Type-level Programming Research Articles

Type-level programming with match types

Type-level programming is becoming more and more popular in the realm of functional programming. However, the combination of type-level programming and subtyping remains largely unexplored in practical programming languages. This paper presents match types , a type-level equivalent of pattern matching. Match types integrate seamlessly into programming languages with subtyping and, despite their simplicity, offer significant additional expressiveness. We formalize the feature of match types in a calculus based on System F sub and prove its soundness. We practically evaluate our system by implementing match types in the Scala 3 reference compiler, thus making type-level programming readily available to a broad audience of programmers.

Structure verification of deep neural networks at compilation time

This paper presents TensorSafe , a Haskell library that makes possible the definition and structural validation of deep neural network architectures. Nowadays, the development process of deep learning models has been notably simplified due to the availability of sophisticated tools in the industry. However, most of these tools do not provide any security controls at compilation time, making the developers deal with unexpected run-time errors and uncertainties. In particular, validating the structure of deep neural networks at compilation time is a complex subject that involves the mathematical validation of all operations that a deep learning model will perform. Moreover, this structural checking requires an advanced usage of types systems theories to manipulate abstract type definitions capable of modeling neural network constructions. Many different programming techniques were involved in the specification of TensorSafe. Primarily, the application of the functional programming paradigm and the use of type-level programming were of great importance for the development process and to probe the correctness of the neural network models. The experimental evaluation showed that by using TensorSafe it is possible to correctly create well-known deep neural network architectures, such as AlexNet or ResNet50 . • Review state of the art of deep learning DSLs that provide structural controls at compile time. • Extensible Haskell library that is capable of defining non-sequential deep learning architectures. • Interaction with external deep learning frameworks like TensorFlow for JavaScript or Keras for Python.

Implementing a Language for Distributed Systems: Choices and Experiences with Type Level and Macro Programming in Scala

Multitier programming languages reduce the complexity of developing distributed systems by developing the distributed system in a single coherent code base. The compiler or the runtime separate the code for the components of the distributed system, enabling abstraction over low level implementation details such as data representation, serialization and network protocols. Our ScalaLoci language allows developers to declare the different components and their architectural relation at the type level, allowing static reasoning about about distribution and remote communication and guaranteeing static type safety across components. The compiler splits the multitier program into the component-specific code and automatically generates the communication boilerplate. Communication between components can be modeled by declaratively specifying data flows between components using reactive programming. In this paper, we report on the implementation of our design and our experience with embedding our language features into Scala as a host language. We show how a combination of Scala's advanced type level programming and its macro system can be used to enrich the language with new abstractions. We comment on the challenges we encountered and the solutions we developed for our current implementation and outline suggestions for an improved macro system to support the such use cases of embedding of domain-specific abstractions.

Fault-tolerant functional reactive programming (extended version)

Abstract Highly critical application domains, like medicine and aerospace, require the use of strict design, implementation, and validation techniques. Functional languages have been used in these domains to develop synchronous dataflow programming languages for reactive systems. Causal stream functions and functional reactive programming (FRP) capture the essence of those languages in a way that is both elegant and robust. To guarantee that critical systems can operate under high stress over long periods of time, these applications require clear specifications of possible faults and hazards, and how they are being handled. Modeling failure is straightforward in functional languages, and many functional reactive abstractions incorporate support for failure or termination. However, handling unknown types of faults, and incorporating fault tolerance into FRP, requires a different construction and remains an open problem. This work demonstrates how to extend an existing functional reactive framework with fault tolerance features. At value level, we tag faulty signals with reliability and probability information and use random testing to inject faults and validate system properties encoded in temporal logic. At type level, we tag components with the kinds of faults they may exhibit and use type-level programming to obtain compile-time guarantees of key aspects of fault tolerance. Our approach is powerful enough to be used in systems with realistic complexity, and flexible enough to be used to guide system analysis and design, validate system properties in the presence of faults, perform runtime monitoring, and study the effects of different fault tolerance mechanisms.

