Handling delimited continuations with dependent types

Abstract

Dependent types are a powerful tool for maintaining program invariants. To take advantage of this aspect in real-world programming, efforts have been put into enriching dependently typed languages with missing constructs, most notably, effects. This paper presents a language that has two practically interesting ingredients: dependent inductive types, and the delimited control constructs shift and reset. When integrating delimited control into a dependently typed language, however, two challenges arise. First, the dynamic nature of control operators, which is the source of their expressiveness, can break fundamental language properties such as logical consistency and subject reduction. Second, CPS translations, which we often use to define the semantics of control operators, do not scale straightforwardly to dependently typed languages. We solve the former issue by restricting dependency of types, and the latter using answer-type polymorphism of pure terms. The main contribution of this paper is to give a sound type system of our language, as well as a type-preserving CPS translation. We also discuss various extensions, which would make our language more like a full-spectrum proof assistant but pose non-trivial issues.

Signature restriction for polymorphic algebraic effects

“…For example, incorporating effects into dependent type theory could easily lead to inconsistency [Pédrot and Tabareau 2020]. This fact encourages dependent type systems to separate term-level computation from types [Ahman 2017;Casinghino et al 2014;Cong and Asai 2018;Sekiyama and Igarashi 2017;Swamy et al 2016;Xi 2007]. For program reasoning, the state transition caused by effectful computations has to be tracked [Ahmed et al 2009;Dreyer et al 2010;Pitts and Stark 1998].…”

A Functional Abstraction of Typed Invocation Contexts

“…By substituting this trail type for the µ 0 in the second compatibility relation, we obtain compatible(µ α , (α → • α), µ α ). This relation does not hold, as we prove in our Agda formalization 15 . As a consequence, we cannot close the above derivation.…”

“…Using the pure judgment and function types, we define the typing rules. There are several possible strategies for designing a fine-grained type system: • Define one rule for each combination of the purity of subexpressions [15]. This allows us to design the most optimized CPS translation, but results in a large number of typing rules (for instance, we would need 2 3 = 8 rules for application).…”

“…The encoding is implemented as shift in lambdaf-fg.agda 15. The impossibility proof c is included in lambdaf.agda.…”

Signature Restriction for Polymorphic Algebraic Effects

“…For example, incorporating computational effects into dependent type theory could lead to inconsistency easily [Pédrot and Tabareau 2020]. This fact encourages dependent type systems to keep term-level computation separate from types [Casinghino et al 2014;Cong and Asai 2018;Sekiyama and Igarashi 2017;Swamy et al 2016;Xi 2007]. Another example is program reasoning, where state transition caused by effectful computations has to be tracked [Ahmed et al 2009;Dreyer et al 2010;Pitts and Stark 1998].…”