Computational reflection is the ability of a system to inspect and modify itself. It has resisted static reasoning, especially in reflective languages, where semantics can change during program execution. We argue that reflection should be revisited in light of progress in verification and large language models. By requiring reflective modifications to carry proofs, it becomes possible to prove system properties even for reflective systems. This "reasonable reflection" could open up new ways of organizing self-improving systems.

Discrete structures are currently second-class in differentiable programming. Since functions over discrete structures lack overt derivatives, differentiable programs do not differentiate through them and limit where they can be used. For example, when programming a neural network, conditionals and iteration cannot be used everywhere; they can break the derivatives necessary for gradient-based learning to work. This limits the class of differentiable algorithms we can directly express, imposing restraints on how we build neural networks and differentiable programs more generally. However, these restraints are not fundamental. Recent work shows conditionals can be first-class, by compiling them into differentiable form as linear neurons. Similarly, this work shows iteration can be first-class---by compiling to linear recurrent neurons. We present a minimal typed, higher-order and linear programming language with iteration called Cajal(N). We prove its programs compile correctly to recurrent neurons, allowing discrete algorithms to be expressed in a differentiable form compatible with gradient-based learning. With our implementation, we conduct two experiments where we link these recurrent neurons against a neural network solving an iterative image transformation task. This determines part of its function prior to learning. As a result, the network learns faster and with greater data-efficiency relative to a neural network programmed without first-class iteration. A key lesson is that recurrent neurons enable a rich interplay between learning and the discrete structures of ordinary programming.

We don’t program neural networks directly. Instead, we rely on an indirect style where learning algorithms, like gradient descent, determine a neural network’s function by learning from data. This indirect style is often a virtue; it empowers us to solve problems that were previously impossible. But it lacks discrete structure. We can’t compile most algorithms into a neural network—even if these algorithms could help the network learn. This limitation occurs because discrete algorithms are not obviously differentiable, making them incompatible with the gradient-based learning algorithms that determine a neural network’s function. To address this, we introduce Cajal ( ⊸ , 𝟚 ): a typed, higher-order and linear programming language intended to be a minimal vehicle for exploring a direct style of programming neural networks. We prove Cajal ( ⊸ , 𝟚 ) programs compile to linear neurons, allowing discrete algorithms to be expressed in a differentiable form compatible with gradient-based learning. With our implementation of Cajal ( ⊸ , 𝟚 ), we conduct several experiments where we link these linear neurons against other neural networks to determine part of their function prior to learning. Linking with these neurons allows networks to learn faster, with greater data-efficiency, and in a way that’s easier to debug. A key lesson is that linear programming languages provide a path towards directly programming neural networks, enabling a rich interplay between learning and the discrete structures of ordinary programming.

We transport multi-stage programming from functional to relational programming, with novel constructs to give programmers control over staging and non-determinism. We stage interpreters written as relations, in which the programs under interpretation can contain holes representing unknown expressions or values. By compiling the known parts without interpretive overhead and deferring interpretation to run time only for the unknown parts, we compound the benefits of staging (e.g., turning interpreters into compilers) and relational interpretation (e.g., turning functions into relations and synthesizing from sketches). We extend miniKanren with staging constructs and apply the resulting multi-stage language to relational interpreters for subsets of Racket and miniKanren as well as a relational recognizer for context-free grammars. We demonstrate significant performance gains across multiple synthesis problems, systematically comparing unstaged and staged computation, as well as indicatively comparing with an existing hand-tuned relational interpreter.

Many obstacles stand in the way of modular, extensible code. Some language constructs, such as pattern matching, are not easily extensible. Inherited code may not be type safe in the presence of extended types. The burden of setting up design patterns can discourage users, and parameter clutter can make the code less readable. Given these challenges, it is no wonder that extensibility often gives way to code duplication. We present our solution: Persimmon, a functional system with nested family polymorphism, extensible variant types, and extensible pattern matching. Most constructs in our language are built-in "extensibility hooks," cutting down on the parameter clutter and user burden associated with extensible code. Persimmon preserves the relationships between nested families upon inheritance, enabling extensibility at a large scale. Since nested family polymorphism can express composable extensions, Persimmon supports mixins via an encoding. We show how Persimmon can be compiled into a functional language without extensible variants with our translation to Scala. Finally, we show that our system is sound by proving the properties of progress and preservation.

We introduce Lurk, a new LISP-based programming language for zk-SNARKs. Traditional approaches to programming over zero-knowledge proofs require compiling the desired computation into a flat circuit, imposing serious constraints on the size and complexity of computations that can be achieved in practice. Lurk programs are instead provided as data to the universal Lurk interpreter circuit, allowing the resulting language to be Turing-complete without compromising the size of the resulting proof artifacts. Our work describes the design and theory behind Lurk, along with detailing how its implementation of content addressing can be used to sidestep many of the usual concerns of programming zero-knowledge proofs.

