Abstract One of the natural problems of operational semantics is to characterise the relationship between eager and lazy evaluation. In the context of $\lambda$ -calculus, this is expressed by the classic theorem that call-by-value evaluation of a program to (weak-head) normal form can always be simulated by a call-by-name evaluation. While the statement and intuition behind it are simple and clear, naive attempts at proof famously fail: the result is usually established as a consequence of the more complex standardisation theorem. In this work, we develop and formalise a novel and lightweight inductive approach to tackle the problem of simulation between two semantics for a single calculus, but with different evaluation orders. We exercise our method on the classic call-by-value and call-by-name example and report on methodological takeaways suggested by our approach, in particular what effect the flavour of semantics chosen has on the proof.

This paper is a contribution to the meta-theory of systems featuring syntax with bindings, such as λ-calculi and logics. It provides a general criterion that targets inductively defined rule-based systems , enabling for them inductive proofs that leverage Barendregt’s variable convention of keeping the bound and free variables disjoint. It improves on the state of the art by (1) achieving high generality in the style of Knaster-Tarski fixed point definitions (as opposed to imposing syntactic formats), (2) capturing systems of interest without modifications, and (3) accommodating infinitary syntax and non-equivariant predicates.

Automatic differentiation (AD) has been a topic of interest for researchers in many disciplines, with increased popularity since its application to machine learning and neural networks. Although many researchers appreciate and know how to apply AD, it remains a challenge to truly understand the underlying processes. From an algebraic point of view, however, AD appears surprisingly natural: it originates from the differentiation laws. In this work we use Algebra of Programming techniques to reason about different AD variants, leveraging Haskell to illustrate our observations. Our findings stem from three fundamental algebraic abstractions: (1) the notion of semimodule, (2) Nagata's construction of the ‘idealization of a module’, and (3) Kronecker's delta function, that together allow us to write a single-line abstract definition of AD. From this single-line definition, and by instantiating our algebraic structures in various ways, we derive different AD variants, that have the same extensional behaviour, but different intensional properties, mainly in terms of (asymptotic) computational complexity. We show the different variants equivalent by means of Kronecker isomorphisms, a further elaboration of our Haskell infrastructure which guarantees correctness by construction. With this framework in place, this paper seeks to make AD variants more comprehensible, taking an algebraic perspective on the matter.

Structural subtyping and parametric polymorphism provide similar flexibility and reusability to programmers. For example, both features enable the programmer to provide a wider record as an argument to a function that expects a narrower one. However, the means by which they do so differs substantially, and the precise details of the relationship between them exists, at best, as folklore in literature. In this paper, we systematically study the relative expressive power of structural subtyping and parametric polymorphism. We focus our investigation on establishing the extent to which parametric polymorphism, in the form of row and presence polymorphism, can encode structural subtyping for variant and record types. We base our study on various Church-style λ-calculi extended with records and variants, different forms of structural subtyping, and row and presence polymorphism. We characterise expressiveness by exhibiting compositional translations between calculi. For each translation we prove a type preservation and operational correspondence result. We also prove a number of non-existence results. By imposing restrictions on both source and target types, we reveal further subtleties in the expressiveness landscape, the restrictions enabling otherwise impossible translations to be defined. More specifically, we prove that full subtyping cannot be encoded via polymorphism, but we show that several restricted forms of subtyping can be encoded via particular forms of polymorphism.

Abstract The syntax of almost every programming language includes a notion of binder and corresponding bound occurrences, along with the accompanying notions of α-equivalence, capture-avoiding substitution, typing contexts, runtime environments, and so on. In the past, implementing and reasoning about programming languages required careful handling to maintain the correct behaviour of bound variables. Modern programming languages include features that enable constraints like scope safety to be expressed in types. Nevertheless, the programmer is still forced to write the same boilerplate over again for each new implementation of a scope-safe operation (e.g., renaming, substitution, desugaring, printing), and then again for correctness proofs. We present an expressive universe of syntaxes with binding and demonstrate how to (1) implement scope-safe traversals once and for all by generic programming; and (2) how to derive properties of these traversals by generic proving. Our universe description, generic traversals and proofs, and our examples have all been formalised in Agda and are available in the accompanying material available online at https://github.com/gallais/generic-syntax .

We present a novel typed language for extensible data types, generalizing and abstracting existing systems of row types and row polymorphism. Extensible data types are a powerful addition to traditional functional programming languages, capturing ideas from OOP-like record extension and polymorphism to modular compositional interpreters. We introduce row theories, a monoidal generalization of row types, giving a general account of record concatenation and projection (dually, variant injection and branching). We realize them via qualified types, abstracting the interpretation of records and variants over different row theories. Our approach naturally types terms untypable in other systems of extensible data types, while maintaining strong metatheoretic properties, such as coherence and principal types. Evidence for type qualifiers has computational content, determining the implementation of record and variant operations; we demonstrate this in giving a modular translation from our calculus, instantiated with various row theories, to polymorphic λ-calculus.

