Hindley-Milner Research Articles

With or Without You: Programming with Effect Exclusion

Type and effect systems have been successfully used to statically reason about effects in many different domains, including region-based memory management, exceptions, and algebraic effects and handlers. Such systems’ soundness is often stated in terms of the absence of effects. Yet, existing systems only admit indirect reasoning about the absence of effects. This is further complicated by effect polymorphism which allows function signatures to abstract over arbitrary, unknown sets of effects. We present a new type and effect system with effect polymorphism as well as union, intersection, and complement effects. The effect system allows us to express effect exclusion as a new class of effect polymorphic functions: those that permit any effects except those in a specific set. This way, we equip programmers with the means to directly reason about the absence of effects. Our type and effect system builds on the Hindley-Milner type system, supports effect polymorphism, and preserves principal types modulo Boolean equivalence. In addition, a suitable extension of Algorithm W with Boolean unification on the algebra of sets enables complete type and effect inference. We formalize these notions in the λ ∁ calculus. We prove the standard progress and preservation theorems as well as a non-standard effect safety theorem: no excluded effect is ever performed. We implement the type and effect system as an extension of the Flix programming language. We conduct a case study of open source projects identifying 59 program fragments that require effect exclusion for correctness. To demonstrate the usefulness of the proposed type and effect system, we recast these program fragments into our extension of Flix.

PureCake: A Verified Compiler for a Lazy Functional Language

We present PureCake, a mechanically-verified compiler for PureLang, a lazy, purely functional programming language with monadic effects. PureLang syntax is Haskell-like and indentation-sensitive, and its constraint-based Hindley-Milner type system guarantees safe execution. We derive sound equational reasoning principles over its operational semantics, dramatically simplifying some proofs. We prove end-to-end correctness for the compilation of PureLang down to machine code---the first such result for any lazy language---by targeting CakeML and composing with its verified compiler. Multiple optimisation passes are necessary to handle realistic lazy idioms effectively. We develop PureCake entirely within the HOL4 interactive theorem prover.

Fluent APIs in Functional Languages

Fluent API is an object-oriented pattern for elegant APIs and embedded DSLs. A smart fluent API can enforce the API protocol or DSL syntax at compile time. Since fluent API implementations typically rely on overloading function names, they are hard to realize in functional programming languages. This work shows how functional fluent APIs can be implemented in the absence of name overloading, by relying on parametric polymorphism and Hindley-Milner type inference. The implementation supports fluent API protocols in the regular- and deterministic context-free language classes, and even beyond.

Constraint-based type inference for FreezeML

FreezeML is a new approach to first-class polymorphic type inference that employs term annotations to control when and how polymorphic types are instantiated and generalised. It conservatively extends Hindley-Milner type inference and was first presented as an extension to Algorithm W. More modern type inference techniques such as HM(X) and OutsideIn(X) employ constraints to support features such as type classes, type families, rows, and other extensions. We take the first step towards modernising FreezeML by presenting a constraint-based type inference algorithm. We introduce a new constraint language, inspired by the Pottier/Rémy presentation of HM(X), in order to allow FreezeML type inference problems to be expressed as constraints. We present a deterministic stack machine for solving FreezeML constraints and prove its termination and correctness.

Static prediction of parallel computation graphs

Many algorithms for analyzing parallel programs, for example to detect deadlocks or data races or to calculate the execution cost, are based on a model variously known as a cost graph, computation graph or dependency graph, which captures the parallel structure of threads in a program. In modern parallel programs, computation graphs are highly dynamic and depend greatly on the program inputs and execution details. As such, most analyses that use these graphs are either dynamic analyses or are specialized static analyses that gather a subset of dependency information for a specific purpose. This paper introduces graph types, which compactly represent all of the graphs that could arise from program execution. Graph types are inferred from a parallel program using a graph type system and inference algorithm, which we present drawing on ideas from Hindley-Milner type inference, affine logic and region type systems. We have implemented the inference algorithm over a subset of OCaml, extended with parallelism primitives, and we demonstrate how graph types can be used to accelerate the development of new graph-based static analyses by presenting proof-of-concept analyses for deadlock detection and cost analysis.

Relational nullable types with Boolean unification

We present a simple, practical, and expressive relational nullable type system. A relational nullable type system captures whether an expression may evaluate to null based on its type, but also based on the type of other related expressions. The type system extends the Hindley-Milner type system with Boolean constraints, supports parametric polymorphism, and preserves principal types modulo Boolean equivalence. We show how to support full Hindley-Milner style type inference with an extension of Algorithm W. We conduct a preliminary study of open source projects showing that there is a need for relational nullable type systems across a wide range of programming languages. The most important findings from the study are: (i) programmers use programming patterns where the nullability of one expression depends on the nullability of other related expressions, (ii) such invariants are commonly enforced with run-time exceptions, and (iii) reasoning about these programming patterns requires not only knowledge of when an expression may evaluate to null, but also when it may evaluate to a non-null value. We incorporate these observations in the design of the proposed relational nullable type system.

