with a redundant arrow. Here we can see that since we deduced
the first two arrows $(\Int\to\Int)\land(\Bool\to\Bool)$, and since
the union of their domain exactly covers the domain the third arrow,
the union of their domain exactly covers the domain of the third arrow,
the latter is not needed. Code~2 shows what happens when the argument
of the function is left unannotated (i.e., it is annotated by the top
type \Any, written \texttt{Any} in our implementation). Here
type \Any, written \texttt{Any} in our implementation). Here
type-checking and refinement also work as expected, but the function
only type checks if all cases for \texttt{x} are covered (which means
that the function must handle the case of inputs that are neither in \Int{}
...
...
@@ -58,17 +58,20 @@ a particular type, are just plain functions with an intersection
type inferred by the system of Section~\ref{sec:refining}. We next define Boolean connectives as overloaded
functions. The \texttt{not\_} connective (Code~4) just tests whether its
argument is the Boolean \texttt{true} by testing that it belongs to
the singleton type \True{} (the type whose only value is \texttt{true}) returning \texttt{false} for it and \texttt{true} for any other value (recall that $\neg\True$ is equivalent to $\texttt{Any\textbackslash}\True$). It works on values of any type, but we could restrict it to Boolean values by simply annotating the parameter by \Bool{} (which in CDuce is syntactic sugar for \True$\vee$\False) yielding the type $(\True{\to}\False)\wedge(\False{\to}\True)$.
The \texttt{or\_} connective (Code~5) is defined as a
curried function, that first type cases on its argument. Depending
on this first type it may return either a constant function that returns
\texttt{true} for every input, or, a function that
discriminates on its argument (which is the second argument of the
\texttt{or\_}) and returns \texttt{true} or \texttt{false}
accordingly. Again we use a generalized version of the
the singleton type \True{} (the type whose only value is
\texttt{true}) returning \texttt{false} for it and \texttt{true} for
any other value (recall that $\neg\True$ is equivalent to
$\texttt{Any\textbackslash}\True$). It works on values of any type,
but we could restrict it to Boolean values by simply annotating the
parameter by \Bool{} (which in CDuce is syntactic sugar for
\True$\vee$\False) yielding the type
$(\True{\to}\False)\wedge(\False{\to}\True)$.
The \texttt{or\_} connective (Code~5) is straightforward as far as the
code goes, but we see that the overloaded type precisely capture all
the possible cases. Again we use a generalized version of the
\texttt{or\_} connective that accepts and treats any value that is not
\texttt{true} as \texttt{false} and again, we could restrict the
domain to \Bool{} if we want.
\texttt{true} as \texttt{false} and again, we could easily restrict the
domain to \Bool{} if desired.
To showcase the power of our type system, and in particular of
the ``$\worra{}{}$''
...
...
@@ -117,8 +120,8 @@ nodes. This splits, at the type level, the case for the \Keyw{Element}
type depending on whether the content of the \texttt{childNodes} field
is the empty list or not.
Our implementation features one last improvement that allows us
further improve the precision of the inferred type.
Our implementation features one last enhancement that allows us
to further improve the precision of the inferred type.
Consider the definition of the \texttt{xor\_} operator (Code~9).
Here the rule~[{\sc AbsInf}+] is not sufficient to precisely type the
function, and using only this rule would yield a type
...
...
@@ -156,7 +159,7 @@ since, \texttt{or\_} has type\\
We consider \True, \Any and $\lnot\True$ as candidate types for
\texttt{x} which, in turn allows us to deduce a precise type given in the table. Finally, thanks to this rule it is no longer necessary to force refinement by using a type case. As a consequence we can define the functions \texttt{and\_} and \texttt{or\_} more naturally as:
\texttt{x} which, in turn allows us to deduce a precise type given in the table. Finally, thanks to this rule it is no longer necessary to force refinement by using a type case. As a consequence we can define the functions \texttt{and\_} and \texttt{xor\_} more naturally as:
\begin{alltt}\color{darkblue}\morecompact
let and_ = fun (x : Any) -> fun (y : Any) -> not_ (or_ (not_ x) (not_ y))
let xor_ = fun (x : Any) -> fun (y : Any) -> and_ (or_ x y) (not_ (and_ x y))
Occurrence typing is for variables in TypeScript, also paths in Flow. STRESS THE USE OF INTERSECTIONS AND UNIONS. ACTUALLY SPEAK OF UNION AND NEGATION (UNION FOR ALTERNATIVES, NEGATION FOR TYPING THE ELSE BRANCH AND THEREFORE WE GET INTERSECTION FOR FREE)
\citet{THF10} extend to selectors, logical connectives, paths, and user
\kim{
I don't really know about these:\\
Occurrence typing is for variables in TypeScript, also paths in Flow. STRESS THE USE OF INTERSECTIONS AND UNIONS. ACTUALLY SPEAK OF UNION AND NEGATION (UNION FOR ALTERNATIVES, NEGATION FOR TYPING THE ELSE BRANCH AND THEREFORE WE GET INTERSECTION FOR FREE)
extend to selectors, logical connectives, paths, and user
defined predicates.
[IMPORTANT: check the differences with what we do here]
State what we capture already, for instance lists since we have product and recursive types.
}
Occurrence typing was introduced by \citet{THF08} and further advanced
in \cite{THF10} in the context of the Typed Racket language. This
latter work in particular is close to ours, with some key differences.
The work of \cite{THF10} defines $\lambda_{TR}$, a core calculus for
typed racket. In this language types are annotated by logical
propositions that records the type of the input depending on the
(Boolean) value of the output. For instance, the type of the
{\tt number?} function states that when the output is {\tt true}, then
the argument has type {\tt Number}, and when the output is false, the
argument does not. Such information is used selectively
in the ``then'' and ``else'' branches of a test. One area where this
work goes further than ours is that the type informations also flows
outside of the tests to the surrounding context. In contrast, our type
system only refine the type of variables strictly in the branches of a
test. However using semantic-subtyping as a foundation we believe our
approach has several merrits over theirs. First, in our case, type
predicates are not built-in. A user may define any type predicate she
wishes by using an overloaded function, as we have shown in
Section~\ref{sec:practical}. Second, in our setting, {\em types\/} play
the role of formulæ. Using set-theoretic types, we can express the
complex types of variables without resorting to a meta-logic. This
allows us to type all but one of their key examples (with the notable
exception of their Example~8 which uses the propgation of type
information outside of the branches of a test). Also, while they
extend their core calculus with pairs, they only provide a simple {\tt
cons?} predicate that allows them to test whether some value is a
pair. It is therefore unclear whether their systems allows one to
write predicates over list types ({\em .e.g\/} test whether the input
is a list of integers), which we can easily do thanks to our support
