Gradual Guarantee Research Articles

Merging Gradual Typing

Programming language mechanisms with a type-directed semantics are nowadays common and widely used. Such mechanisms include gradual typing, type classes, implicits and intersection types with a merge operator. While sharing common challenges in their design and having complementary strengths, type-directed mechanisms have been mostly independently studied. This paper studies a new calculus, called λM⋆, which combines two type-directed mechanisms: gradual typing and a merge operator based on intersection types. Gradual typing enables a smooth transition between dynamically and statically typed code, and is available in languages such as TypeScript or Flow. The merge operator generalizes record concatenation to allow merges of values of any two types. Recent work has shown that the merge operator enables modelling expressive OOP features like first-class traits/classes and dynamic inheritance with static type-checking. These features are not found in mainstream statically typed OOP languages, but they can be found in dynamically or gradually typed languages such as JavaScript or TypeScript. In λM⋆, by exploiting the complementary strengths of gradual typing and the merge operator, we obtain a foundation for modelling gradually typed languages with both first-class classes and dynamic inheritance. We study a static variant of λM⋆ (called λM); prove the type-soundness of λM⋆; show that λM⋆ can encode gradual rows and all well-typed terms in the GTFL≲ calculus; and show that λM⋆ satisfies gradual typing criteria. The dynamic gradual guarantee (DGG) is challenging due to the possibility of ambiguity errors. We establish a variant of the DGG using a semantic notion of precision based on a step-indexed logical relation.

A Gradual Probabilistic Lambda Calculus

Probabilistic programming languages have recently gained a lot of attention, in particular due to their applications in domains such as machine learning and differential privacy. To establish invariants of interest, many such languages include some form of static checking in the form of type systems. However, adopting such a type discipline can be cumbersome or overly conservative. Gradual typing addresses this problem by supporting a smooth transition between static and dynamic checking, and has been successfully applied for languages with different constructs and type abstractions. Nevertheless, its benefits have never been explored in the context of probabilistic languages. In this work, we present and formalize GPLC, a gradual source probabilistic lambda calculus. GPLC includes a binary probabilistic choice operator and allows programmers to gradually introduce/remove static type–and probability–annotations. The static semantics of GPLC heavily relies on the notion of probabilistic couplings, as required for defining several relations, such as consistency, precision, and consistent transitivity. The dynamic semantics of GPLC is given via elaboration to the target language TPLC, which features a distribution-based semantics interpreting programs as probability distributions over final values. Regarding the language metatheory, we establish that TPLC–and therefore also GPLC–is type safe and satisfies two of the so-called refined criteria for gradual languages, namely, that it is a conservative extension of a fully static variant and that it satisfies the gradual guarantee, behaving smoothly with respect to type precision.

Label dependent lambda calculus and gradual typing

Dependently-typed programming languages are gaining importance, because they can guarantee a wide range of properties at compile time. Their use in practice is often hampered because programmers have to provide very precise types. Gradual typing is a means to vary the level of typing precision between program fragments and to transition smoothly towards more precisely typed programs. The combination of gradual typing and dependent types seems promising to promote the widespread use of dependent types. We investigate a gradual version of a minimalist value-dependent lambda calculus. Compile-time calculations and thus dependencies are restricted to labels, drawn from a generic enumeration type. The calculus supports the usual Pi and Sigma types as well as singleton types and subtyping. It is sufficiently powerful to provide flexible encodings of variant and record types with first-class labels. We provide type checking algorithms for the underlying label-dependent lambda calculus and its gradual extension. The gradual type checker drives the translation into a cast calculus, which extends the original language. The cast calculus comes with several innovations: refined typing for casts in the presence of singletons, type reduction in casts, and fully dependent Sigma types. Besides standard metatheoretical results, we establish the gradual guarantee for the gradual language.

Gradual type theory

AbstractGradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. Sound gradually typed languages dynamically check types at runtime at the boundary between statically typed and dynamically typed modules. However, there is much disagreement in the gradual typing literature over how to enforce complex types such as tuples, lists, functions and objects. In this paper, we propose a new perspective on the design of runtime gradual type enforcement: runtime type casts exist precisely to ensure the correctness of certain type-based refactorings and optimizations. For instance, for simple types, a language designer might desire that beta-eta equality is valid. We show that this perspective is useful by demonstrating that a cast semantics can be derived from beta-eta equality. We do this by providing an axiomatic account program equivalence in a gradual cast calculus in a logic we callgradual type theory(GTT). Based on Levy’s call-by-push-value, GTT allows us to axiomatize both call-by-value and call-by-name gradual languages. We then show that we can derive the behavior of casts for simple types from the corresponding eta equality principle and the assumption that the language satisfies a property calledgraduality, also known as the dynamic gradual guarantee. Since we can derive the semantics from the assumption of eta equality, we also receive a useful contrapositive: any observably different cast semantics that satisfies gradualitymustviolate the eta equality. We show the consistency and applicability of our axiomatic theory by proving that a contract-based implementation using the lazy cast semantics gives a logical relations model of our type theory, where equivalence in GTT implies contextual equivalence of the programs. Since GTT also axiomatizes the dynamic gradual guarantee, our model also establishes this central theorem of gradual typing. The model is parameterized by the implementation of the dynamic types, and so gives a family of implementations that validate type-based optimization and the gradual guarantee.

