Selective applicative functors & probabilistic programming

Dissertação de mestrado integrado em Informatics EngineeringIn functional programming, selective applicative functors (SAF) are an abstraction between applicative functors and monads. This abstraction requires all effects to be statically declared, but provides a way to select which effects to execute dynamically. SAF have been shown to be a useful abstraction in several examples, including two industrial case studies. Selective functors have been used for their static analysis capabilities. The collection of information about all possible effects in a computation and the fact that they enable speculative execution make it possible to take advantage to describe probabilistic computations instead of using monads. In particular, selective functors appear to provide a way to obtain a more efficient implementation of probability distributions than monads. This dissertation addresses a probabilistic interpretation for the arrow and selective abstractions in the light of the linear algebra of programming discipline, as well as exploring ways of offering SAF capabilities to probabilistic programming, by exposing sampling as a concurrency problem. As a result, provides a Haskell type-safe matrix library capable of expressing probability distributions and probabilistic computations as typed matrices, and a probabilistic programming eDSL that explores various techniques in order to offer a novel, performant solution to probabilistic functional programming.Em programação funcional, os functores aplicativos seletivos (FAS) são uma abstração entre functores aplicativos e monades. Essa abstração requer que todos os efeitos sejam declarados estaticamente, mas fornece uma maneira de selecionar quais efeitos serão executados dinamicamente. FAS têm se mostrado uma abstração útil em vários exemplos, incluindo dois estudos de caso industriais. Functores seletivos têm sido usados pela suas capacidade de análise estática. O conjunto de informações sobre todos os efeitos possíveis numa computação e o facto de que eles permitem a execução especulativa tornam possível descrever computações probabilísticas. Em particular, functores seletivos parecem oferecer uma maneira de obter uma implementação mais eficiente de distribuições probabilisticas do que monades. Esta dissertação aborda uma interpretação probabilística para as abstrações Arrow e Selective à luz da disciplina da álgebra linear da programação, bem como explora formas de oferecer as capacidades dos FAS para programação probabilística, expondo sampling como um problema de concorrência. Como resultado, fornece uma biblioteca de matrizes em Haskell, capaz de expressar distribuições de probabilidade e cálculos probabilísticos como matrizes tipadas e uma eDSL de programação probabilística que explora várias técnicas, com o obejtivo de oferecer uma solução inovadora e de alto desempenho para a programação funcional probabilística

A functional approach to heterogeneous computing in embedded systems

Developing programs for embedded systems presents quite a challenge; not only should programs be resource efficient, as they operate under memory and timing constraints, but they should also take full advantage of the hardware to achieve maximum performance. Since performance is such a significant factor in the design of embedded systems, modern systems typically incorporate more than one kind of processing element to benefit from specialized processing capabilities. For such heterogeneous systems the challenge in developing programs is even greater.In this thesis we explore a functional approach to heterogeneous system development as a means to address many of the modularity problems that are typically found in the application of low-level imperative programming for embedded systems. In particular, we explore a staged hardware software co-design language that we name Co-Feldspar and embed in Haskell. The staged approach enables designers to build their applications from reusable components and skeletons while retaining control over much of the generated source code. Furthermore, by embedding the language in Haskell we can exploit its type classes to write not only hardware and software programs, but also generic programs with overloaded instructions and expressions. We demonstrate the usefulness of the functional approach for co-design on a cryptographic example and signal processing filters, and benchmark software and mixed hardware-software implementations. Co-Feldspar currently adopts a monadic interface, which provides an imperative functional programming style that is suitable for explicit memory management and algorithms that rely on a certain evaluation order. For algorithms that are better defined as pure functions operating on immutable values, we provide a signal and array library that extends a monadic language, like Co-Feldspar. These extensions permit a functional style of programming by composing high-level combinators. Our compiler transforms such high-level code into efficient programs with mutating code. In particular, we show how to execute an FFT safely in-place, and how to describe a FIR and IIR filter efficiently as streams. Co-Feldspar’s monadic interface is however quite invasive; not only is the burden of explicit memory management quite heavy on the user, it is also quite easy to shoot on eself in the foot. It is for these reasons that we also explore a dynamic memory management discipline that is based on regions but predictable enough to be of use for embedded systems. Specifically, this thesis introduces a program analysis which annotates values with dynamically allocated memory regions. By limiting our efforts to functional languages that target embedded software, we manage to define a region inference algorithm that is considerably simpler than traditional approaches

