This paper addresses the question “what is a digital circuit?” in relation to the fundamentally analog nature of actual (physical) circuits. A simple informal definition is given and then formalized in the proof assistant Agda. At the heart of this definition is the timely embedding of discrete information in temporally continuous signals. Once this embedding is defined (in constructive logic, i.e., type theory), it is extended in a generic fashion from one signal to many and from simple boolean operations (logic gates) to arbitrarily sophisticated sequential and parallel compositions, i.e., to computational circuits. Rather than constructing circuits and then trying to prove their correctness, a compositionally correct methodology maintains specification, implementation, timing, and correctness proofs at every step. Compositionality of each aspect and of their combination is supported by a single, shared algebraic vocabulary and related by homomorphisms. After formally defining and proving these notions, a few key transformations are applied to reveal the linearity of circuit timing (over a suitable semiring), thus enabling practical, modular, and fully verified timing analysis as linear maps over higher-dimensional time intervals. An emphasis throughout the paper is simplicity and generality of specification, minimizing circuit-specific definitions and proofs while highlighting a broadly applicable methodology of scalable, compositionally correct engineering through simple denotations and homomorphisms.

Formal languages are usually defined in terms of set theory. Choosing type theory instead gives us languages as type-level predicates over strings. Applying a language to a string yields a type whose elements are language membership proofs describing how a string parses in the language. The usual building blocks of languages (including union, concatenation, and Kleene closure) have precise and compelling specifications uncomplicated by operational strategies and are easily generalized to a few general domain-transforming and codomain-transforming operations on predicates. A simple characterization of languages (and indeed functions from lists to any type) captures the essential idea behind language “differentiation” as used for recognizing languages, leading to a collection of lemmas about type-level predicates. These lemmas are the heart of two dual parsing implementations—using (inductive) regular expressions and (coinductive) tries—each containing the same code but in dual arrangements (with representation and primitive operations trading places). The regular expression version corresponds to symbolic differentiation, while the trie version corresponds to automatic differentiation. The relatively easy-to-prove properties of type-level languages transfer almost effortlessly to the decidable implementations. In particular, despite the inductive and coinductive nature of regular expressions and tries respectively, we need neither inductive nor coinductive/bisimulation arguments to prove algebraic properties.

Automatic differentiation (AD) in reverse mode (RAD) is a central component of deep learning and other uses of large-scale optimization. Commonly used RAD algorithms such as backpropagation, however, are complex and stateful, hindering deep understanding, improvement, and parallel execution. This paper develops a simple, generalized AD algorithm calculated from a simple, natural specification. The general algorithm is then specialized by varying the representation of derivatives. In particular, applying well-known constructions to a naive representation yields two RAD algorithms that are far simpler than previously known. In contrast to commonly used RAD implementations, the algorithms defined here involve no graphs, tapes, variables, partial derivatives, or mutation. They are inherently parallel-friendly, correct by construction, and usable directly from an existing programming language with no need for new data types or programming style, thanks to use of an AD-agnostic compiler plugin.

Parallel programming, whether imperative or functional, has long focused on arrays as the central data type. Meanwhile, typed functional programming has explored a variety of data types, including lists and various forms of trees. Generic functional programming decomposes these data types into a small set of fundamental building blocks: sum, product, composition, and their associated identities. Definitions over these few fundamental type constructions then automatically assemble into algorithms for an infinite variety of data types—some familiar and some new. This paper presents generic functional formulations for two important and well-known classes of parallel algorithms: parallel scan (generalized prefix sum) and fast Fourier transform (FFT). Notably, arrays play no role in these formulations. Consequent benefits include a simpler and more compositional style, much use of common algebraic patterns and freedom from possibility of run-time indexing errors. The functional generic style also clearly reveals deep commonality among what otherwise appear to be quite different algorithms. Instantiating the generic formulations, two well-known algorithms for each of parallel scan and FFT naturally emerge, as well as two possibly new algorithms.