Effekt: Capability-passing style for type- and effect-safe, extensible effect handlers in Scala

AbstractEffect handlers are a promising way to structure effectful programs in a modular way. We present the Scala libraryEffekt, which is centered around capability passing and implemented in terms of a monad for multi-prompt delimited continuations.Effektis the first library implementation of effect handlers that supports effect safety and effect polymorphism without resorting to type-level programming. We describe a novel way of achieving effect safety using intersection types and path-dependent types. The effect system of our library design fits well into the programming paradigm of capability passing and is inspired by the effect system of Zhang & Myers (2019,Proc. ACM Program. Lang.3(POPL), 5:1-5:29). Capabilities carry an abstract type member, which represents an individual effect type and reflects the use of the capability on the type level. We represent effect rows as the contravariant intersection of effect types. Handlers introduce capabilities and remove components of the intersection type. Reusing the existing type system of Scala, we get effect subtyping and effect polymorphism for free.

Higher-order type-level programming in Haskell

Type family applications in Haskell must be fully saturated. This means that all type-level functions have to be first-order, leading to code that is both messy and longwinded. In this paper we detail an extension to GHC that removes this restriction. We augment Haskell’s existing type arrow, |->|, with an unmatchable arrow, | >|, that supports partial application of type families without compromising soundness. A soundness proof is provided. We show how the techniques described can lead to substantial code-size reduction (circa 80%) in the type-level logic of commonly-used type-level libraries whilst simultaneously improving code quality and readability.

Fault tolerant functional reactive programming (functional pearl)

Highly critical application domains, like medicine and aerospace, require the use of strict design, implementation and validation techniques. Functional languages have been used in these domains to develop synchronous dataflow programming languages for reactive systems. Causal stream functions and Functional Reactive Programming capture the essence of those languages in a way that is both elegant and robust. To guarantee that critical systems can operate under high stress over long periods of time, these applications require clear specifications of possible faults and hazards, and how they are being handled. Modeling failure is straightforward in functional languages, and many Functional Reactive abstractions incorporate support for failure or termination. However, handling unknown types of faults , and incorporating fault tolerance into Functional Reactive Programming, requires a different construction and remains an open problem. This work presents extensions to an existing functional reactive abstraction to facilitate tagging reactive transformations with hazard tags or confidence levels. We present a prototype framework to quantify the reliability of a reactive construction, by means of numeric factors or probability distributions, and demonstrate how to aid the design of fault-tolerant systems, by constraining the allowed reliability to required boundaries. By applying type-level programming, we show that it is possible to improve static analysis and have compile-time guarantees of key aspects of fault tolerance. Our approach is powerful enough to be used in systems with realistic complexity, and flexible enough to be used to guide their analysis and design, to test system properties, to verify fault tolerance properties, to perform runtime monitoring, to implement fault tolerance during execution and to address faults during runtime. We present implementations in Haskell and in Idris.

A Simple and Practical Linear Algebra Library Interface with Static Size Checking

Linear algebra is a major field of numerical computation and is widely applied. Most linear algebra libraries (in most programming languages) do not statically guarantee consistency of the dimensions of vectors and matrices, causing runtime errors. While advanced type systems--specifically, dependent types on natural numbers--can ensure consistency among the sizes of collections such as lists and arrays, such type systems generally require non-trivial changes to existing languages and application programs, or tricky type-level programming. We have developed a linear algebra library interface that verifies the consistency (with respect to dimensions) of matrix operations by means of generative phantom types, implemented via fairly standard ML types and module system. To evaluate its usability, we ported to it a practical machine learning library from a traditional linear algebra library. We found that most of the changes required for the porting could be made mechanically, and changes that needed human thought are minor.

Injective type families for Haskell

Haskell, as implemented by the Glasgow Haskell Compiler (GHC), allows expressive type-level programming. The most popular type-level programming extension is TypeFamilies, which allows users to write functions on types. Yet, using type functions can cripple type inference in certain situations. In particular, lack of injectivity in type functions means that GHC can never infer an instantiation of a type variable appearing only under type functions. In this paper, we describe a small modification to GHC that allows type functions to be annotated as injective. GHC naturally must check validity of the injectivity annotations. The algorithm to do so is surprisingly subtle. We prove soundness for a simplification of our algorithm, and state and prove a completeness property, though the algorithm is not fully complete. As much of our reasoning surrounds functions defined by a simple pattern-matching structure, we believe our results extend beyond just Haskell. We have implemented our solution on a branch of GHC and plan to make it available to regular users with the next stable release of the compiler.

