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1. Uniform Strong Normalization for Multi-discipline Calculi

   https://doi.org/10.1007/978-3-319-99840-4_12  

   Downen P., Johnson-Freyd P. (January 2018, Rewriting Logic and Its Applications. WRLA 2018.)

   Modern programming languages have effects and mix multiple calling conventions, and their core calculi should too. We characterize calling conventions by their “substitution discipline” that says what variables stand for, and design calculi for mixing disciplines in a single program. Building on variations of the reducibility candidates method, including biorthogonality and symmetric candidates which are both specialized for one discipline, we develop a single uniform framework for strong normalization encompassing call-by-name, call-by-value, call-by-need, call-by-push-value, non-deterministic disciplines, and any others satisfying some simple criteria. We explicate commonalities of previous methods and show they are special cases of the uniform framework and they extend to multi-discipline programs. 

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2. Call-by-Unboxed-Value

   https://doi.org/10.1145/3674654  

   Downen, Paul (August 2024, Proceedings of the ACM on Programming Languages)

   Call-By-Push-Value has famously subsumed both call-by-name and call-by-value by decomposing programs along the axis of values versus computations. Here, we introduce Call-By-Unboxed-Value which further decomposes programs along an orthogonal axis separating atomic versus complex. As the name suggests, these two dimensions make it possible to express the representations of values as boxed or unboxed, so that functions pass unboxed values as inputs and outputs. More importantly, Call-By-Unboxed-Value allows for an unrestricted mixture of polymorphism and unboxed types, giving a foundation for studying compilation techniques for polymorphism based on representation irrelevance. In this regard, we use Call-By-Unboxed-Value to formalize representation polymorphism independently of types; for the first time compiling untyped representation-polymorphic code, while nonetheless preserving types all the way to the machine. 

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3. Gradual type theory

   https://doi.org/10.1017/S0956796821000125  

   NEW, MAX S.; LICATA, DANIEL R.; AHMED, AMAL (January 2021, Journal of Functional Programming)

   null (Ed.)

   Abstract Gradually 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 call gradual 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 called graduality , 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 graduality must violate 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. 

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4. Sensitivity by Parametricity

   https://doi.org/10.1145/3689726  

   Lobo-Vesga, Elisabet; Russo, Alejandro; Gaboardi, Marco; Cortiñas, Carlos Tomé (October 2024, Proceedings of the ACM on Programming Languages)

   The work of Fuzz has pioneered the use of functional programming languages where types allow reasoning about the sensitivity of programs. Fuzz and subsequent work (e.g., DFuzz and Duet) use advanced technical devices like linear types, modal types, and partial evaluation. These features usually require the design of a new programming language from scratch—a significant task on its own! While these features are part of the classical toolbox of programming languages, they are often unfamiliar to non-experts in this field. Fortunately, recent studies (e.g., Solo) have shown that linear and complex types in general, are not strictly needed for the task of determining programs’ sensitivity since this can be achieved by annotating base types with static sensitivity information. In this work, we take a different approach. We propose to enrich base types with information about the metric relation between values, and we present the novel idea of applyingparametricityto derive direct proofs for the sensitivity of functions. A direct consequence of our result is thatcalculating and provingthe sensitivity of functions is reduced to simply type-checking in a programming language with support for polymorphism and type-level naturals. We formalize our main result in a calculus, prove its soundness, and implement a software library in the programming language Haskell–where we reason about the sensitivity of canonical examples. We show that the simplicity of our approach allows us to exploit the type inference of the host language to support a limited form of sensitivity inference. Furthermore, we extend the language with a privacy monad to showcase how our library can be used in practical scenarios such as the implementation of differentially private programs, where the privacy guarantees depend on the sensitivity of user-defined functions. Our library, called Spar, is implemented in less than 500 lines of code. 

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5. Kind inference for datatypes

   https://doi.org/10.1145/3371121  

   Xie, Ningning; Eisenberg, Richard_A; Oliveira, Bruno_C_d_S (December 2019, Proceedings of the ACM on Programming Languages)

   In recent years, languages like Haskell have seen a dramatic surge of new features that significantly extends the expressive power of their type systems. With these features, the challenge of kind inference for datatype declarations has presented itself and become a worthy research problem on its own. This paper studies kind inference for datatypes. Inspired by previous research on type-inference, we offer declarative specifications for what datatype declarations should be accepted, both for Haskell98 and for a more advanced system we call PolyKinds, based on the extensions in modern Haskell, including a limited form of dependent types. We believe these formulations to be novel and without precedent, even for Haskell98. These specifications are complemented with implementable algorithmic versions. We study soundness, completeness and the existence of principal kinds in these systems, proving the properties where they hold. This work can serve as a guide both to language designers who wish to formalize their datatype declarations and also to implementors keen to have principled inference of principal types. 

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