It is well-known that the simply typed lambda-calculus is modeled by any cartesian closed category (CCC). This correspondence suggests giving typed functional programs a variety of interpretations, each corresponding to a different category. A convenient way to realize this idea is as a collection of meaning-preserving transformations added to an existing compiler, such as GHC for Haskell. This paper describes such an implementation and demonstrates its use for a variety of interpretations including hardware circuits, automatic differentiation, incremental computation, and interval analysis. Each such interpretation is a category easily defined in Haskell (outside of the compiler). The general technique appears to provide a compelling alternative to deeply embedded domain-specific languages.

Functional reactive programming (FRP) has simple and powerful semantics, but has resisted efficient implementation. In particular, most past implementations have used demand-driven sampling, which accommodates FRP's continuous time semantics and fits well with the nature of functional programming. Consequently, values are wastefully recomputed even when inputs don't change, and reaction latency can be as high as the sampling period. This paper presents a way to implement FRP that combines data- and demand-driven evaluation, in which values are recomputed only when necessary, and reactions are nearly instantaneous. The implementation is rooted in a new simple formulation of FRP and its semantics and so is easy to understand and reason about. On the road to a new implementation, we'll meet some old friends (monoids, functors, applicative functors, monads, morphisms, and improving values) and make some new friends (functional future values, reactive normal form, and concurrent unambiguous choice).

Automatic differentiation (AD) is a precise, efficient, and convenient method for computing derivatives of functions. Its forward-mode implementation can be quite simple even when extended to compute all of the higher-order derivatives as well. The higher-dimensional case has also been tackled, though with extra complexity. This paper develops an implementation of higher-dimensional, higher-order, forward-mode AD in the extremely general and elegant setting of calculus on manifolds and derives that implementation from a simple and precise specification. In order to motivate and discover the implementation, the paper poses the question "What does AD mean, independently of implementation?" An answer arises in the form of naturality of sampling a function and its derivative. Automatic differentiation flows out of this naturality condition, together with the chain rule. Graduating from first-order to higher-order AD corresponds to sampling all derivatives instead of just one. Next, the setting is expanded to arbitrary vector spaces, in which derivative values are linear maps. The specification of AD adapts to this elegant and very general setting, which even simplifies the development.

We present a user-friendly approach to unifying program creation and execution, based on a notion of "tangible values" (TVs), which are visual and interactive manifestations of pure values, including functions. Programming happens by gestural composition of TVs. Our goal is to give end-users the ability to create parameterized, composable content without imposing the usual abstract and linguistic working style of programmers. We hope that such a system will put the essence of programming into the hands of many more people, and in particular people with artistic/visual creative style.

Graphics cards for personal computers have recently undergone a radical transformation from fixed-function graphics pipelines to multi-processor, programmable architectures. Multi-processor architectures are clearly advantageous for graphics for the simple reason that graphics computations are naturally concurrent, mapping well to stateless stream processing. They therefore parallelize easily and need no random access to memory with its problematic latencies.This paper presents Vertigo, a purely functional, Haskell-embedded language for 3D graphics and an optimizing compiler that generates graphics processor code. The language integrates procedural surface modeling, shading, and texture generation, and the compiler exploits the unusual processor architecture. The shading sub-language is based on a simple and precise semantic model, in contrast to previous shading languages. Geometry and textures are also defined via a very simple denotational semantics. The formal semantics yields not only programs that are easy to understand and reason about, but also very efficient implementation, thanks to a compiler based on partial evaluation and symbolic optimization, much in the style of Pan [2].Haskell's overloading facility is extremely useful throughout Vertigo. For instance, math operators are used not just for floating point numbers, but also expressions (for differentiation and compilation), tuples, and functions. Typically, these overloadings cascade, as in the case of surfaces, which may be combined via math operators, though they are really functions over tuples of expressions on floating point numbers. Shaders may be composed with the same notational convenience. Functional dependencies are exploited for vector spaces, cross products, and derivatives.

