Structural induction is pervasively used by functional programmers and researchers for both informal reasoning as well as formal methods for program verification and semantics. In this paper, we promote its dual—structural coinduction—as a technique for understanding corecursive programs only in terms of the logical structure of their context. We illustrate this technique as an informal method of proofs which closely match the style of informal inductive proofs, where it is straightforward to check that all cases are covered and the coinductive hypotheses are used correctly. This intuitive idea is then formalized through a syntactic theory for deriving program equalities, which is justified purely in terms of the computational behavior of abstract machines and proved sound with respect to observational equivalence.

This theoretical pearl shows how a graphical, relational, point-free, and calculational approach to linear algebra, known as graphical linear algebra, can be used to reason not only about matrices (and matrix algebra, as can be found in the literature) but also vector spaces and more generally linear relations. Linear algebra is usually seen as the study of vector spaces and linear transformations. However, to reason effectively with subspaces in a point-free and calculational manner, both can be generalized to an unifying concept: linear relations, much like relational algebra. While the semantics is relational, the syntax is graphical and uses string diagrams, 2-dimensional formal diagrams, which represent the linear relations. Most importantly, in a number of cases, the relational semantics allows algorithms and properties to be derived calculationally instead of just verified. Our approach is to proceed primarily by examples which involve finding inverses, switching from an implicit basis to an explicit basis (solving a homogeneous linear system), exploring both the exchange lemma and the Zassenhaus’ algorithm.

OCaml Blockly is a block-based programming environment for a subset of the functional language OCaml, developed based on Google Blockly. The distinct feature of OCaml Blockly is that it knows the scoping and typing rules of OCaml. As such, for any complete program in OCaml Blockly, its OCaml counterpart compiles: it is free from syntax errors, scoping errors, and type errors. OCaml Blockly supports introductory constructs of OCaml that are sufficient to write the shortest path problem for the Tokyo metro network. This paper describes the design of OCaml Blockly and how it is used in a CS-major course on functional programming.

The formal theory of monads shows that much of the theory of monads can be developed in the abstract at the level of 2-categories. This means that results about monads can be established once and for all and simply instantiated in settings such as enriched category theory.

Unfortunately, these results can be hard to reason about as they involve more abstract machinery. In this paper, we present the formal theory of monads in terms of string diagrams — a graphical language for 2-categorical calculations. Using this perspective, we show that many aspects of the theory of monads, such as the Eilenberg–Moore and Kleisli resolutions of monads, liftings, and distributive laws, can be understood in terms of systematic graphical calculational reasoning.

This paper will serve as an introduction both to the formal theory of monads and to the use of string diagrams, in particular, their application to calculations in monad theory.

The setting is a tutorial on program verification in Agda. Please consult the programme for further details. [See also Appendix A.]

The tensor notation used in several areas of mathematics is a useful one, but it is not widely available to the functional programming community. In a practical sense, the (embedded) domain-specific languages (dsls) that are currently in use for tensor algebra are either 1. array-oriented languages that do not enforce or take advantage of tensor properties and algebraic structure or 2. follow the categorical structure of tensors but require the programmer to manipulate tensors in an unwieldy point-free notation. A deeper issue is that for tensor calculus, the dominant pedagogical paradigm assumes an audience which is either comfortable with notational liberties which programmers cannot afford, or focus on the applied mathematics of tensors, largely leaving their linguistic aspects (behaviour of variable binding, syntax and semantics, etc.) for the reader to figure out by themselves. This state of affairs is hardly surprising, because, as we highlight, several properties of standard tensor notation are somewhat exotic from the perspective of lambda calculi. We bridge the gap by defining a dsl, embedded in Haskell, whose syntax closely captures the index notation for tensors in wide use in the literature. The semantics of this edsl is defined in terms of the algebraic structures which define tensors in their full generality. This way, we believe that our edsl can be used both as a tool for scientific computing, but also as a vehicle to express and present the theory and applications of tensors.

Recursive types and bounded quantification are prominent features in many modern programming languages, such as Java, C#, Scala, or TypeScript. Unfortunately, the interaction between recursive types, bounded quantification, and subtyping has shown to be problematic in the past. Consequently, defining a simple foundational calculus that combines those features and has desirable properties, such as decidability, transitivity of subtyping, conservativity, and a sound and complete algorithmic formulation, has been a long-time challenge.