Abstracting gradual references

Gradual typing is an effective approach to integrate static and dynamic typing, which supports the smooth transition between both extremes via the imprecision of type annotations. Gradual typing has been applied in many scenarios such as objects, subtyping, effects, ownership, typestates, information-flow typing, parametric polymorphism, etc. In particular, the combination of gradual typing and mutable references has been explored by different authors, giving rise to four different semantics—invariant, guarded, monotonic and permissive references. These semantics were specially crafted to reflect different design decisions with respect to precision and efficiency tradeoffs. Since then, progress has been made in the formulation of methodologies to systematically derive gradual counterparts of statically-typed languages, but these have not been applied to study mutable references. In this article, we explore how the Abstracting Gradual Typing (AGT) methodology, which has been shown to be effective in a variety of settings, applies to mutable references. Starting from a standard statically-typed language with references, we systematically derive with AGT a novel gradual language, called λ REF ˜ . We establish the properties of λ REF ˜ ; in particular, it is the first gradual language with mutable references that is proven to satisfy the gradual guarantee. We then compare λ REF ˜ with the main four existing approaches to gradual references, and show that the application of AGT does justify one of the proposed semantics: we formally prove that the treatment of references in λ REF ˜ corresponds to the guarded semantics, by presenting a bisimilation with the coercion semantics of Herman et al. In the process, we uncover that any direct application of AGT yields a gradual language that is not space-efficient. We consequently adjust the dynamic semantics of λ REF ˜ to recover space efficiency. We then show how to extend λ REF ˜ to support both monotonic and permissive references as well. Finally, we provide the first proof of the dynamic gradual guarantee for monotonic references. 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. • Detailed application of the Abstracting Gradual Typing methodology (AGT) to a language with mutable references. • Establishes formal equivalence with guarded semantics of Herman et al. • Shows how to achieve space efficiency in the AGT-derived language. • Combines different gradual reference semantics in the same language. • First proof of the dynamic gradual guarantee for both guarded and monotonic references.

Consistent Subtyping for All

Consistent subtyping is employed in some gradual type systems to validate type conversions. The original definition by Siek and Taha serves as a guideline for designing gradual type systems with subtyping. Polymorphic types à la System F also induce a subtyping relation that relates polymorphic types to their instantiations. However, Siek and Taha’s definition is not adequate for polymorphic subtyping. The first goal of this article is to propose a generalization of consistent subtyping that is adequate for polymorphic subtyping and subsumes the original definition by Siek and Taha. The new definition of consistent subtyping provides novel insights with respect to previous polymorphic gradual type systems, which did not employ consistent subtyping. The second goal of this article is to present a gradually typed calculus for implicit (higher-rank) polymorphism that uses our new notion of consistent subtyping. We develop both declarative and (bidirectional) algorithmic versions for the type system. The algorithmic version employs techniques developed by Dunfield and Krishnaswami for higher-rank polymorphism to deal with instantiation. We prove that the new calculus satisfies all static aspects of the refined criteria for gradual typing. We also study an extension of the type system with static and gradual type parameters, in an attempt to support a variant of the dynamic criterion for gradual typing. Assuming a coherence conjecture for the extended calculus, we show that the dynamic gradual guarantee of our source language can be reduced to that of λ B, which, at the time of writing, is still an open question. Most of the metatheory of this article, except some manual proofs for the algorithmic type system and extensions, has been mechanically formalized using the Coq proof assistant.