Functional languages are particularly well-suited to the interpretive implementations of Domain-Specific Embedded Languages (DSELs). We describe an implemented technique for producing optimizing compilers for DSELs, based on Kamin's idea of DSELs for program generation. The technique uses a data type of syntax for basic types, a set of smart constructors that perform rewriting over those types, some code motion transformations, and a back-end code generator. Domain-specific optimization results from chains of domain-independent rewrites on basic types. New DSELs are defined directly in terms of the basic syntactic types, plus host language functions and tuples. This definition style makes compilers easy to write and, in fact, almost identical to the simplest embedded interpreters. We illustrate this technique with a language Pan for the computationally intensive domain of image synthesis and manipulation.

Events play an important role in the construction of most software that involves interaction or simulation. Typically, programmers make use of a fixed set of low level events supplied by a window system, possibly augmented with timers and UI components. Event handling generally involves some interpretation of these event occurrences, followed by external actions or modifications to program state. It is possible to extend the event paradigm by using event interpretation to synthesize new kinds of events tailored specifically for a domain or application. In turn, these new events may be used to synthesize yet others, and so on, to an arbitrarily sophisticated degree. This programming paradigm, which we call event-oriented programming, aids in the factoring of programs into understandable and reusable pieces. We propose a declarative approach to event-oriented programming, based on a powerfully expressive event language with a lightweight notation. We illustrate this new approach throug...

While interactive multimedia animation is a very compelling medium, few people are able to express themselves in it. There are too many low-level details that have to do not with the desired content-e.g., shapes, appearance and behavior-but rather how to get a computer to present the content. For instance, behavior such as motion and growth are generally gradual, continuous phenomena. Moreover, many such behaviors go on simultaneously. Computers, on the other hand, cannot directly accommodate either of these basic properties, because they do their work in discrete steps rather than continuously, and they only do one thing at a time. Graphics programmers have to spend much of their effort bridging the gap between what an animation is and how to present it on a computer. We propose that this situation can be improved by a change of language, and present Fran, synthesized by complementing an existing declarative host language, Haskell, with an embedded domain-specific vocabulary for modeled animation. As demonstrated in a collection of examples, the resulting animation descriptions are not only relatively easy to write, but also highly composable.

Fran (Functional Reactive Animation) is a collection of data types and functions for composing richly interactive, multi-media animations. The key ideas in Fran are its notions of behaviors and events. Behaviors are time-varying, reactive values, while events are sets of arbitrarily complex condi-tions, carrying possibly rich information. Most traditional values can be treated as behaviors, and when images are thus treated, they become animations. Although these no-tions are captured as data types rather than a programming language, we provide them with a denotational semantics, including a proper treatment of real time, to guide reason-ing and implementation. A method to eectively and ef-ciently perform event detection using interval analysis is also described, which relies on the partial information struc-ture on the domain of event times. Fran has been imple-mented in Hugs, yielding surprisingly good performance for an interpreter-based system. Several examples are given, in-cluding the ability to describe physical phenomena involving

We present a paradigm and toolkit for rapid prototyping of interactive, animated 3D graphics programs. The paradigm has its roots in declarative programming, emphasizing immutable values, first class functions, and relations, applying these concepts to a broad range of types, including points, vectors, planes, colors, transforms, geometry, and sound. The narrow role of modifiable state in this paradigm allows applications to be run in a collaborative setting (multi-user and multi-computer) without modification.

We describe motivation, design, use, and implementation of higher-order abstract syntax as a central representation for programs, formulas, rules, and other syntactic objects in program manipulation and other formal systems where matching and substitution or unification are central operations. Higher-order abstract syntax incorporates name binding information in a uniform and language generic way. Thus it acts as a powerful link integrating diverse tools in such formal environments. We have implemented higher-order abstract syntax, a supporting matching and unification algorithm, and some clients in Common Lisp in the framework of the Ergo project at Carnegie Mellon University.