This paper shows how to extend $F_{\le}$ with iso-recursive types in a new calculus called $F_{\le}^{\mu}$. $F_{\le}$ is a well-known polymorphic calculus with bounded quantification. In $F_{\le}^{\mu}$, we add iso-recursive types and correspondingly extend the subtyping relation with iso-recursive subtyping using the recently proposed nominal unfolding rules. In addition, we use so-called structural folding/unfolding rules for typing iso-recursive expressions, inspired by the structural unfolding rule proposed by Abadi et al. (1996). The structural rules add expressive power to the more conventional folding/unfolding rules in the literature, and they enable additional applications. We present several results, including: type soundness; transitivity; the conservativity of $F_{\le}^{\mu}$ over $F_{\le}$; and a sound and complete algorithmic formulation of $F_{\le}^{\mu}$. We study two variants of $F_{\le}^{\mu}$. The first one uses an extension of the $\textrm{kernel}~F_{\le}$ (a well-known decidable variant of $F_{\le}$). This extension accepts equivalent rather than equal bounds and is shown to preserve decidable subtyping. The second variant employs the $\textrm{full}~F_{\le}$ rule for bounded quantification and has undecidable subtyping. Moreover, we also study an extension of the kernel version of $F_{\le}^{\mu}$, called $F_{\le\ge}^{\mu\wedge}$, with a form of intersection types and lower bounded quantification. All the properties from the kernel version of $F_{\le}^{\mu}$ are preserved in $F_{\le\ge}^{\mu\wedge}$. All the results in this paper have been formalized in the Coq theorem prover.

Experience in teaching functional programming (FP) on a relational basis has led the author to focus on a graphical style of expression and reasoning in which a geometric construct shines: the (semi) commutative square. In the classroom this is termed the “magic square” (MS), since virtually everything that we do in logic, FP, database modeling, formal semantics and so on fits in some MS geometry. The sides of each magic square are binary relations and the square itself is a comparison of two paths, each involving two sides. MSs compose and have a number of useful properties. Among several examples given in the paper ranging over different application domains, free-theorem MSs are shown to be particularly elegant and productive. Helped by a little bit of Galois connections, a generic, induction-free theory for ${\mathsf{foldr}}$ and $\mathsf{foldl}$ is given, showing in particular that ${\mathsf{foldl} \, {{s}}{}\mathrel{=}\mathsf{foldr}{({flip} \unicode{x005F}{s})}{}}$ holds under conditions milder than usually advocated.

Functional programmers have many things for which to thank the late David Turner: design decisions he made in his languages SASL, KRC, and Miranda over the last 50 years are still influential and inspirational now. In particular, Turner was a strong advocate of lazy evaluation and of list comprehensions. As an illustration of these techniques, he popularized a one-line recursive “sieve” to generate the infinite list of prime numbers.

Turner called this algorithm The Sieve of Eratosthenes. In a lovely paper called “The Genuine Sieve of Eratosthenes”, Melissa O’Neill argued that Turner’s program is not in fact a faithful implementation of the algorithm, and gave a detailed presentation using priority queues of the real thing. She included a variation by Richard Bird, which uses only lists but makes clever use of circular programming. Bird describes his circular program again in his textbook “Thinking Functionally with Haskell”, and sets its proof of correctness as an exercise. In particular, why is this circular program productive? Unfortunately, Bird’s hint for a solution is incorrect. So what should a proof look like?

One of the last projects Turner worked on was the notion of “Total Functional Programming”. He observed that most programs are already structurally recursive or corecursive, therefore guaranteed respectively terminating or productive; he conjectured that “with more practice we will find this is always true”. We explore Bird’s circular Sieve of Eratosthenes as a challenge problem for Turner’s Total Functional Programming.

This paper reports on the experiences of using an early assessment intervention, specifically employing a Use-Modify-Create scaffold, to teach first-year undergraduate functional programming. The particular intervention that was trialled was the use of an early assessment instrument, in which students had to use code given to them, or slightly modify it, to achieve certain goals. The intended outcome was that the students would thus engage earlier with the functional language, enabling them to be better prepared for the second piece of assessment, where they create code to solve given problems. This intervention showed promise: the difference between a student’s score on the Create assignment improved by an average of 9% in the year after the intervention was implemented, a small effect.

Fenwick trees, also known as binary indexed trees are a clever solution to the problem of maintaining a sequence of values while allowing both updates and range queries in sublinear time. Their implementation is concise and efficient—but also somewhat baffling, consisting largely of nonobvious bitwise operations on indices. We begin with segment trees, a much more straightforward, easy-to-verify, purely functional solution to the problem, and use equational reasoning to explain the implementation of Fenwick trees as an optimized variant, making use of a Haskell EDSL for operations on infinite two’s complement binary numbers.

One can perform equational reasoning about computational effects with a purely functional programming language thanks to monads. Even though equational reasoning for effectful programs is desirable, it is not yet mainstream. This is partly because it is difficult to maintain pencil-and-paper proofs of large examples. We propose a formalization of a hierarchy of effects using monads in the Coq proof assistant that makes monadic equational reasoning practical. Our main idea is to formalize the hierarchy of effects and algebraic laws as interfaces like it is done when formalizing hierarchy of algebras in dependent-type theory. Thanks to this approach, we clearly separate equational laws from models. We can then take advantage of the sophisticated rewriting capabilities of Coq and build libraries of lemmas to achieve concise proofs of programs. We can also use the resulting framework to leverage on Coq’s mathematical theories and formalize models of monads. In this article, we explain how we formalize a rich hierarchy of effects (nondeterminism, state, probability, etc.), how we mechanize examples of monadic equational reasoning from the literature, and how we apply our framework to the design of equational laws for a subset of ML with references.