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تسجيل مستخدم جديدExamples and Results from a BSc-level Course on Domain Specific Languages of Mathematics
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Patrik Jansson
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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.
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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 co
mplex 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.
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 synthes
is 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.
These are the lecture notes of an introductory course on ordinal analysis. Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical independence result: Kruskals theorem for binary trees is unprovable in cons
ervative extensions of Peano arithmetic (note that much stronger results of this type are due to Harvey Friedman). Concerning prerequisites, we assume a solid introduction to mathematical logic but no specialized knowledge of proof theory. The material in these notes is intended for 14 lectures and 7 exercise sessions of 90 minutes each.
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor a
ny of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field. Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. From the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Finally, monads and comonads are treated on an equal footing, differently to most literature in which comonads are often overlooked as just the dual to monads. Theorems, interpretations and concrete examples are given for monads as well as for comonads.
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uch as Statecharts and Simulink, Synoptic features hierarchical decomposition of application and control modules in synchronous block diagrams and state machines. Its semantics is described in the polychronous model of computation, which is that of the synchronous language Signal.
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