In his seminal paper “A Natural Semantics for Lazy Evaluation”, John Launchbury proves his semantics correct with respect to a denotational semantics, and outlines a proof of adequacy. Previous attempts to rigorize the adequacy proof, which involves an intermediate natural semantics and an intermediate resourced denotational semantics, have failed. We devised a new, direct proof that skips the intermediate natural semantics. It is the first rigorous adequacy proof of Launchbury's semantics. We have modeled our semantics in the interactive theorem prover Isabelle and machine-checked our proofs. This does not only provide a maximum level of rigor, but also serves as a tool for further work, such as a machine-checked correctness proof of a compiler transformation.

When type inference fails, it is often difficult to pinpoint the cause of the type error among many potential candidates. Generating informative messages to remove the type error is another difficult task due to the limited availability of type information. Over the last three decades many approaches have been developed to help debug type errors. However, most of these methods suffer from one or more of the following problems: (1) Being incomplete, they miss the real cause. (2) They cover many potential causes without distinguishing them. (3) They provide little or no information for how to remove the type error. Any one of this problems can turn the type-error debugging process into a tedious and ineffective endeavor. To address this issue, we have developed a method named counter-factual typing, which (1) finds a comprehensive set of error causes in AST leaves, (2) computes an informative message on how to get rid of the type error for each error cause, and (3) ranks all messages and iteratively presents the message for the most likely error cause. The biggest technical challenge is the efficient generation of all error messages, which seems to be exponential in the size of the expression. We address this challenge by employing the idea of variational typing that systematically reuses computations for shared parts and generates all messages by typing the whole ill-typed expression only once. We have evaluated our approach over a large set of examples collected from previous publications in the literature. The evaluation result shows that our approach outperforms previous approaches and is computationally feasible.

We present a model of computation that heavily emphasizes the concept of duality and the interaction between opposites–production interacts with consumption. The symmetry of this framework naturally explains more complicated features of programming languages through relatively familiar concepts. For example, binding a value to a variable is dual to manipulating the flow of control in a program. By looking at the computational interpretation of the sequent calculus, we find a language that lets us speak about duality, control flow, and evaluation order in programs as first-class concepts.

We begin by reviewing Gentzen's LK sequent calculus and show how the Curry–Howard isomorphism still applies to give us a different basis for expressing computation. We then illustrate how the fundamental dilemma of computation in the sequent calculus gives rise to a duality between evaluation strategies: strict languages are dual to lazy languages. Finally, we discuss how the concept of focusing, developed in the setting of proof search, is related to the idea of type safety for computation expressed in the sequent calculus. In this regard, we compare and contrast two different methods of focusing that have appeared in the literature, static and dynamic focusing, and illustrate how they are two means to the same end.

Contract systems have come to play a vital role in many aspects of software engineering. This has resulted in a wide variety of approaches to enforcing contracts—ranging from the straightforward pre-condition and post-condition checking of Eiffel to lazy, optional, and parallel enforcement strategies. Each of these approaches has its merits, but each has required ground-up development of an entire contract monitoring system. We present a unified approach to understanding this variety, while also opening the door to as-yet-undiscovered strategies. By observing that contracts are fundamentally about communication between a program and a monitor, we reframe contract checking as communication between concurrent processes. This brings out the underlying relations between widely studied enforcement strategies, including strict and lazy enforcement as well as concurrent approaches, including new contracts and strategies. We show how each of these can be embedded into a core calculus, and demonstrate a proof (via simulation) of correctness for one such encoding. Finally, we show that our approach suggests new monitoring approaches and contracts not previously expressible.