With the growing practice of mechanizing language metatheories, it has become ever more pressing that interactive theorem provers make it easy to write reusable, extensible code and proofs. This paper presents a novel language design geared towards extensible metatheory mechanization in a proof assistant. The new design achieves reuse and extensibility via a form of family polymorphism, an object-oriented idea, that allows code and proofs to be polymorphic to their enclosing families. Our development addresses technical challenges that arise from the underlying language of a proof assistant being simultaneously functional, dependently typed, a logic, and an interactive tool. Our results include (1) a prototypical implementation of the language design as a Coq plugin, (2) a dependent type theory capturing the essence of the language mechanism and its consistency and canonicity results, and (3) case studies showing how the new expressiveness naturally addresses real programming challenges in metatheory mechanization.

Metareasoning can be achieved in probabilistic programming languages (PPLs) using agent models that recursively nest inference queries inside inference queries. However, the semantics of this powerful, reflection-like language feature has defied an operational treatment, much less reasoning principles for contextual equivalence. We give formal semantics to a core PPL with continuous distributions, scoring, general recursion, and nested queries. Unlike prior work, the presence of nested queries and general recursion makes it impossible to stratify the definition of a sampling-based operational semantics and that of a measure-theoretic semantics—the two semantics must be defined mutually recursively. A key yet challenging property we establish is that probabilistic programs have well-defined meanings: limits exist for the step-indexed measures they induce. Beyond a semantics, we offer relational reasoning principles for probabilistic programs making nested queries. We construct a step-indexed, biorthogonal logical-relations model. A soundness theorem establishes that logical relatedness implies contextual equivalence. We demonstrate the usefulness of the reasoning principles by proving novel equivalences of practical relevance—in particular, game-playing and decisionmaking agents. We mechanize our technical developments leading to the soundness proof using the Coq proof assistant. Nested queries are an important yet theoretically underdeveloped linguistic feature in PPLs; we are first to give them semantics in the presence of general recursion and to provide them with sound reasoning principles for contextual equivalence.

Abstract We present the design and implementation of a SQL query processor that outperforms existing database systems and is written in just about 500 lines of Scala code – a convincing case study that high-level functional programming can handily beat C for systems-level programming where the last drop of performance matters. The key enabler is a shift in perspective toward generative programming. The core of the query engine is an interpreter for relational-algebra operations, written in Scala. Using the open-source lightweight modular staging framework, we turn this interpreter into a query compiler with very low effort. To do so, we capitalize on an old and widely known result from partial evaluation: the first Futamura projection, which states that a process that can specialize an interpreter to any given input program is equivalent to a compiler. In this context, we discuss lightweight modular staging programming patterns such as mixed-stage data structures (e.g., data records with static schema and dynamic field components) and techniques to generate low-level C code, including specialized data structures and data loading primitives.

Given a tower of interpreters, i.e., a sequence of multiple interpreters interpreting one another as input programs, we aim to collapse this tower into a compiler that removes all interpretive overhead and runs in a single pass. In the real world, a use case might be Python code executed by an x86 runtime, on a CPU emulated in a JavaScript VM, running on an ARM CPU. Collapsing such a tower can not only exponentially improve runtime performance, but also enable the use of base-language tools for interpreted programs, e.g., for analysis and verification. In this paper, we lay the foundations in an idealized but realistic setting. We present a multi-level lambda calculus that features staging constructs and stage polymorphism: based on runtime parameters, an evaluator either executes source code (thereby acting as an interpreter) or generates code (thereby acting as a compiler). We identify stage polymorphism, a programming model from the domain of high-performance program generators, as the key mechanism to make such interpreters compose in a collapsible way. We present Pink, a meta-circular Lisp-like evaluator on top of this calculus, and demonstrate that we can collapse arbitrarily many levels of self-interpretation, including levels with semantic modifications. We discuss several examples: compiling regular expressions through an interpreter to base code, building program transformers from modi ed interpreters, and others. We develop these ideas further to include reflection and reification, culminating in Purple, a reflective language inspired by Brown, Blond, and Black, which realizes a conceptually infinite tower, where every aspect of the semantics can change dynamically. Addressing an open challenge, we show how user programs can be compiled and recompiled under user-modified semantics.

We present the first language design to uniformly express variants of n -way joins over asynchronous event streams from different domains, e.g., stream-relational algebra, event processing, reactive and concurrent programming. We model asynchronous reactive programs and joins in direct style, on top of algebraic effects and handlers. Effect handlers act as modular interpreters of event notifications, enabling fine-grained control abstractions and customizable event matching. Join variants can be considered as cartesian product computations with ”degenerate” control flow, such that unnecessary tuples are not materialized a priori. Based on this computational interpretation, we decompose joins into a generic, naive enumeration procedure of the cartesian product, plus variant-specific extensions, represented in terms of user-supplied effect handlers. Our microbenchmarks validate that this extensible design avoids needless materialization. Alongside a formal semantics for joining and prototypes in Koka and multicore OCaml, we contribute a systematic comparison of the covered domains and features.

