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A universe of propositions equipped with definitional proof irrelevance constitutes a convenient medium to express properties and proofs in type-theoretic proof assistants such as Lean, Rocq, and Agda. However, allowing accessibility predicates - used to establish semantic termination arguments - to inhabit such a universe yields undecidable typechecking, hampering the predictability and foundational bases of a proof assistant. To effectively reconcile definitional proof irrelevance and accessibility predicates with both theoretical foundations and practicality in mind, we describe a type theory that extends the Calculus of Inductive Constructions featuring observational equality in a universe of strict propositions, with two variants for handling the elimination principle of accessibility predicates: one variant safeguards decidability by sticking to propositional unfolding, and the other variant favors flexibility with definitional unfolding, at the expense of a potentially diverging typechecking procedure. Crucially, the metatheory of this dual approach establishes that any proof made in the definitional variant of the theory can be translated into a proof of the same statement in the propositional variant, all while preserving the decidability of the latter. Moreover, we prove the two variants to be consistent and to satisfy forms of canonicity, ensuring that programs can indeed be properly evaluated. We present an implementation in Rocq and compare it with existing approaches. Overall, this work introduces an effective technique that informs the design of proof assistants with strict propositions, enabling local computation with accessibility predicates without compromising the ambient type theory.

Static analysis tools typically address the problem of excessive false positives by requiring programmers to explicitly annotate their code. However, when faced with incomplete annotations, many analysis tools are either too conservative, yielding false positives, or too optimistic, resulting in unsound analysis results. In order to flexibly and soundly deal with partially-annotated programs, we propose to build upon and adapt the gradual typing approach to abstract-interpretation-based program analyses. Specifically, we focus on null-pointer analysis and demonstrate that a gradual null-pointer analysis hits a sweet spot, by gracefully applying static analysis where possible and relying on dynamic checks where necessary for soundness. In addition to formalizing a gradual null-pointer analysis for a core imperative language, we build a prototype using the Infer static analysis framework, and present preliminary evidence that the gradual null-pointer analysis reduces false positives compared to two existing null-pointer checkers for Infer. Further, we discuss ways in which the gradualization approach used to derive the gradual analysis from its static counterpart can be extended to support more domains. This work thus provides a basis for future analysis tools that can smoothly navigate the tradeoff between human effort and run-time overhead to reduce the number of reported false positives.