A Computer Science introduction course ought to focus on exciting students about the subject matter and on problem solving through the methodical design of programs. An effective way to achieve both is through the development of functional video games. As most students are interested in video games, their development adds an exciting domain to any introduction to programming. This article advocates that an exciting crowning achievement for students in such a course is the design and implementation of a multiplayer distributed video game. By exploiting a domain that is popular with students, they are taught about design principles, communication protocols, and pitfalls in distributed programming. This article puts forth a successfully used design recipe that places distributed programming well within the reach of beginning students and outlines the use of this design recipe in the classroom. For those teaching beginners, this article presents a model for developing their own distributed programming module. The success of the presented methodology is measured through student feedback on their distributed programming experience. The empirical results suggest that the design and implementation of distributed functional video games is effective and well-received by students. Furthermore, the data suggests that the presented methodology fails to exhibit the gender gap common in Computer Science and is effective regardless of the programming experience CS1 students bring to the classroom.

Functional languages are uniquely suited to providing programmers with a programming model for parallel and concurrent computing. This is reflected in the wide range of work that is currently underway, both on parallel and concurrent functional languages, as well as on bringing functional language features to other programming languages. This has resulted in a rapidly growing number of practical applications. The Journal of Functional Programming decided to dedicate a special issue to this field to showcase the state of the art in how functional languages and functional concepts currently assist programmers with the task of managing the challenges of creating parallel and concurrent systems.

The most successful systems for “big data” processing have all adopted functional APIs. We present a new programming model, we call function passing, designed to provide a more principled substrate, or middleware, upon which to build data-centric distributed systems like Spark. A key idea is to build up a persistent functional data structure representing transformations on distributed immutable data by passing well-typed serializable functions over the wire and applying them to this distributed data. Thus, the function passing model can be thought of as a persistent functional data structure that is distributed, where transformations performed on distributed data are stored in its nodes rather than the distributed data itself. One advantage of this model is that failure recovery is simplified by design – data can be recovered by replaying function applications atop immutable data loaded from stable storage. Deferred evaluation is also central to our model; by incorporating deferred evaluation into our design only at the point of initiating network communication, the function passing model remains easy to reason about while remaining efficient in time and memory. Moreover, we provide a complete formalization of the programming model in order to study the foundations of lineage-based distributed computation. In particular, we develop a theory of safe, mobile lineages based on a subject reduction theorem for a typed core language. Furthermore, we formalize a progress theorem that guarantees the finite materialization of remote, lineage-based data. Thus, the formal model may serve as a basis for further developments of the theory of data-centric distributed programming, including aspects such as fault tolerance. We provide an open-source implementation of our model in and for the Scala programming language, along with a case study of several example frameworks and end-user programs written atop this model.

Ideas from programming languages play an important role in a range of advanced applications of databases, in database system implementation, distributed programming (MapReduce), streaming computation, and high-performance (GPU/multicore) computation. This creative research area is broadening into a subfield of data-centric computation. Although the interaction of databases and programming has a long history (the 16th biennial Database Programming Languages symposium was held in 2017), there has been a recent renewal of interest and broadening of programming language techniques for dealing with data from several quarters in the last few years, including workshops at Microsoft Research (RADICAL 2010), ICFP (XLDI 2012), POPL (DDFP 2013, DCM 2014) and a Dagstuhl Seminar on Programming Languages for Big Data (December 2014). This special issue recognises and encourages the publication of mature research contributions in this area.

Full-spectrum dependent types promise to enable the development of correct-by-construction software. However, even certified software needs to interact with simply-typed or untyped programs, be it to perform system calls, or to use legacy libraries. Trading static guarantees for runtime checks, the dependent interoperability framework provides a mechanism by which simply-typed values can safely be coerced to dependent types and, conversely, dependently-typed programs can defensively be exported to a simply-typed application. In this article, we give a semantic account of dependent interoperability. Our presentation relies on and is guided by a pervading notion of type equivalence, whose importance has been emphasized in recent work on homotopy type theory. Specifically, we develop the notions of type-theoretic partial Galois connections as a key foundation for dependent interoperability, which accounts for the partiality of the coercions between types. We explore the applicability of both type-theoretic Galois connections and anticonnections in the setting of dependent interoperability. A partial Galois connection enforces a translation of dependent types to runtime checks that are both sound and complete with respect to the invariants encoded by dependent types. Conversely, picking an anticonnection instead lets us induce weaker, sound conditions that can amount to more efficient runtime checks.

