Mœbius: metaprogramming using contextual types: the stage where system f can pattern match on itself

We describe the foundation of the metaprogramming language, Mœbius, which supports the generation of polymorphic code and, more importantly, the analysis of polymorphic code via pattern matching. Mœbius has two main ingredients: 1) we exploit contextual modal types to describe open code together with the context in which it is meaningful. In Mœbius, open code can depend on type and term variables (level 0) whose values are supplied at a later stage, as well as code variables (level 1) that stand for code templates supplied at a later stage. This leads to a multi-level modal lambda-calculus that supports System-F style polymorphism and forms the basis for polymorphic code generation. 2) we extend the multi-level modal lambda-calculus to support pattern matching on code. As pattern matching on polymorphic code may refine polymorphic type variables, we extend our type-theoretic foundation to generate and track typing constraints that arise. We also give an operational semantics and prove type preservation. Our multi-level modal foundation for Mœbius provides the appropriate abstractions for both generating and pattern matching on open code without committing to a concrete representation of variable binding and contexts. Hence, our work is a step towards building a general type-theoretic foundation for multi-staged metaprogramming that, on the one hand, enforces strong type guarantees and, on the other hand, makes it easy to generate and manipulate code. This will allow us to exploit the full potential of metaprogramming without sacrificing the reliability of and trust in the code we are producing and running.

Monotone recursive types and recursive data representations in Cedille

Guided by Tarksi’s fixpoint theorem in order theory, we show how to derive monotone recursive types with constant-time roll and unroll operations within Cedille, an impredicative, constructive, and logically consistent pure typed lambda calculus. This derivation takes place within the preorder on Cedille types induced by type inclusions, a notion which is expressible within the theory itself. As applications, we use monotone recursive types to generically derive two recursive representations of data in lambda calculus, the Parigot and Scott encoding. For both encodings, we prove induction and examine the computational and extensional properties of their destructor, iterator, and primitive recursor in Cedille. For our Scott encoding in particular, we translate into Cedille a construction due to Lepigre and Raffalli (2019) that equips Scott naturals with primitive recursion, then extend this construction to derive a generic induction principle. This allows us to give efficient and provably unique (up to function extensionality) solutions for the iteration and primitive recursion schemes for Scott-encoded data.

Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

Label dependent lambda calculus and gradual typing

Dependently-typed programming languages are gaining importance, because they can guarantee a wide range of properties at compile time. Their use in practice is often hampered because programmers have to provide very precise types. Gradual typing is a means to vary the level of typing precision between program fragments and to transition smoothly towards more precisely typed programs. The combination of gradual typing and dependent types seems promising to promote the widespread use of dependent types. We investigate a gradual version of a minimalist value-dependent lambda calculus. Compile-time calculations and thus dependencies are restricted to labels, drawn from a generic enumeration type. The calculus supports the usual Pi and Sigma types as well as singleton types and subtyping. It is sufficiently powerful to provide flexible encodings of variant and record types with first-class labels. We provide type checking algorithms for the underlying label-dependent lambda calculus and its gradual extension. The gradual type checker drives the translation into a cast calculus, which extends the original language. The cast calculus comes with several innovations: refined typing for casts in the presence of singletons, type reduction in casts, and fully dependent Sigma types. Besides standard metatheoretical results, we establish the gradual guarantee for the gradual language.

Solver-based gradual type migration

Gradually typed languages allow programmers to mix statically and dynamically typed code, enabling them to incrementally reap the benefits of static typing as they add type annotations to their code. However, this type migration process is typically a manual effort with limited tool support. This paper examines the problem of automated type migration: given a dynamic program, infer additional or improved type annotations. Existing type migration algorithms prioritize different goals, such as maximizing type precision, maintaining compatibility with unmigrated code, and preserving the semantics of the original program. We argue that the type migration problem involves fundamental compromises: optimizing for a single goal often comes at the expense of others. Ideally, a type migration tool would flexibly accommodate a range of user priorities. We present TypeWhich, a new approach to automated type migration for the gradually-typed lambda calculus with some extensions. Unlike prior work, which relies on custom solvers, TypeWhich produces constraints for an off-the-shelf MaxSMT solver. This allows us to easily express objectives, such as minimizing the number of necessary syntactic coercions, and constraining the type of the migration to be compatible with unmigrated code. We present the first comprehensive evaluation of GTLC type migration algorithms, and compare TypeWhich to four other tools from the literature. Our evaluation uses prior benchmarks, and a new set of "challenge problems." Moreover, we design a new evaluation methodology that highlights the subtleties of gradual type migration. In addition, we apply TypeWhich to a suite of benchmarks for Grift, a programming language based on the GTLC. TypeWhich is able to reconstruct all human-written annotations on all but one program.

Durable functions: semantics for stateful serverless

Serverless, or Functions-as-a-Service (FaaS), is an increasingly popular paradigm for application development, as it provides implicit elastic scaling and load based billing. However, the weak execution guarantees and intrinsic compute-storage separation of FaaS create serious challenges when developing applications that require persistent state, reliable progress, or synchronization. This has motivated a new generation of serverless frameworks that provide stateful abstractions. For instance, Azure's Durable Functions (DF) programming model enhances FaaS with actors, workflows, and critical sections. As a programming model, DF is interesting because it combines task and actor parallelism, which makes it suitable for a wide range of serverless applications. We describe DF both informally, using examples, and formally, using an idealized high-level model based on the untyped lambda calculus. Next, we demystify how the DF runtime can (1) execute in a distributed unreliable serverless environment with compute-storage separation, yet still conform to the fault-free high-level model, and (2) persist execution progress without requiring checkpointing support by the language runtime. To this end we define two progressively more complex execution models, which contain the compute-storage separation and the record-replay, and prove that they are equivalent to the high-level model.

Coherence for bicategorical cartesian closed structure

We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.