Gradual type theory

Gradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. While existing gradual type soundness theorems for these languages aim to show that type-based reasoning is preserved when moving from the fully static setting to a gradual one, these theorems do not imply that correctness of type-based refactorings and optimizations is preserved. Establishing correctness of program transformations is technically difficult, because it requires reasoning about program equivalence, and is often neglected in the metatheory of gradual languages. In this paper, we propose an axiomatic account of program equivalence in a gradual cast calculus, which we formalize in a logic we call gradual type theory (GTT). Based on Levy’s call-by-push-value, GTT gives an axiomatic account of both call-by-value and call-by-name gradual languages. Based on our axiomatic account we prove many theorems that justify optimizations and refactorings in gradually typed languages. For example, uniqueness principles for gradual type connectives show that if the βη laws hold for a connective, then casts between that connective must be equivalent to the so-called “lazy” cast semantics. Contrapositively, this shows that “eager” cast semantics violates the extensionality of function types. As another example, we show that gradual upcasts are pure functions and, dually, gradual downcasts are strict functions. We show the consistency and applicability of our axiomatic theory by proving that a contract-based implementation using the lazy cast semantics gives a logical relations model of our type theory, where equivalence in GTT implies contextual equivalence of the programs. Since GTT also axiomatizes the dynamic gradual guarantee, our model also establishes this central theorem of gradual typing. The model is parametrized by the implementation of the dynamic types, and so gives a family of implementations that validate type-based optimization and the gradual guarantee.

Dynamic type inference for gradual Hindley–Milner typing

Garcia and Cimini study a type inference problem for the ITGL, an implicitly and gradually typed language with let-polymorphism, and develop a sound and complete inference algorithm for it. Soundness and completeness mean that, if the algorithm succeeds, the input term can be translated to a well-typed term of an explicitly typed blame calculus by cast insertion and vice versa. However, in general, there are many possible translations depending on how type variables that were left undecided by static type inference are instantiated with concrete static types. Worse, the translated terms may behave differently—some evaluate to values but others raise blame. In this paper, we propose and formalize a new blame calculus λ B DTI that avoids such divergence as an intermediate language for the ITGL. A main idea is to allow a term to contain type variables (that have not been instantiated during static type inference) and defer instantiation of these type variables to run time. We introduce dynamic type inference (DTI) into the semantics of λ B DTI so that type variables are instantiated along reduction. The DTI-based semantics not only avoids the divergence described above but also is sound and complete with respect to the semantics of fully instantiated terms in the following sense: if the evaluation of a term succeeds (i.e., terminates with a value) in the DTI-based semantics, then there is a fully instantiated version of the term that also succeeds in the explicitly typed blame calculus and vice versa. Finally, we prove the gradual guarantee, which is an important correctness criterion of a gradually typed language, for the ITGL.

Gradual parametricity, revisited

Bringing the benefits of gradual typing to a language with parametric polymorphism like System F, while preserving relational parametricity, has proven extremely challenging: first attempts were formulated a decade ago, and several designs were recently proposed. Among other issues, these proposals can however signal parametricity errors in unexpected situations, and improperly handle type instantiations when imprecise types are involved. These observations further suggest that existing polymorphic cast calculi are not well suited for supporting a gradual counterpart of System F. Consequently, we revisit the challenge of designing a gradual language with explicit parametric polymorphism, exploring the extent to which the Abstracting Gradual Typing methodology helps us derive such a language, GSF. We present the design and metatheory of GSF, and provide a reference implementation. In addition to avoiding the uncovered semantic issues, GSF satisfies all the expected properties of a gradual parametric language, save for one property: the dynamic gradual guarantee, which was left as conjecture in all prior work, is here proven to be simply incompatible with parametricity. We nevertheless establish a weaker property that allows us to disprove several claims about gradual free theorems, clarifying the kind of reasoning supported by gradual parametricity.

Type-Driven Gradual Security with References

In security-typed programming languages, types statically enforce noninterference between potentially conspiring values, such as the arguments and results of functions. But to adopt static security types, like other advanced type disciplines, programmers face a steep wholesale transition, often forcing them to refactor working code just to satisfy their type checker. To provide a gentler path to security typing that supports safe and stylish but hard-to-verify programming idioms, researchers have designed languages that blend static and dynamic checking of security types. Unfortunately, most of the resulting languages only support static, type-based reasoning about noninterference if a program is entirely statically secured. This limitation substantially weakens the benefits that dynamic enforcement brings to static security typing. Additionally, current proposals are focused on languages with explicit casts and therefore do not fulfill the vision of gradual typing, according to which the boundaries between static and dynamic checking only arise from the (im)precision of type annotations and are transparently mediated by implicit checks. In this article, we present GSL Ref , a gradual security-typed higher-order language with references. As a gradual language, GSL Ref supports the range of static-to-dynamic security checking exclusively driven by type annotations, without resorting to explicit casts. Additionally, GSL Ref lets programmers use types to reason statically about termination-insensitive noninterference in all programs, even those that enforce security dynamically. We prove that GSL Ref satisfies all but one of Siek et al.’s criteria for gradually-typed languages, which ensure that programs can seamlessly transition between simple typing and security typing. A notable exception regards the dynamic gradual guarantee, which some specific programs must violate if they are to satisfy noninterference; it remains an open question whether such a language could fully satisfy the dynamic gradual guarantee. To realize this design, we were led to draw a sharp distinction between syntactic type safety and semantic type soundness , each of which constrains the design of the gradual language.