The idea of a context lemma spans a range of programming-language models: from Milner’s original through the CIU theorem to ‘CIU-like’ results for multiple language features. Each shows that to prove observational equivalence between program terms it is enough to test only some restricted class of contexts: applicative, evaluation, reduction, etc.

Almost every programming language’s syntax includes a notion of binder and corresponding bound occurrences, along with the accompanying notions of α-equivalence, capture avoiding substitution, typing contexts, runtime environments, and so on. In the past, implementing and reasoning about programming languages required careful handling to maintain the correct behaviour of bound variables. Modern programming languages include features that enable constraints like scope safety to be expressed in types. Nevertheless, the programmer is still forced to write the same boilerplate over again for each new implementation of a scope safe operation (e.g., renaming, substitution, desugaring, printing, etc.), and then again for correctness proofs. We present an expressive universe of syntaxes with binding and demonstrate how to (1) implement scope safe traversals once and for all by generic programming; and (2) how to derive properties of these traversals by generic proving. Our universe description, generic traversals and proofs, and our examples have all been formalised in Agda and are available in the accompanying material. NB. we recommend printing the paper in colour to benefit from syntax highlighting in code fragments.

We abstract the common type-and-scope safe structure from computations on λ-terms that deliver, e.g., renaming, substitution, evaluation, CPS-transformation, and printing with a name supply. By exposing this structure, we can prove generic simulation and fusion lemmas relating operations built this way. This work has been fully formalised in Agda.

Language designers have in recent years proposed a wealth of richer type systems for programming which seek to extend the range of statically enforced guarantees on data and code. Most such proposals have been evolutionary extensions of ML or Haskell, offering programmers a balanced compromise between expressive strength and existing well-understood technology. Typically they revolve around type- or kind-indexed types such as GADTs, supported by limited equality reasoning at the type-checking level, thus separating the dynamic behaviour of programs from the (simpler) static behaviour of indexing information occurring in their types.I want to argue in this talk for a more radical departure from such practice by examining full spectrum type dependency, lifting such restrictions on the data upon which types may depend. Conor McBride and I designed the language EPIGRAM for experiments in programming with inductive families of data (of which GADTs are a special case). Using it for illustration, I will explore some of the possibilities and challenges afforded by full spectrum type dependency at the static and dynamic level:

In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [5] for bound variables. By doing so, we retain the advantages of both representations: naming supports easy, arithmetic-free manipulation of terms; de Bruijn indices eliminate the need for α-conversion. Further, we have ensured that not only the user but also the implementation need never deal with de Bruijn indices, except within key basic operations.Moreover, we give a hierarchical representation for names which naturally reflects the structure of the operations we implement. Name choice is safe and straightforward. Our technology combines easily with an approach to syntax manipulation inspired by Huet's 'zippers'[10].Without the ideas in this paper, we would have struggled to implement EPIGRAM [19]. Our example-constructing inductive elimination operators for datatype families-is but one of many where it proves invaluable.

Pattern matching has proved an extremely powerful and durable notion in functional programming. This paper contributes a new programming notation for type theory which elaborates the notion in various ways. First, as is by now quite well-known in the type theory community, definition by pattern matching becomes a more discriminating tool in the presence of dependent types, since it refines the explanation of types as well as values. This becomes all the more true in the presence of the rich class of datatypes known as inductive families (Dybjer, 1991). Secondly, as proposed by Peyton Jones (1997) for Haskell, and independently rediscovered by us, subsidiary case analyses on the results of intermediate computations, which commonly take place on the right-hand side of definitions by pattern matching, should rather be handled on the left. In simply-typed languages, this subsumes the trivial case of Boolean guards; in our setting it becomes yet more powerful. Thirdly, elementary pattern matching decompositions have a well-defined interface given by a dependent type; they correspond to the statement of an induction principle for the datatype. More general, user-definable decompositions may be defined which also have types of the same general form. Elementary pattern matching may therefore be recast in abstract form, with a semantics given by translation. Such abstract decompositions of data generalize Wadler's (1987) notion of ‘view’. The programmer wishing to introduce a new view of a type $\mathit{T}$ , and exploit it directly in pattern matching, may do so via a standard programming idiom. The type theorist, looking through the Curry–Howard lens, may see this as proving a theorem , one which establishes the validity of a new induction principle for $\mathit{T}$ . We develop enough syntax and semantics to account for this high-level style of programming in dependent type theory. We close with the development of a typechecker for the simply-typed lambda calculus, which furnishes a view of raw terms as either being well-typed, or containing an error. The implementation of this view is ipso facto a proof that typechecking is decidable.