arXiv subjects

Philip Wadler

Publications and source records attributed to Philip Wadler.

14 recordsLinked to original sources

arXiv2026

UC, Categorically: Rigorous Diagrammatic Proofs

Category theory is a mathematical theory of composition, widely used in logic, computing, and physics. Here we apply it to give a theory of secure composition. In particular, we provide a categorical treatment of Canetti's Universal Composability (UC) framework for systems with a static number of parties and sessions, often termed UC for static systems, yielding four benefits. First, we present our results graphically yet retain rigor by applying a standard categorical technique known as string diagrams. In particular, our formulation of the composition theorem can be graphically verified with a short sequence of diagrams, while remaining translatable to equations and amenable to formal verification. Second, categories let us generalize so that our results extend beyond interactive Turing machines to other forms of computation, such as quantum computation or domain-specific languages. Third, categories help us drop some unnecessary restrictions of UC (e.g., our adversary can be a computational network rather than a single Turing machine); we prove equivalence between our variant and the usual UC, showing no expressiveness is lost. Finally, the categorical perspective leads us to identify and correct some minor technical oversights in the standard formulation of simple UC.

cs.LO↗

arXiv2026

Welterweight Go: Boxing, Structural Subtyping, and Generics (Extended Version)

Go's unique combination of structural subtyping between generics and types with non-uniform runtime representations presents significant challenges for formalising the language. We introduce WG (Welterweight Go), a core model of Go that captures key features excluded by prior work, including underlying types, type unions and type sets, and proposed new features, such as generic methods. We also develop LWG, a lower-level language that models Go's runtime mechanisms, notably the distinction between raw struct values and interface values that carry runtime type information (RTTI). We give a type-directed compilation from WG to LWG that demonstrates how the proposed features can be implemented while observing important design and implementation goals for Go: compatibility with separate compilation, and no runtime code generation. Unlike existing approaches based on static monomorphisation, our compilation strategy uses runtime type conversions and adaptor methods to handle the complex interactions between structural subtyping, generics, and Go's runtime infrastructure.

cs.PL↗

arXiv2025

Substitution Without Copy and Paste

Defining substitution for a language with binders like the simply typed $λ$-calculus requires repetition, defining substitution and renaming separately. To verify the categorical properties of this calculus, we must repeat the same argument many times. We present a lightweight method that avoids repetition and that gives rise to a simply typed category with families (CwF) isomorphic to the initial simply typed CwF. Our paper is a literate Agda script.

cs.LO↗

arXiv2024

Explicit Weakening

I present a novel formulation of substitution, where facts about substitution that previously required tens or hundreds of lines to justify in a proof assistant now follow immediately - they can be justified by writing the four letters "refl". The paper is an executable literate Agda script, and source of the paper is available as an artifact in the file Weaken.lagda.md. Not all consequences of the pandemic have been awful. For the last three years, I've had the great pleasure of meeting with Peter Thiemann and Jeremy Siek for a couple of hours every week, via Zoom, exploring topics including core calculi, gradual typing, and formalisation in Agda. The work reported here arose from those discussions, and is dedicated to Peter on the occasion of his 60th birthday.

cs.PL↗

arXiv2020

Towards Races in Linear Logic

Process calculi based in logic, such as $π$DILL and CP, provide a foundation for deadlock-free concurrent programming, but exclude non-determinism and races. HCP is a reformulation of CP which addresses a fundamental shortcoming: the fundamental operator for parallel composition from the $π$-calculus does not correspond to any rule of linear logic, and therefore not to any term construct in CP. We introduce non-deterministic HCP, which extends HCP with a novel account of non-determinism. Our approach draws on bounded linear logic to provide a strongly-typed account of standard process calculus expressions of non-determinism. We show that our extension is expressive enough to capture many uses of non-determinism in untyped calculi, such as non-deterministic choice, while preserving HCP's meta-theoretic properties, including deadlock freedom.

cs.LO↗

arXiv2020

Featherweight Go

We describe a design for generics in Go inspired by previous work on Featherweight Java by Igarashi, Pierce, and Wadler. Whereas subtyping in Java is nominal, in Go it is structural, and whereas generics in Java are defined via erasure, in Go we use monomorphisation. Although monomorphisation is widely used, we are one of the first to formalise it. Our design also supports a solution to The Expression Problem.

cs.PL↗

arXiv2020

Unifying graded and parameterised monads

Monads are a useful tool for structuring effectful features of computation such as state, non-determinism, and continuations. In the last decade, several generalisations of monads have been suggested which provide a more fine-grained model of effects by replacing the single type constructor of a monad with an indexed family of constructors. Most notably, graded monads (indexed by a monoid) model effect systems and parameterised monads (indexed by pairs of pre- and post-conditions) model program logics. This paper studies the relationship between these two generalisations of monads via a third generalisation. This third generalisation, which we call category-graded monads, arises by generalising a view of monads as a particular special case of lax functors. A category-graded monad provides a family of functors T f indexed by morphisms f of some other category. This allows certain compositions of effects to be ruled out (in the style of a program logic) as well as an abstract description of effects (in the style of an effect system). Using this as a basis, we show how graded and parameterised monads can be unified, studying their similarities and differences along the way.