Graduality from embedding-projection pairs

Gradually typed languages allow statically typed and dynamically typed code to interact while maintaining benefits of both styles. The key to reasoning about these mixed programs is Siek-Vitousek-Cimini-Boyland’s (dynamic) gradual guarantee, which says that giving components of a program more precise types only adds runtime type checking, and does not otherwise change behavior. In this paper, we give a semantic reformulation of the gradual guarantee called graduality . We change the name to promote the analogy that graduality is to gradual typing what parametricity is to polymorphism. Each gives a local-to-global, syntactic-to-semantic reasoning principle that is formulated in terms of a kind of observational approximation. Utilizing the analogy, we develop a novel logical relation for proving graduality. We show that embedding-projection pairs (ep pairs) are to graduality what relations are to parametricity. We argue that casts between two types where one is “more dynamic” (less precise) than the other necessarily form an ep pair, and we use this to cleanly prove the graduality cases for casts from the ep-pair property. To construct ep pairs, we give an analysis of the type dynamism relation—also known as type precision or naïve subtyping—that interprets the rules for type dynamism as compositional constructions on ep pairs, analogous to the coercion interpretation of subtyping.

On polymorphic gradual typing

We study an extension of gradual typing—a method to integrate dynamic typing and static typing smoothly in a single language—to parametric polymorphism and its theoretical properties, including conservativity of typing and semantics over both statically and dynamically typed languages, type safety, blame-subtyping theorem, and the gradual guarantee—the so-called refined criteria, advocated by Siek et al. We develop System F G , which is a gradually typed extension of System F with the dynamic type and a new type consistency relation, and translation to a new polymorphic blame calculus System F C , which is based on previous polymorphic blame calculi by Ahmed et al. The design of System F G and System F C , geared to the criteria, is influenced by the distinction between static and gradual type variables, first observed by Garcia and Cimini. This distinction is also useful to execute statically typed code without incurring additional overhead to manage type names as in the prior calculi. We prove that System F G satisfies most of the criteria: all but the hardest property of the gradual guarantee on semantics. We show that a key conjecture to prove the gradual guarantee leads to the Jack-of-All-Trades property, conjectured as an important property of the polymorphic blame calculus by Ahmed et al.

Big types in little runtime: open-world soundness and collaborative blame for gradual type systems

Gradual typing combines static and dynamic typing in the same language, offering programmers the error detection and strong guarantees of static types and the rapid prototyping and flexible programming idioms of dynamic types. Many gradually typed languages are implemented by translation into an untyped target language (e.g., Typed Clojure, TypeScript, Gradualtalk, and Reticulated Python). For such languages, it is desirable to support arbitrary interaction between translated code and legacy code in the untyped language while maintaining the type soundness of the translated code. In this paper we formalize this goal in the form of the open-world soundness criterion. We discuss why it is challenging to achieve open-world soundness using the traditional proxy-based approach for higher-order casts. However, the transient design satisfies open-world soundness. Indeed, we present a formal semantics for the transient design and prove that our semantics satisfies open-world soundness. In this paper we also solve a challenging problem for the transient design: how to provide blame tracking without proxies. We define a semantics for blame and prove the Blame Theorem. We also prove that the Gradual Guarantee holds for this system, ensuring that programs can be evolved freely between static and dynamic typing. Finally, we demonstrate that the runtime overhead of the transient approach is low in the context of Reticulated Python, an implementation of gradual typing for Python.

Crisis and Reform of the Western World

T HE wide national and territorial market areas of the European states and their colonial empires were, from the end of the eighteenth century, opened first for internal and later for international free trade. Considerable national and international uniformity of law grew under the influence of the law of the Roman Empire which was favorable not only to national but also to international intercourse. This Roman law had been pretty generally established by the dawn of modern times in practically all leading countries and it had been molded into the national conceptions of law. The gradual guarantee of personal rights even in otherwise absolutist states of Europe, and the abolition of the rest of mediaeval feudalism in favor of personal freedom, legal equality and free individual property and contract rights, opened the way for the activities of the growing bourgeois class. This class, the most active exponent of the spirit of the age, became more and more conscious of its value and importance. The individual, freed from mediaeval ties, immediately and personally confronted with responsibility for the social universe, and armed with weapons of modern science, buoyantly started out for new horizons. The technique of transportation of persons, materials, and news very soon overcame the resistance of time and space to a degree that made the whole world in many respects into one market area and one system of national and international division of labor. This world market was served by an equally widely developed monetary and credit system.