Type families with class, type classes with family

Type classes and type families are key ingredients in Haskell programming. Type classes were introduced to deal with ad-hoc polymorphism, although with the introduction of functional dependencies, their use expanded to type-level programming. Type families also allow encoding type-level functions, but more directly in the form of rewrite rules. In this paper we show that type families are powerful enough to simulate type classes (without overlapping instances), and we provide a formal proof of the soundness and completeness of this simulation. Encoding instance constraints as type families eases the path to proposed extensions to type classes, like closed sets of instances, instance chains, and control over the search procedure. The only feature which type families cannot simulate is elaboration, that is, generating code from the derivation of a rewriting. We look at ways to solve this problem in current Haskell, and propose an extension to allow elaboration during the rewriting phase.

Promoting functions to type families in Haskell

Haskell, as implemented in the Glasgow Haskell Compiler (GHC), is enriched with many extensions that support type-level programming, such as promoted datatypes, kind polymorphism, and type families. Yet, the expressiveness of the type-level language remains limited. It is missing many features present at the term level, including case expressions, anonymous functions, partially-applied functions, and let expressions. In this paper, we present an algorithm - with a proof of correctness - to encode these term-level constructs at the type level. Our approach is automated and capable of promoting a wide array of functions to type families. We also highlight and discuss those term-level features that are not promotable. In so doing, we offer a critique on GHC's existing type system, showing what it is already capable of and where it may want improvement. We believe that delineating the mismatch between GHC's term level and its type level is a key step toward supporting dependently typed programming.

An EDSL approach to high performance Haskell programming

This paper argues for a new methodology for writing high performance Haskell programs by using Embedded Domain Specific Languages. We exemplify the methodology by describing a complete library, meta-repa, which is a reimplementation of parts of the repa library. The paper describes the implementation of meta-repa and contrasts it with the standard approach to writing high performance libraries. We conclude that even though the embedded language approach has an initial cost of defining the language and some syntactic overhead it gives a more tailored programming model, stronger performance guarantees, better control over optimizations, simpler implementation of fusion and inlining and allows for moving type level programming down to value level programming in some cases. We also provide benchmarks showing that meta-repa is as fast, or faster, than repa. Furthermore, meta-repa also includes push arrays and we demonstrate their usefulness for writing certain high performance kernels such as FFT.

Sneaking around concatMap

We present a framework of dynamic programming combinators that provides a high-level environment to describe the recursions typical of dynamic programming over sequence data in a style very similar to algebraic dynamic programming (ADP). Using a combination of type-level programming and stream fusion leads to a substantial increase in performance, without sacrificing much of the convenience and theoretical underpinnings of ADP. We draw examples from the field of computational biology, more specifically RNA secondary structure prediction, to demonstrate how to use these combinators and what differences exist between this library, ADP, and other approaches. The final version of the combinator library allows writing algorithms with performance close to hand-optimized C code.

Attribute grammars fly first-class

Attribute Grammars (AGs), a general-purpose formalism for describing recursive computations over data types, avoid the trade-off which arises when building software incrementally: should it be easy to add new data types and data type alternatives or to add new operations on existing data types? However, AGs are usually implemented as a pre-processor, leaving e.g. type checking to later processing phases and making interactive development, proper error reporting and debugging difficult. Embedding AG into Haskell as a combinator library solves these problems. Previous attempts at embedding AGs as a domain-specific language were based on extensible records and thus exploiting Haskell's type system to check the well formedness of the AG, but fell short in compactness and the possibility to abstract over oft occurring AG patterns. Other attempts used a very generic mapping for which the AG well-formedness could not be statically checked. We present a typed embedding of AG in Haskell satisfying all these requirements. The key lies in using HList-like typed heterogeneous collections (extensible polymorphic records) and expressing AG well-formedness conditions as type-level predicates (i.e., type-class constraints). By further type-level programming we can also express common programming patterns, corresponding to the typical use cases of monads such as Reader, Writer and State. The paper presents a realistic example of type-class-based type-level programming in Haskell.