Gradual typing is an effective approach to integrate static and dynamic typing, which supports the smooth transition between both extremes via the (programmer-controlled) precision of type annotations [Jeremy Siek and Walid Taha, 2006; Siek et al., 2015]. Imprecision is normally introduced via the unknown type ?, e.g. function type Int → Bool is more precise than ? → ?, and both more precise than ?. Gradual typing relates types of different precision using consistent type relations, such as type consistency (resp. consistent subtyping), the gradual counterpart of type equality (resp. subtyping). For instance, ? → Int is consistent with Bool → ?. This approach has been applied in a number of settings, such as objects [Jeremy Siek and Walid Taha, 2007], subtyping [Jeremy Siek and Walid Taha, 2007; Ronald Garcia et al., 2016], effects [Bañados Schwerter et al., 2014; Bañados Schwerter et al., 2016], ownership [Ilya Sergey and Dave Clarke, 2012], typestates [Roger Wolff et al., 2011; Ronald Garcia et al., 2014], information-flow typing [Tim Disney and Cormac Flanagan, 2011; Luminous Fennell and Peter Thiemann, 2013; Matías Toro et al., 2018], session types [Igarashi et al., 2017], refinements [Nico Lehmann and {É}ric Tanter, 2017], set-theoretic types [Castagna and Lanvin, 2017], Hoare logic [Johannes Bader et al., 2018], parametric polymorphism [Amal Ahmed et al., 2011; Ahmed et al., 2017; Ina and Igarashi, 2011; Igarashi et al., 2017; Ningning Xie et al., 2018; Matías Toro et al., 2019], and references [Jeremy Siek and Walid Taha, 2006; Herman et al., 2010; Siek et al., 2015]. In particular, gradual typing for mutable references has seen the elaboration of various possible semantics: invariant references [Jeremy Siek and Walid Taha, 2006], guarded references [Herman et al., 2010], monotonic references [Siek et al., 2015], and permissive references [Siek et al., 2015]. Invariant references are a form of references where reference types are invariant with respect to type consistency. Guarded references admit variance thanks to systematic runtime checks on reference reads and writes; the runtime type of an allocated cell never changes during execution. Guarded references have been formulated in a space-efficient coercion calculus, which ensures that gradual programs do not accumulate unbounded pending checks during execution. Hereafter, we refer to this language as HCC. Monotonic references favor efficiency over flexibility by only allowing reference cells to vary towards more precise types. This allows reference operations in statically-typed regions to safely proceed without any runtime checks. Permissive references are the most flexible approach, in which reference cells can be initialized and updated to any value of any type at any time. These four developments reflect different design decisions with respect to gradual references: is the reference type constructor variant under consistency? Can the programmer specify a precise bound on the static type of a reference, and hence on the corresponding heap cell type? Can the heap cell type evolve its precision at runtime, and if yes, how? There is obviously no absolute answer to these questions, as they reflect different tradeoffs such as in efficiency and precision. This work explores the semantics that results from the application of a systematic methodology to gradualize static type systems. Currently we can find in the literature two methodologies to gradualize statically-typed languages: Abstracting Gradual Typing (AGT) [Ronald Garcia et al., 2016], and the Gradualizer [Matteo Cimini and Jeremy Siek, 2016]. In this work, we consider the AGT methodology as it naturally scales to auxiliary structures such as a mutable heap. The AGT methodology helps to systematically construct gradually-typed languages by using abstract interpretation [Cousot and Cousot, 1977] at the type level. In brief, AGT interprets gradual types as an abstraction of sets of possible static types, formally captured through a Galois connection. The static semantics of a gradual language are then derived by lifting the semantics of a statically-typed language through this connection, and the dynamic semantics follow by Curry-Howard from proof normalization of the type safety argument. The AGT methodology has been shown to be effective in many contexts: records and subtyping [Ronald Garcia et al., 2016], type-and-effects [Bañados Schwerter et al., 2014; Bañados Schwerter et al., 2016], refinement types [Nico Lehmann and Éric Tanter, 2017; Niki Vazou et al., 2018], set-theoretic and union types [Castagna and Lanvin, 2017; Matías Toro and Éric Tanter, 2017], information-flow typing [Matías Toro et al., 2018], and parametric polymorphism [Matías Toro et al., 2019]. However, this methodology has never been applied to mutable references in isolation. Although Toro et al. [Matías Toro et al., 2018] apply AGT to a language with references, they only gradualize security levels of types (e.g. Ref Int_?), not whole types (e.g. Ref ? is not supported). In this article we answer the following open questions: Which semantics for gradually-type references follows by systematically applying AGT? Does AGT justify one of the existing approaches, or does it suggest yet another design? Can we recover other semantics for gradual references, if yes, how? This article first reviews the different existing gradual approaches to mutable references through examples. It then presents the semantics for gradual references that is obtained by applying AGT, and how to accommodate the other semantics. More specifically, this work makes the following contributions: - We present λ_REF~, a gradual language with support for mutable references. We derive λ_REF~ by applying the AGT methodology to a fully-static simple language with mutable references called λ_REF. This is the first application of AGT that focuses on gradually-typed mutable references. - We prove that λ_REF~ satisfies the gradual guarantee of Siek et al. [Siek et al., 2015]. We also present the first formal statement and proof of the conservative extension of the dynamic semantics of the static language [Siek et al., 2015], for a gradual language derived using AGT. - We prove that the derived language, λ_REF~, corresponds to the semantics of guarded references from HCC. Formally, given a λ_REF~ term and its compilation to HCC^+ (an adapted version of HCC extended with conditionals and binary operations) we prove that both terms are bisimilar, and that consequently they either both terminate, both fail, or both diverge. - We observe that λ_REF~ and HCC^+ differ in the order of combination of runtime checks. As a result, HCC is space efficient whereas λ_REF~ is not: we can write programs in λ_REF~ that may accumulate an unbounded number of checks. We formalize the changes needed in the dynamic semantics of λ_REF~ to achieve space efficiency. This technique to recover space efficiency is in fact independent from mutable references, and is therefore applicable to other gradual languages derived with AGT. - We formally describe how to support other gradual reference semantics in λ_REF~ by presenting λ_REF~^𝗉𝗆, an extension that additionally supports both permissive and monotonic references. Finally, we prove for the first time that monotonic references satisfy the dynamic gradual guarantee, a non-trivial result that requires careful consideration of updates to the store. Additionally, we implemented λ_REF~ as an interactive prototype that displays both typing derivations and reduction traces. All the examples mentioned in this paper are readily available in the online prototype available at https://pleiad.cl/grefs. As a result, this paper sheds further light on the design space of gradual languages with mutable references and contributes to deepening the understanding of the AGT methodology.

This artifact is a web interpreter for the ObSec language defined in the companion paper. ObSec is a simple object-oriented language that supports type-based declassification. Type-base declassification exploits the familiar notion of type abstraction to support expressive declassification policies in a simple and expressive manner.

Information-flow security typing statically prevents confidential information to leak to public channels. The fundamental information flow property, known as noninterference, states that a public observer cannot learn anything from private data. As attractive as it is from a theoretical viewpoint, noninterference is impractical: real systems need to intentionally declassify some information, selectively. Among the different information flow approaches to declassification, a particularly expressive approach was proposed by Li and Zdancewic, enforcing a notion of relaxed noninterference by allowing programmers to specify declassification policies that capture the intended manner in which public information can be computed from private data. This paper shows how we can exploit the familiar notion of type abstraction to support expressive declassification policies in a simpler, yet more expressive manner. In particular, the type-based approach to declassification---which we develop in an object-oriented setting---addresses several issues and challenges with respect to prior work, including a simple notion of label ordering based on subtyping, support for recursive declassification policies, and a local, modular reasoning principle for relaxed noninterference. This work paves the way for integrating declassification policies in practical security-typed languages.