takes 4
milliseconds (on a laptop with an Intel Core i7-8750H processor). We
added running times in the prototype (it is `js_ocaml`, so they will be
much slower) and can add them in the table.
DONE
_**Reviewer B:** No theoretical guarantee for the extensions is mentioned_
......
......@@ -13,7 +13,10 @@ systems (even though some systems such as Flow and TypeScript
can type some of these examples by adding explicit type annotations, the code 6,
7, 9, and 10 in Table~\ref{tab:implem} and, even more, the \code{and\_} and \code{xor\_} functions at the end of this
section are out of reach of current systems, even when using the right
explicit annotations). These and other examples can be tested in the
explicit annotations). It should be noted that for all the examples we discuss,
the the time for the type inference process is less than 5ms, hence we do not
report precise timings in the table.
These and other examples can be tested in the
\ifsubmission
anonymized
\else
......
......@@ -161,7 +161,7 @@ We introduce the new syntactic category of \emph{type schemes} which are the ter
\]
Type schemes denote sets of types, as formally stated by the following definition:
\begin{definition}[Interpretation of type schemes]
We define the function $\tsint {\_}$ that maps type schemes into sets of types.\vspace{-2.5mm}
We define the function $\tsint {\_}$ that maps type schemes into sets of types.\svvspace{-2.5mm}
\begin{align*}
\begin{array}{lcl}
\tsint t &=& \{s\alt t \leq s\}\\
......@@ -204,13 +204,13 @@ We also need to perform intersections of type schemes so as to intersect the sta
\end{lemma}
Finally, given a type scheme $\ts$ it is straightforward to choose in its interpretation a type $\tsrep\ts$ which serves as the canonical representative of the set (i.e., $\tsrep \ts \in \tsint \ts$):
\begin{definition}[Representative]
We define a function $\tsrep {\_}$ that maps every non-empty type scheme into a type, \textit{representative} of the set of types denoted by the scheme.\vspace{-2mm}
We define a function $\tsrep {\_}$ that maps every non-empty type scheme into a type, \textit{representative} of the set of types denoted by the scheme.\svvspace{-2mm}
\begin{align*}
\begin{array}{lcllcl}
\tsrep t &=& t & \tsrep {\ts_1 \tstimes \ts_2} &=& \pair {\tsrep {\ts_1}} {\tsrep {\ts_2}}\\
\tsrep {\tsfunone {t_i} {s_i}_{i\in I}} &=& \bigwedge_{i\in I} \arrow {t_i} {s_i} \qquad& \tsrep {\ts_1 \tsor \ts_2} &=& \tsrep {\ts_1} \vee \tsrep {\ts_2}\\
\tsrep \tsempty && \textit{undefined}
\end{array}\vspace{-4mm}
\end{array}\svvspace{-4mm}
\end{align*}
\end{definition}
......
......@@ -257,7 +257,7 @@ with
Expression substitutions, ranged over by $\rho$, map an expression into another expression. The application of an expressions substitution $\rho$ to an expression $e$, noted $e\rho$ is the capture avoiding replacement defined as follows:
\begin{itemize}
\item If $e'\equiv_\alpha e''$, then $e''\subst{e'}e = e$.\vspace{1mm}
\item If $e'\not\equiv_\alpha e''$, then $e''\subst{e'}e$ is inductively defined as \vspace{-1.5mm}
\item If $e'\not\equiv_\alpha e''$, then $e''\subst{e'}e$ is inductively defined as \svvspace{-1.5mm}
\begin{align*}
c\subst{e'}e & = c\\
x\subst{e'}{e} & = x\\
......
\newcommand{\negspace}{\vspace{-.5mm}}
\newcommand{\negspace}{\svvspace{-.5mm}}
As we explained in the introduction, both TypeScript and Flow deduce the type
\code{(number$\vee$string) $\to$ (number$\vee$string)} for the first definition of the function \code{foo} in~\eqref{foo}, and the more precise type\vspace{-3pt}
\code{(number$\vee$string) $\to$ (number$\vee$string)} for the first definition of the function \code{foo} in~\eqref{foo}, and the more precise type\svvspace{-3pt}
\begin{equation}\label{tyinter}
\code{(number$\to$number)$\,\wedge\,$(string$\to$string)}\vspace{-3pt}
\code{(number$\to$number)$\,\wedge\,$(string$\to$string)}\svvspace{-3pt}
\end{equation}
can be deduced by these languages only if they are instructed to do so: the
programmer has to explicitly annotate \code{foo} with the
......@@ -59,7 +59,7 @@ the set that contains the types of all the occurrences of $x$ in $e$. This
judgement can be deduced by the following deduction system
that collects type information on the variables that are $\lambda$-abstracted
(i.e., those in the domain of $\Gamma$, since lambdas are our only
binders):\vspace{-1.5mm}
binders):\svvspace{-1.5mm}
\begin{mathpar}
\Infer[Var]
{
......@@ -79,7 +79,7 @@ binders):\vspace{-1.5mm}
\Gamma\vdash\lambda x:s.e\triangleright\psi\setminus\{x\}
}