cs.PL↗

arXiv2019

Gradual Session Types

Session types are a rich type discipline, based on linear types, that lifts the sort of safety claims that come with type systems to communications. However, web-based applications and microservices are often written in a mix of languages, with type disciplines in a spectrum between static and dynamic typing. Gradual session types address this mixed setting by providing a framework which grants seamless transition between statically typed handling of sessions and any required degree of dynamic typing. We propose Gradual GV as a gradually typed extension of the functional session type system GV. Following a standard framework of gradual typing, Gradual GV consists of an external language, which relaxes the type system of GV using dynamic types, and an internal language with casts, for which operational semantics is given, and a cast-insertion translation from the former to the latter. We demonstrate type and communication safety as well as blame safety, thus extending previous results to functional languages with session-based communication. The interplay of linearity and dynamic types requires a novel approach to specifying the dynamics of the language.

cs.PL↗

arXiv2017

Refinement Reflection: Complete Verification with SMT

We introduce Refinement Reflection, a new framework for building SMT-based deductive verifiers. The key idea is to reflect the code implementing a user-defined function into the function's (output) refinement type. As a consequence, at uses of the function, the function definition is instantiated in the SMT logic in a precise fashion that permits decidable verification. Reflection allows the user to write equational proofs of programs just by writing other programs using pattern-matching and recursion to perform case-splitting and induction. Thus, via the propositions-as-types principle, we show that reflection permits the specification of arbitrary functional correctness properties. Finally, we introduce a proof-search algorithm called Proof by Logical Evaluation that uses techniques from model checking and abstract interpretation, to completely automate equational reasoning. We have implemented reflection in Liquid Haskell and used it to verify that the widely used instances of the Monoid, Applicative, Functor, and Monad typeclasses actually satisfy key algebraic laws required to make the clients safe, and have used reflection to build the first library that actually verifies assumptions about associativity and ordering that are crucial for safe deterministic parallelism.

cs.PL↗

arXiv2017

Mixing Metaphors: Actors as Channels and Channels as Actors (Extended Version)

Channel- and actor-based programming languages are both used in practice, but the two are often confused. Languages such as Go provide anonymous processes which communicate using buffers or rendezvous points---known as channels---while languages such as Erlang provide addressable processes---known as actors---each with a single incoming message queue. The lack of a common representation makes it difficult to reason about translations that exist in the folklore. We define a calculus $λ_{\textrm{ch}}$ for typed asynchronous channels, and a calculus $λ_{\textrm{act}}$ for typed actors. We define translations from $λ_{\textrm{act}}$ into $λ_{\textrm{ch}}$ and $λ_{\textrm{ch}}$ into $λ_{\textrm{act}}$ and prove that both are type- and semantics-preserving. We show that our approach accounts for synchronisation and selective receive in actor systems and discuss future extensions to support guarded choice and behavioural types.

cs.PL↗

arXiv2016

Embedding by Normalisation

This paper presents the insight that practical embedding techniques, commonly used for implementing Domain-Specific Languages, correspond to theoretical Normalisation-By-Evaluation (NBE) techniques, commonly used for deriving canonical form of terms with respect to an equational theory. NBE constitutes of four components: a syntactic domain, a semantic domain, and a pair of translations between the two. Embedding also often constitutes of four components: an object language, a host language, encoding of object terms in the host, and extraction of object code from the host. The correspondence is deep in that all four components in embedding and NBE correspond to each other. Based on this correspondence, this paper introduces Embedding-By-Normalisation (EBN) as a principled approach to study and structure embedding. The correspondence is useful in that solutions from NBE can be borrowed to solve problems in embedding. In particular, based on NBE techniques, such as Type-Directed Partial Evaluation, this paper presents a solution to the problem of extracting object code from embedded programs involving sum types, such as conditional expressions, and primitives, such as literals and operations on them.

cs.PL↗

arXiv2015

Everything old is new again: Quoted Domain Specific Languages

We describe a new approach to domain specific languages (DSLs), called Quoted DSLs (QDSLs), that resurrects two old ideas: quotation, from McCarthy's Lisp of 1960, and the subformula property, from Gentzen's natural deduction of 1935. Quoted terms allow the DSL to share the syntax and type system of the host language. Normalising quoted terms ensures the subformula property, which guarantees that one can use higher-order types in the source while guaranteeing first-order types in the target, and enables using types to guide fusion. We test our ideas by re-implementing Feldspar, which was originally implemented as an Embedded DSL (EDSL), as a QDSL; and we compare the QDSL and EDSL variants.

cs.PL↗

arXiv2014

Query shredding: Efficient relational evaluation of queries over nested multisets (extended version)

Nested relational query languages have been explored extensively, and underlie industrial language-integrated query systems such as Microsoft's LINQ. However, relational databases do not natively support nested collections in query results. This can lead to major performance problems: if programmers write queries that yield nested results, then such systems typically either fail or generate a large number of queries. We present a new approach to query shredding, which converts a query returning nested data to a fixed number of SQL queries. Our approach, in contrast to prior work, handles multiset semantics, and generates an idiomatic SQL:1999 query directly from a normal form for nested queries. We provide a detailed description of our translation and present experiments showing that it offers comparable or better performance than a recent alternative approach on a range of examples.

cs.DB↗

arXiv2014

Effective Quotation: relating approaches to language-integrated query

Language-integrated query techniques have been explored in a number of different language designs. We consider two different, type-safe approaches employed by Links and F#. Both approaches provide rich dynamic query generation capabilities, and thus amount to a form of heterogeneous staged computation, but to date there has been no formal investigation of their relative expressiveness. We present two core calculi Eff and Quot, respectively capturing the essential aspects of language-integrated querying using effects in Links and quotation in LINQ. We show via translations from Eff to Quot and back that the two approaches are equivalent in expressiveness. Based on the translation from Eff to Quot, we extend a simple Links compiler to handle queries.

cs.PL↗