Formally justified and modular Bayesian inference for probabilistic programs

Probabilistic modelling offers a simple and coherent framework to describe the real world in the face of uncertainty. Furthermore, by applying Bayes' rule it is possible to use probabilistic models to make inferences about the state of the world from partial observations. While traditionally probabilistic models were constructed on paper, more recently the approach of probabilistic programming enables users to write the models in executable languages resembling computer programs and to freely mix them with deterministic code. It has long been recognised that the semantics of programming languages is complicated and the intuitive understanding that programmers have is often inaccurate, resulting in difficult to understand bugs and unexpected program behaviours. Programming languages are therefore studied in a rigorous way using formal languages with mathematically defined semantics. Traditionally formal semantics of probabilistic programs are defined using exact inference results, but in practice exact Bayesian inference is not tractable and approximate methods are used instead, posing a question of how the results of these algorithms relate to the exact results. Correctness of such approximate methods is usually argued somewhat less rigorously, without reference to a formal semantics. In this dissertation we formally develop denotational semantics for probabilistic programs that correspond to popular sampling algorithms often used in practice. The semantics is defined for an expressive typed lambda calculus with higher-order functions and inductive types, extended with probabilistic effects for sampling and conditioning, allowing continuous distributions and unbounded likelihoods. It makes crucial use of the recently developed formalism of quasi-Borel spaces to bring all these elements together. We provide semantics corresponding to several variants of Markov chain Monte Carlo and Sequential Monte Carlo methods and formally prove a notion of correctness for these algorithms in the context of probabilistic programming. We also show that the semantic construction can be directly mapped to an implementation using established functional programming abstractions called monad transformers. We develop a compact Haskell library for probabilistic programming closely corresponding to the semantic construction, giving users a high level of assurance in the correctness of the implementation. We also demonstrate on a collection of benchmarks that the library offers performance competitive with existing systems of similar scope. An important property of our construction, both the semantics and the implementation, is the high degree of modularity it offers. All the inference algorithms are constructed by combining small building blocks in a setup where the type system ensures correctness of compositions. We show that with basic building blocks corresponding to vanilla Metropolis-Hastings and Sequential Monte Carlo we can implement more advanced algorithms known in the literature, such as Resample-Move Sequential Monte Carlo, Particle Marginal Metropolis-Hastings, and Sequential Monte Carlo squared. These implementations are very concise, reducing the effort required to produce them and the scope for bugs. On top of that, our modular construction enables in some cases deterministic testing of randomised inference algorithms, further increasing reliability of the implementation.Engineering and Physical Sciences Research Council, Cambridge Trust, Cambridge-Tuebingen programm

The constrained-monad problem

In Haskell, there are many data types that would form monads were it not for the presence of type-class constraints on the operations onthat data type. This is a frustrating problem in practice, because there is a considerable amount of support and infrastructure for monads that these data types cannot use. Using several examples,we show that a monadic computation can be restructured into a normal form such that the standard monad class can be used. The technique is not specific to monads, and we show how it can also be applied to other structures, such as applicative functors. One significant use case for this technique is domain-specific languages,where it is often desirable to compile a deep embedding of a computation to some other language, which requires restricting the types that can appear in that computation

Developing and Measuring Parallel Rule-Based Systems in a Functional Programming Environment

This thesis investigates the suitability of using functional programming for building parallel rule-based systems. A functional version of the well known rule-based system OPS5 was implemented, and there is a discussion on the suitability of functional languages for both building compilers and manipulating state. Functional languages can be used to build compilers that reflect the structure of the original grammar of a language and are, therefore, very suitable. Particular attention is paid to the state requirements and the state manipulation structures of applications such as a rule-based system because, traditionally, functional languages have been considered unable to manipulate state. From the implementation work, issues have arisen that are important for functional programming as a whole. They are in the areas of algorithms and data structures and development environments. There is a more general discussion of state and state manipulation in functional programs and how theoretical work, such as monads, can be used. Techniques for how descriptions of graph algorithms may be interpreted more abstractly to build functional graph algorithms are presented. Beyond the scope of programming, there are issues relating both to the functional language interaction with the operating system and to tools, such as debugging and measurement tools, which help programmers write efficient programs. In both of these areas functional systems are lacking. To address the complete lack of measurement tools for functional languages, a profiling technique was designed which can accurately measure the number of calls to a function , the time spent in a function, and the amount of heap space used by a function. From this design, a profiler was developed for higher-order, lazy, functional languages which allows the programmer to measure and verify the behaviour of a program. This profiling technique is designed primarily for application programmers rather than functional language implementors, and the results presented by the profiler directly reflect the lexical scope of the original program rather than some run-time representation. Finally, there is a discussion of generally available techniques for parallelizing functional programs in order that they may execute on a parallel machine. The techniques which are easier for the parallel systems builder to implement are shown to be least suitable for large functional applications. Those techniques that best suit functional programmers are not yet generally available and usable

Relay: A New IR for Machine Learning Frameworks

Machine learning powers diverse services in industry including search, translation, recommendation systems, and security. The scale and importance of these models require that they be efficient, expressive, and portable across an array of heterogeneous hardware devices. These constraints are often at odds; in order to better accommodate them we propose a new high-level intermediate representation (IR) called Relay. Relay is being designed as a purely-functional, statically-typed language with the goal of balancing efficient compilation, expressiveness, and portability. We discuss the goals of Relay and highlight its important design constraints. Our prototype is part of the open source NNVM compiler framework, which powers Amazon's deep learning framework MxNet

Smart test data generators via logic programming

We present a novel counterexample generator for the interactive theorem prover Isabelle based on a compiler that synthesizes test data generators for functional programming languages (e.g. Standard ML, OCaml) from specifications in Isabelle. In contrast to naive type-based test data generators, the smart generators take the preconditions into account and only generate tests that fulfill the preconditions. The smart generators are constructed by a compiler that reformulates the preconditions as logic programs and analyzes them by an enriched mode inference. From this inference, the compiler can construct the desired generators in the functional programming language. These test data generators are applied to find errors in specifications, as we show in a case study of a hotel key card system