Our framework is developed in Coq; it is thus constructive and verified in the strictest sense of the terms. Using our library, users can specify domain-specific partial connections between data structures. Our library then takes care of the (sometimes, heavy) lifting that leads to interoperable programs. It thus becomes possible, as we shall illustrate, to internalize and hand-tune the extraction of dependently-typed programs to interoperable OCaml programs within Coq itself.

Database query engines use pull-based or push-based approaches to avoid the materialization of data across query operators. In this paper, we study these two types of query engines in depth and present the limitations and advantages of each engine. Similarly, the programming languages community has developed loop fusion techniques to remove intermediate collections in the context of collection programming. We draw parallels between databases (DB) and programming language (PL) research by demonstrating the connection between pipelined query engines and loop fusion techniques. Based on this connection, we propose a new type of pull-based engine, inspired by a loop fusion technique, which combines the benefits of both approaches. Then, we experimentally evaluate the various engines, in the context of query compilation, for the first time in a fair environment, eliminating the biasing impact of ancillary optimizations that have traditionally only been used with one of the approaches. We show that for realistic analytical workloads, there is no considerable advantage for either form of pipelined query engine, as opposed to what recent research suggests. Also, by using micro-benchmarks, which demonstrate certain edge cases on which one approach or the other performs better, we show that our proposed engine dominates the existing engines by combining the benefits of both.

We provide a general and modular criterion for the termination of simply typed λ-calculus extended with function symbols defined by user-defined rewrite rules. Following a work of Hughes, Pareto and Sabry for functions defined with a fixpoint operator and pattern matching, several criteria use typing rules for bounding the height of arguments in function calls. In this paper, we extend this approach to rewriting-based function definitions and more general user-defined notions of size.

Dependently typed languages such as Agda, Coq, and Idris use a syntactic first-order unification algorithm to check definitions by dependent pattern matching. However, standard unification algorithms implicitly rely on principles such as uniqueness of identity proofs and injectivity of type constructors. These principles are inadmissible in many type theories, particularly in the new and promising branch known as homotopy type theory. As a result, programs and proofs in these new theories cannot make use of dependent pattern matching or other techniques relying on unification, and are as a result much harder to write, modify, and understand. This paper proposes a proof-relevant framework for reasoning formally about unification in a dependently typed setting. In this framework, unification rules compute not just a unifier but also a corresponding soundness proof in the form of an equivalence between two sets of equations. By rephrasing the standard unification rules in a proof-relevant manner, they are guaranteed to preserve soundness of the theory. In addition, it enables us to safely add new rules that can exploit the dependencies between the types of equations, such as rules for eta-equality of record types and higher dimensional unification rules for solving equations between equality proofs. Using our framework, we implemented a complete overhaul of the unification algorithm used by Agda. As a result, we were able to replace previous ad-hoc restrictions with formally verified unification rules, fixing a substantial number of bugs in the process. In the future, we may also want to integrate new principles with pattern matching, for example, the higher inductive types introduced by homotopy type theory. Our framework also provides a solid basis for such extensions to be built on.

Static type errors are a common stumbling block for newcomers to typed functional languages. We present a dynamic approach to explaining type errors by generating counterexample witness inputs that illustrate how an ill-typed program goes wrong. First, given an ill-typed function, we symbolically execute the body to synthesize witness values that make the program go wrong. We prove that our procedure synthesizes general witnesses in that if a witness is found, then for all inhabited input types, there exist values that can make the function go wrong. Second, we show how to extend this procedure to produce a reduction graph that can be used to interactively visualize and debug witness executions. Third, we evaluate the coverage of our approach on two data sets comprising over 4,500 ill-typed student programs. Our technique is able to generate witnesses for around 85% of the programs, our reduction graph yields small counterexamples for over 80% of the witnesses, and a simple heuristic allows us to use witnesses to locate the source of type errors with around 70% accuracy. Finally, we evaluate whether our witnesses help students understand and fix type errors, and find that students presented with our witnesses show a greater understanding of type errors than those presented with a standard error message.