While type soundness proofs are taught in every graduate PL class, the gap between realistic languages and what is accessible to formal proofs is large. In the case of Scala, it has been shown that its formal model, the Dependent Object Types (DOT) calculus, cannot simultaneously support key metatheoretic properties such as environment narrowing and subtyping transitivity, which are usually required for a type soundness proof. Moreover, Scala and many other realistic languages lack a general substitution property.

Performance critical software is almost always developed in C, as programmers do not trust high-level languages to deliver the same reliable performance. This is bad because low-level code in unsafe languages attracts security vulnerabilities and because development is far less productive, with PL advances mostly lost on programmers operating under tight performance constraints. High-level languages provide memory safety out of the box, but they are deemed too slow and unpredictable for serious system software.

We present short programs that demonstrate the unsoundness of Java and Scala's current type systems. In particular, these programs provide parametrically polymorphic functions that can turn any type into any type without (down)casting. Fortunately, parametric polymorphism was not integrated into the Java Virtual Machine (JVM), so these examples do not demonstrate any unsoundness of the JVM. Nonetheless, we discuss broader implications of these findings on the field of programming languages.

Scala’s type system unifies aspects of ML modules, object- oriented, and functional programming. The Dependent Object Types (DOT) family of calculi has been proposed as a new theoretic foundation for Scala and similar expressive languages. Unfortunately, type soundness has only been established for restricted subsets of DOT. In fact, it has been shown that important Scala features such as type refinement or a subtyping relation with lattice structure break at least one key metatheoretic property such as environment narrowing or invertible subtyping transitivity, which are usually required for a type soundness proof. The main contribution of this paper is to demonstrate how, perhaps surprisingly, even though these properties are lost in their full generality, a rich DOT calculus that includes recursive type refinement and a subtyping lattice with intersection types can still be proved sound. The key insight is that subtyping transitivity only needs to be invertible in code paths executed at runtime, with contexts consisting entirely of valid runtime objects, whereas inconsistent subtyping contexts can be permitted for code that is never executed.

We present the design and implementation of a SQL query processor that outperforms existing database systems and is written in just about 500 lines of Scala code -- a convincing case study that high-level functional programming can handily beat C for systems-level programming where the last drop of performance matters. The key enabler is a shift in perspective towards generative programming. The core of the query engine is an interpreter for relational algebra operations, written in Scala. Using the open-source LMS Framework (Lightweight Modular Staging), we turn this interpreter into a query compiler with very low effort. To do so, we capitalize on an old and widely known result from partial evaluation known as Futamura projections, which state that a program that can specialize an interpreter to any given input program is equivalent to a compiler. In this pearl, we discuss LMS programming patterns such as mixed-stage data structures (e.g. data records with static schema and dynamic field components) and techniques to generate low-level C code, including specialized data structures and data loading primitives.

A scalable programming language is one in which the same concepts can describe small as well as large parts. Towards this goal, Scala unifies concepts from object and module systems. An essential ingredient of this unification is the concept of objects with type members, which can be referenced through path-dependent types. Unfortunately, path-dependent types are not well-understood, and have been a roadblock in grounding the Scala type system on firm theory.

This article presents a modular, flow-insensitive type-and-effect system for purity with lightweight annotations. It does not enforce a global programming discipline and allows arbitrary effects to occur in impure parts of the program. The system is designed to support higher-order languages that mix functional and imperative code like Scala or C#. We show that it can express purity of non-local programming patterns which involve mutable state such as those used in the Scala collections library. We formalize the type system using a functional language with mutable records and define type and effect soundness.

High level data structures are a cornerstone of modern programming and at the same time stand in the way of compiler optimizations. In order to reason about user- or library-defined data structures compilers need to be extensible. Common mechanisms to extend compilers fall into two categories. Frontend macros, staging or partial evaluation systems can be used to programmatically remove abstraction and specialize programs before they enter the compiler. Alternatively, some compilers allow extending the internal workings by adding new transformation passes at different points in the compile chain or adding new intermediate representation (IR) types. None of these mechanisms alone is sufficient to handle the challenges posed by high level data structures. This paper shows a novel way to combine them to yield benefits that are greater than the sum of the parts.

Microfluidic chips are emerging as a powerful platform for automating biology experiments. As it becomes possible to integrate tens of thousands of components on a single chip, researchers will require design automation tools to push the scale and complexity of their designs to match the capabilities of the substrate. However, to date such tools have focused only on droplet-based devices, leaving out the popular class of chips that are based on multilayer soft lithography. In this paper, we develop design automation techniques for microfluidic chips based on multilayer soft lithography. We focus our attention on the control layer, which is driven by pressure actuators to invoke the desired flows on chip. We present a language in which designers can specify the Instruction Set Architecture (ISA) of a microfluidic device. Given an ISA, we automatically infer the locations of valves needed to implement the ISA. We also present novel algorithms for minimizing the number of control lines needed to drive the valves, as well as for routing valves to control ports while admitting sharing between the control lines. To the microfluidic community, we offer a free computer-aided design tool, Micado, which implements a subset of our algorithms as a practical plug-in to AutoCAD. Micado is being used successfully by microfluidic designers. We demonstrate its performance on three realistic chips.