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Register a new userDomain-Specific Languages of Mathematics: Presenting Mathematical Analysis Using Functional Programming
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Publication date
2016
fields
Informatics Engineering
and research's language is
English
Authors
Cezar Ionescu
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We present the approach underlying a course on Domain-Specific Languages of Mathematics, currently being developed at Chalmers in response to difficulties faced by third-year students in learning and applying classical mathematics (mainly real and complex analysis). The main idea is to encourage the students to approach mathematical domains from a functional programming perspective: to identify the main functions and types involved and, when necessary, to introduce new abstractions; to give calculational proofs; to pay attention to the syntax of the mathematical expressions; and, finally, to organise the resulting functions and types in domain-specific languages.
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At the workshop on Trends in Functional Programming in Education (TFPIE) in 2015 Ionescu and Jansson presented the approach underlying the Domain Specific Languages of Mathematics (DSLsofMath) course even before the first course instance. We were then encouraged to come back to present our experience and the student results. Now, three years later, we have seen three groups of learners attend the course, and the first two groups have also continued on to take challenging courses in the subsequent year. In this paper we present three examples from the course material to set the scene, and we present an evaluation of the student results showing improvements in the pass rates and grades in later courses.
We propose a runtime architecture that can be used in the development of a quantum programming language and its programming environment. The proposed runtime architecture enables dynamic interaction between classical and quantum data following the restriction that a quantum computer is available in the cloud as a batch computer, with no interaction with the classical computer during its execution. It is done by leaving the quantum code generation for the runtime and introducing the concept of futures for quantum measurements. When implemented in a quantum programming language, those strategies aim to facilitate the development of quantum applications, especially for beginning programmers and students. Being suitable for the current Noisy Intermediate-Scale Quantum (NISQ) Computers, the runtime architecture is also appropriate for simulation and future Fault-Tolerance Quantum Computers.
We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixed-variance datatypes. Our semantics justifies important program equivalences including commutativity. Our new semantic model is based on `quasi-Borel predomains. These are a mixture of chain-complete partial orders (cpos) and quasi-Borel spaces. Quasi-Borel spaces are a recent model of probability theory that focuses on sets of admissible random elements. Probability is traditionally treated in cpo models using probabilistic powerdomains, but these are not known to be commutative on any class of cpos with higher order functions. By contrast, quasi-Borel predomains do support both a commutative probabilistic powerdomain and higher-order functions. As we show, quasi-Borel predomains form both a model of Fiores axiomatic domain theory and a model of Kocks synthetic measure theory.
Program synthesis from input-output examples has been a long-standing challenge, and recent works have demonstrated some success in designing deep neural networks for program synthesis. However, existing efforts in input-output neural program synthesis have been focusing on domain-specific languages, thus the applicability of previous approaches to synthesize code in full-fledged popular programming languages, such as C, remains a question. The main challenges lie in two folds. On the one hand, the program search space grows exponentially when the syntax and semantics of the programming language become more complex, which poses higher requirements on the synthesis algorithm. On the other hand, increasing the complexity of the programming language also imposes more difficulties on data collection, since building a large-scale training set for input-output program synthesis require random program generators to sample programs and input-output examples. In this work, we take the first step to synthesize C programs from input-output examples. In particular, we propose LaSynth, which learns the latent representation to approximate the execution of partially generated programs, even if their semantics are not well-defined. We demonstrate the possibility of synthesizing elementary C code from input-output examples, and leveraging learned execution significantly improves the prediction performance over existing approaches. Meanwhile, compared to the randomly generated ground-truth programs, LaSynth synthesizes more concise programs that resemble human-written code. We show that training on these synthesized programs further improves the prediction performance for both Karel and C program synthesis, indicating the promise of leveraging the learned program synthesizer to improve the dataset quality for input-output program synthesis.
The goal of TFPIE is to gather researchers, professors, teachers, and all professionals interested in functional programming in education. This includes the teaching of functional programming, but also the application of functional programming as a tool for teaching other topics. The post-workshop review process received 13 submissions, which were vetted by the program committee, assuming scientific journal standards of publication. The six articles in this volume were selected for publication as the result of this process.
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