Sound and Complete Type Inference for Closed Effect Rows

Abstract

Koka is a functional programming language that has algebraic effect handlers and a row-based effect system. The row-based effect system infers types by naively applying the Hindley-Milner type inference. However, it infers effect-polymorphic types for many functions, which are hard to read by the programmers and have a negative runtime performance impact to the evidence-passing translation. In order to improve readability and runtime efficiency, we aim to infer closed effect rows when possible, and open those closed effect rows automatically at instantiation to avoid loss of typability. This paper gives a type inference algorithm with the ${\mathsf {{open}}}$ and ${\mathsf {{close}}}$ mechanisms. In this paper, we define a type inference algorithm with the open and close constructs.

Appendices

Appendix

Fig. 11.

Translation to system F${^{\epsilon }}$+${\mathsf {{restrict}}}$

Fig. 12.

Types of system F${^{\epsilon }}$+${\mathsf {{restrict}}}$

Fig. 13.

Expressions and operational semantics of F${^{\epsilon }}$+${\mathsf {{restrict}}}$

Fig. 14.

Typing rules of F${^{\epsilon }}$+${\mathsf {{restrict}}}$

In this appendix, we present an extension of System F${^{\epsilon }}$, which we call System F${^{\epsilon }}$ + ${\mathsf {{restrict}}}$, and a type-directed translation from ${\mathsf {{ImpKoka}}}$ to System F${^{\epsilon }}$ + ${\mathsf {{restrict}}}$. The translation allows us to prove the type soundness of ${\mathsf {{ImpKoka}}}$ without directly defining an operational semantics for ${\mathsf {{ImpKoka}}}$. This is a well-known technique, and is used in [10], for instance. The target calculus of the translation has a new construct ${\mathsf {{restrict}}}$, which is necessary for establishing soundness of the translation.

A System F${^{\epsilon }}$ + ${\mathsf {{restrict}}}$

In Figs. 12, 13 and 14, we define the syntax, operational semantics, and typing rules of System F ${^{\epsilon }}$ + ${\mathsf {{restrict}}}$. The typing rule ${{[\textsc {{restrict}}]}}$ extends the closed effect row ${\langle {} l _{1},.., l _ n \rangle {}}$ of the expression ${ e }$ to ${\langle {} l _1,{\dots } l _ n \mid {}{\epsilon }\rangle {}}$, where ${{\epsilon }}$ is an arbitrary effects.

We can prove the type soundness of System F${^{\epsilon }}$ + ${\mathsf {{restrict}}}$ by showing the following theorems.

Theorem 13

(progress) $\textrm{If }\, {\emptyset \vdash {}~ e _{1}~\,:\,{}~{\sigma }\mid {}\langle \rangle {}}\ \mathrm {then\ either}\ { e _{1}} \ \mathrm {is\ a\ value\ or}\ { e _{1}~{\longmapsto }~ e _{2}}.$

Theorem 14

(preservation) $\textrm{If }\, {\emptyset \vdash {}~ e _{1}~\,:\,{}~{\sigma }\mid {}\langle \rangle {}} \ \textrm{and}\ { e _{1}~{\longmapsto }~ e _{2}} \ \textrm{then}\ {\emptyset \vdash {}~ e _{2}~\,:\,{}~{\sigma }\mid {}\langle \rangle {}}.$

B Type-Directed Translation to System F${^{\epsilon }+\mathsf {{restrict}}}$

In Fig. 11, we next define the type-directed translation from ${\mathsf {{ImpKoka}}}$ to System F${^{\epsilon }}$ + ${\mathsf {{restrict}}}$. The judgment ${\varXi {}\mid {}\varGamma {}\mid {}\varDelta \vdash {}~ e ~\,:\,{}~{\sigma }\mid {}{\epsilon }~{\;\rightsquigarrow }\;~ e '}$ states that an expression ${ e }$ has type ${{\sigma }}$ and effect ${{\epsilon }}$ under the type variable context ${\varXi {}}$ and typing contexts ${\varGamma {}}$ and ${\varDelta }$, and translates to an expression ${ e '}$ of System F${^{\epsilon }}$ + ${\mathsf {{restrict}}}$. We can easily prove the soundness of the type-directed translation.

Theorem 15

(soundness of type-directed translation)

$\textrm{If}~{\varXi {}\mid {}\varGamma {}\mid {}\varDelta \vdash {}~ e ~\,:\,{}{\sigma }\mid {}}{{\epsilon }~{\;\rightsquigarrow }\;~ e '}\ \textrm{then}\ {\varGamma {},~\varDelta \vdash {}~ e '~\,:\,{}~{\sigma }\mid {}{\epsilon }}$.

Proof

By straightforward induction on the typing derivation.