Polymorphic types and effects with Boolean unification

We present a simple, practical, and expressive type and effect system based on Boolean constraints. The effect system extends the Hindley-Milner type system, supports parametric polymorphism, and preserves principal types modulo Boolean equivalence. We show how to support type inference by extending Algorithm W with Boolean unification based on the successive variable elimination algorithm. We implement the type and effect system in the Flix programming language. We perform an in-depth evaluation on the impact of Boolean unification on type inference time and end-to-end compilation time. While the computational complexity of Boolean unification is NP-hard, the experimental results demonstrate that it works well in practice. We find that the impact on type inference time is on average a 1.4x slowdown and the overall impact on end-to-end compilation time is a 1.1x slowdown.

The simple essence of algebraic subtyping: principal type inference with subtyping made easy (functional pearl)

MLsub extends traditional Hindley-Milner type inference with subtyping while preserving compact principal types, an exciting new development. However, its specification in terms of biunification is difficult to understand, relying on the new concepts of bisubstitution and polar types, and making use of advanced notions from abstract algebra. In this paper, we show that these are in fact not essential to understanding the mechanisms at play in MLsub. We propose an alternative algorithm called Simple-sub, which can be implemented efficiently in under 500 lines of code (including parsing, simplification, and pretty-printing), looks more familiar, and is easier to understand. We present an experimental evaluation of Simple-sub against MLsub on a million randomly-generated well-scoped expressions, showing that the two systems agree. The mutable automaton-based implementation of MLsub is quite far from its algebraic specification, leaving a lot of space for errors; in fact, our evaluation uncovered several bugs in it. We sketch more straightforward soundness and completeness arguments for Simple-sub, based on a syntactic specification of the type system. This paper is meant to be light in formalism, rich in insights, and easy to consume for prospective designers of new type systems and programming languages. In particular, no abstract algebra is inflicted on readers.

Chemoinformatics and structural bioinformatics in OCaml

BackgroundOCaml is a functional programming language with strong static types, Hindley–Milner type inference and garbage collection. In this article, we share our experience in prototyping chemoinformatics and structural bioinformatics software in OCaml.ResultsFirst, we introduce the language, list entry points for chemoinformaticians who would be interested in OCaml and give code examples. Then, we list some scientific open source software written in OCaml. We also present recent open source libraries useful in chemoinformatics. The parallelization of OCaml programs and their performance is also shown. Finally, tools and methods useful when prototyping scientific software in OCaml are given.ConclusionsIn our experience, OCaml is a programming language of choice for method development in chemoinformatics and structural bioinformatics.

Proof-relevant Horn Clauses for Dependent Type Inference and Term Synthesis

AbstractFirst-order resolution has been used for type inference for many years, including in Hindley-Milner type inference, type-classes, and constrained data types. Dependent types are a new trend in functional languages. In this paper, we show that proof-relevant first-order resolution can play an important role in automating type inference and term synthesis for dependently typed languages. We propose a calculus that translates type inference and term synthesis problems in a dependently typed language to a logic program and a goal in the proof-relevant first-order Horn clause logic. The computed answer substitution and proof term then provide a solution to the given type inference and term synthesis problem. We prove the decidability and soundness of our method.

Преобразование типизированных функций в реляционную форму

Relational programming is an approach that allows you to execute programs in different directions to get different behaviors from one relational specification. The direct development of relational programs is a complex task, requiring the developer to have certain skills. However, in many cases, the required relational program can be obtained from a certain functional program automatically. In this paper, the problem of automatic conversion of functional programs into relational ones is considered. The main contribution of the paper is the method of converting typed functions into a relational form, as well as proving its static and dynamic correctness. This method can be applied to typed programs of a general kind. To describe these programs, a compact ML-like language (a subset of OCaml) is used, equipped with a Hindley-Milner type system with let-polymorphism Also, the paper discusses the limitations of the proposed method, presents an implementation for a subset of the OCaml language, and evaluates the method on a number of realistic examples.

Generating Code with Polymorphic let: A Ballad of Value Restriction, Copying and Sharing

Getting polymorphism and effects such as mutation to live together in the same language is a tale worth telling, under the recurring refrain of copying vs. sharing. We add new stanzas to the tale, about the ordeal to generate code with polymorphism and effects, and be sure it type-checks. Generating well-typed-by-construction polymorphic let-expressions is impossible in the Hindley-Milner type system: even the author believed that. The polymorphic-let generator turns out to exist. We present its derivation and the application for the lightweight implementation of quotation via a novel and unexpectedly simple source-to-source transformation to code-generating combinators. However, generating let-expressions with polymorphic functions demands more than even the relaxed value restriction can deliver. We need a new deal for let-polymorphism in ML. We conjecture the weaker restriction and implement it in a practically-useful code-generation library. Its formal justification is formulated as the research program.