Comparison networks offer an attractive framework for studying parallel sorting algorithms. The visual presentation of such hard-wired networks makes processes easy to conceptualize without the overhead of programming syntax or the semantic complications of concurrency. Though simple in appearance they have a surprisingly rich structure. In this pearl we delve a little into their theory.

A bidirectional transformation is a pair of mappings between source and view data objects, one in each direction. When the view is modified, the source is updated accordingly with respect to some laws. One way to reduce the development and maintenance effort of bidirectional transformations is to have specialized languages in which the resulting programs are bidirectional by construction—giving rise to the paradigm of bidirectional programming. In this paper, we develop a framework for applicative-style and higher-order bidirectional programming, in which we can write bidirectional transformations as unidirectional programs in standard functional languages, opening up access to the bundle of language features previously only available to conventional unidirectional languages. Our framework essentially bridges two very different approaches of bidirectional programming, namely the lens framework and Voigtländer's semantic bidirectionalization, creating a new programming style that is able to obtain benefits from both.

Generalized Algebraic Data Types, or simply GADTs, can encode non-trivial properties in the types of the constructors. Once such properties are encoded in a datatype, however, all code manipulating that datatype must provide proof that it maintains these properties in order to typecheck. In this paper, we take a step toward gradualizing these obligations. We introduce a tool, Ghostbuster, that produces simplified versions of GADTs which elide selected type parameters, thereby weakening the guarantees of the simplified datatype in exchange for reducing the obligations necessary to manipulate it. Like ornaments, these simplified datatypes preserve the recursive structure of the original, but unlike ornaments, we focus on information-preserving bidirectional transformations. Ghostbuster generates type-safe conversion functions between the original and simplified datatypes, which we prove are the identity function when composed. We evaluate a prototype tool for Haskell against thousands of GADTs found on the Hackage package database, generating simpler Haskell'98 datatypes and round-trip conversion functions between the two.

Many students complete PhDs in functional programming each year. As a service to the community, the Journal of Functional Programming publishes the abstracts from PhD dissertations completed during the previous year.

In higher order settings, control-flow analysis aims to model the propagation of both data and control by finitely approximating program behaviors across all possible executions. The polyvariance of an analysis describes the number of distinct abstract representations, or variants, for each syntactic entity (e.g., functions, variables, or intermediate expressions). Monovariance, one of the most basic forms of polyvariance, maintains only a single abstract representation for each variable or expression. Other polyvariant strategies allow a greater number of distinct abstractions and increase analysis complexity with the aim of increasing analysis precision. For example, k-call sensitivity distinguishes flows by the most recent k call sites, k-object sensitivity by a history of allocation points, and argument sensitivity by a tuple of dynamic argument types. From this perspective, even a concrete operational semantics may be thought of as an unboundedly polyvariant analysis. In this paper, we develop a unified methodology that fully captures this design space. It is easily tunable and guarantees soundness regardless of how tuned. We accomplish this by extending the method of abstracting abstract machines, a systematic approach to abstract interpretation of operational abstract-machine semantics. Our approach permits arbitrary instrumentation of the underlying analysis and arbitrary tuning of an abstract-allocation function. We show that the design space of abstract allocators both unifies and generalizes existing notions of polyvariance. Simple changes to the behavior of this function recapitulate classic styles of analysis and yield novel combinations and variants.

We consider the task of constructing 2-3 trees. Given a sequence of elements we seek to build a 2-3 tree–in linear time–that contains the elements in symmetric order. We discuss three approaches: top-down, bottom-up, and incremental. The incremental approach is more flexible than the other two in that it allows us to interleave the construction work with other operations, for example, queries.

Iris is a framework for higher-order concurrent separation logic, which has been implemented in the Coq proof assistant and deployed very effectively in a wide variety of verification projects. Iris was designed with the express goal of simplifying and consolidating the foundations of modern separation logics, but it has evolved over time, and the design and semantic foundations of Iris itself have yet to be fully written down and explained together properly in one place. Here, we attempt to fill this gap, presenting a reasonably complete picture of the latest version of Iris (version 3.1), from first principles and in one coherent narrative.