Victor Lanvin committed Jun 27, 2019 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 Occurrence typing and gradual typing are two complementary disciplines which have a lot to gain to be integrated, although we are not aware of any work in this sense. In a sense occurrence typing is a discipline designed to push forward the frontiers beyond which gradual typing is needed. For instance, the example at the beginning can be typed by using gradual typing: \begin{alltt}\color{darkblue} function foo(x\textcolor{darkred}{ : \dyn}) \{ (typeof(x) === "number")? x++ : x.length \} \end{alltt} Using standard'' gradual typing this is compiled into: \begin{alltt}\color{darkblue} function foo(x) \{ (typeof(x) === "number")? (\Cast{number}{x})++ : (\Cast{string}{x}).length \} \end{alltt} where {\Cast{$t$}{$e$}} is a type-cast.\footnote{Intuitively, \code{\Cast{$t$}{$e$}} is syntactic sugar for \code{(typeof($e$)==="$t$")? $e$ : (throw "Type error")}. Not exactly though, since to implement compilation \emph{à la} sound gradual typing we need cast on function types.} % We have already seen in the introduction that by using occurrence typing combined with a union type instead of the gradual type \dyn for parameter annotation, we can avoid the insertion of any cast, at the cost of some additional type annotations. But occurrence typing can be used also on the gradually typed code. If we use occurrence typing to type the gradually-typed version of \code{foo}, this allows the system to avoid inserting the first cast \code{\Cast{number}{x}} since, thanks to occurrence typing, the occurrence of \code{x} at issue is given type \code{number} (the second cast is still necessary however). But removing this cast is far from being satisfactory, since when this function is applied to an integer there are some casts that still need to be inserted outside of the function. The reason is that the compiled version of the function has type \code{\dyn$\to$number}, that is, it expects an argument of type \dyn, and thus we have to apply a cast (either to the argument or to the function) whenever this is not the case. In particular, the application \code{foo(42)} will be compiled as \code{foo(\Cast{\dyn}{42})}. Now the main problem with such a cast is not that it produces some unnecessary overhead by performing useless checks (a cast to \dyn can easily be detected and safely ignored at runtime). The main problem is that the combination of such a cast with type-cases will lead to unintuitive results under the standard operational semantics of type-cases and casts. Indeed, consider the standard semantics of the type-case \code{(typeof($e$)==="$t$")} which consists in reducing $e$ to a value and checking whether the type of the value is a subtype of $t$. In standard gradual semantics, \code{\Cast{\dyn}{42}} is a value. And this value is of type \code{\dyn}, which is not a subtype of \code{number}. Therefore the check in \code{foo} would fail for \code{\Cast{\dyn}{42}}, and so would the whole function call. Although this behavior is sound, this is the opposite of what every programmer would expect: one would expect the test \code{(typeof($e$)==="number")} to return true for \code{\Cast{\dyn}{42}} and false for, say, \code{\Cast{\dyn}{true}}, whereas the standard semantics of type-cases would return false in both cases. A solution is to modify the semantics of type-cases, and in particular of \code{typeof}, to strip off all the casts in a value, even nested ones. This however adds a new overhead at runtime. Another solution is to simply accept this counter-intuitive result, which has the additional benefit of promoting the dynamic type to a first class type, instead of just considering it as a directive to the front-end. Indeed, this approach allows to dynamically check whether some argument has the dynamic type \code{\dyn} (i.e., whether it was applied to a cast to such a type, simply by \code{(typeof($e$)==="\dyn")}. Whatever solution we choose it is clear that in both cases it would be much better if the application \code{foo(42)} were compiled as is, thus getting rid of a cast that at best is useless and at worse gives a counter-intuitive and unexpected semantics. This is where the previous section about refining function types comes in handy. To get rid of all superfluous casts, we have to fully exploit the information provided to us by occurrence typing and deduce for the compiled function the type \code{(number$\to$number)$\wedge$((\dyn\textbackslash number)$\to$number)}, so that no cast is inserted when the function is applied to a number. To achieve this, we simply modify the typing rule for functions that we defined in the previous section to accommodate for gradual typing. For every gradual type $\tau$, we define $\tau^*$ as the type obtained from $\tau$ by replacing all covariant occurrences of \dyn by \Any\ and all contravariant ones by \Empty. The type $\tau^*$ can be seen as the \emph{maximal} interpretation of $\tau$, that is, any expression that can safely be cast to $\tau$ is of type $\tau^*$. In other words, if a function expects an argument of type $\tau$ but can be typed under the hypothesis that the argument has type $\tau^*$, then no casts are needed, since every cast that succeeds will always be to a subtype of $\tau^*$. Taking advantage of this property, we modify the typing rule for functions as follows: \begin{mathpar} \Infer[Abs] {\Gamma,x:\tau\vdash e\triangleright\psi\and \forall i\in I\quad \Gamma,x:\sigma_i\vdash e:\tau_i \and \forall j \in J \subseteq I\quad \Gamma,x:\sigma_j^*\vdash e:\tau_j} { \Gamma\vdash\lambda x:\tau.e:\textstyle\bigwedge_{i\in I}\sigma_i\to \tau_i \land \bigwedge_{j\in J}\sigma_j^*\to \tau_j } {\psi(x)=\{\sigma_i\alt i\in I\}} \end{mathpar}