Set-theoretic types for polymorphic variants

Polymorphic variants are a useful feature of the OCaml language whose current definition and implementation rely on kinding constraints to simulate a subtyping relation via unification. This yields an awkward formalization and results in a type system whose behaviour is in some cases unintuitive and/or unduly restrictive. In this work, we present an alternative formalization of polymorphic variants, based on set-theoretic types and subtyping, that yields a cleaner and more streamlined system. Our formalization is more expressive than the current one (it types more programs while preserving type safety), it can internalize some meta-theoretic properties, and it removes some pathological cases of the current implementation resulting in a more intuitive and, thus, predictable type system. More generally, this work shows how to add full-fledged union types to functional languages of the ML family that usually rely on the Hindley-Milner type system. As an aside, our system also improves the theory of semantic subtyping, notably by proving completeness for the type reconstruction algorithm.

Finding minimum type error sources

Automatic type inference is a popular feature of functional programming languages. If a program cannot be typed, the compiler typically reports a single program location in its error message. This location is the point where the type inference failed, but not necessarily the actual source of the error. Other potential error sources are not even considered. Hence, the compiler often misses the true error source, which increases debugging time for the programmer. In this paper, we present a general framework for automatic localization of type errors. Our algorithm finds all minimum error sources, where the exact definition of minimum is given in terms of a compiler-specific ranking criterion. Compilers can use minimum error sources to produce more meaningful error reports, and for automatic error correction. Our approach works by reducing the search for minimum error sources to an optimization problem that we formulate in terms of weighted maximum satisfiability modulo theories (MaxSMT). The reduction to weighted MaxSMT allows us to build on SMT solvers to support rich type systems and at the same time abstract from the concrete criterion that is used for ranking the error sources. We have implemented an instance of our framework targeted at Hindley-Milner type systems and evaluated it on existing OCaml benchmarks for type error localization. Our evaluation shows that our approach has the potential to significantly improve the quality of type error reports produced by state of the art compilers.

Hindley-milner elaboration in applicative style

Type inference - the problem of determining whether a program is well-typed - is well-understood. In contrast, elaboration - the task of constructing an explicitly-typed representation of the program - seems to have received relatively little attention, even though, in a non-local type inference system, it is non-trivial. We show that the constraint-based presentation of Hindley-Milner type inference can be extended to deal with elaboration, while preserving its elegance. This involves introducing a new notion of "constraint with a value", which forms an applicative functor.

Programming with binders and indexed data-types

We show how to combine a general purpose type system for an existing language with support for programming with binders and contexts by refining the type system of ML with a restricted form of dependent types where index objects are drawn from contextual LF. This allows the user to specify formal systems within the logical framework LF and index ML types with contextual LF objects. Our language design keeps the index language generic only requiring decidability of equality of the index language providing a modular design. To illustrate the elegance and effectiveness of our language, we give programs for closure conversion and normalization by evaluation. Our three key technical contribution are: 1) We give a bi-directional type system for our core language which is centered around refinement substitutions instead of constraint solving. As a consequence, type checking is decidable and easy to trust, although constraint solving may be undecidable. 2) We give a big-step environment based operational semantics with environments which lends itself to efficient implementation. 3) We prove our language to be type safe and have mechanized our theoretical development in the proof assistant Coq using the fresh approach to binding.

HMF

HMF is a conservative extension of Hindley-Milner type inference with first-class polymorphism. In contrast to other proposals, HML uses regular System F types and has a simple type inference algorithm that is just a small extension of the usual Damas-Milner algorithm W. Given the relative simplicity and expressive power, we feel that HMF can be an attractive type system in practice. There is a reference implementation of the type system available online together with a technical report containing proofs (Leijen 2007a,b).

A type directed translation of MLF to system F

The MLF type system by Le Botlan and Rémy is a natural extension of Hindley-Milner type inference that supports full first-class polymorphism, where types can be of higher-rank and impredicatively instantiated. Even though MLF is theoretically very attractive, it has not seen widespread adoption. We believe that this partly because it is unclear how the rich language of MLF types relate to standard System F types. In this article we give the first type directed translation of MLF terms to System F terms. Based on insight gained from this translation, we also define "Rigid MLF" (MLF = ), a restriction of MLF where all bound values have a System F type. The expressiveness of MLF = is the same as that of boxy types, but MLF = needs fewer annotations and we give a detailed comparison between them.

Dependent ML An approach to practical programming with dependent types

AbstractWe present an approach to enriching the type system of ML with a restricted form of dependent types, where type index terms are required to be drawn from a given type index language${\cal L}$that is completely separate from run-time programs, leading to the DML(${\cal L}$) language schema. This enrichment allows for specification and inference of significantly more precise type information, facilitating program error detection and compiler optimization. The primary contribution of the paper lies in our language design, which can effectively support the use of dependent types in practical programming. In particular, this design makes it both natural and straightforward to accommodate dependent types in the presence of effects such as references and exceptions.

Phantom types and subtyping

We investigate a technique from the literature, called the phantom-types technique, that uses parametric polymorphism, type constraints, and unification of polymorphic types to model a subtyping hierarchy. Hindley-Milner type systems, such as the one found in Standard ML, can be used to enforce the subtyping relation, at least for first-order values. We show that this technique can be used to encode any finite subtyping hierarchy (including hierarchies arising from multiple interface inheritance). We formally demonstrate the suitability of the phantom-types technique for capturing first-order subtyping by exhibiting a type-preserving translation from a simple calculus with bounded polymorphism to a calculus embodying the type system of SML.