Domain-Specific Languages of Mathematics

GitHub repository for open source material related the book "Domain-Specific Languages of Mathematics" published in 2022 by College Publications.

The book and the repository are used in a BSc-level course at Chalmers and GU.

The main course homepage is in the Canvas LMS:

• Main course page including links to zoom lectures and other media
• Lecture media

Course codes: DAT326 / DIT982

News

• Tuesday 2022-01-18: First lecture of the 2022 course instance (zoom links, etc)
• 2022: book available from Amazon and from other sources. Bibtex entry.
• Lecture note snapshots with drafts of the full course book
• YouTube playlist collecting the 2022 lectures (all in English)
  • Also available: the recorded lectures from the 2021 course instance (most in Swedish, some in English).

with

Contributors

• Main author, examiner, lecturer: Patrik Jansson (patrikj AT)
• First version (and continued support): Cezar Ionescu (cezar AT)
• Book co-author: Jean-Philippe Bernardy
• Teaching assistants:
  • 2022: Sólrún Einarsdóttir (slrn AT), David Wärn (warnd AT), and Felix Cherubini (felixche AT)
  • 2021: Maximilian Algehed (algehed AT) and Víctor López Juan (lopezv AT)
  • 2020: Sólrún Einarsdóttir (slrn AT) and Víctor López Juan (lopezv AT)
  • 2019: Maximilian Algehed (algehed AT) and Abhiroop Sarkar (sarkara AT)
  • 2018: Daniel Schoepe (schoepe AT)
  • 2017: Frederik Hanghøj Iversen (hanghj AT student) and Daniel Schoepe (schoepe AT)
  • 2016: Irene Lobo Valbuena (lobo AT)
• Project assistants: Daniel Heurlin, Sólrún Einarsdóttir, Adam Sandberg Ericsson (saadam AT)

where AT = @chalmers.se

Course material

This repository is mainly the home of the DSLsofMath book (originating from the course lecture notes), also available in print from Amazon.

Comments and contributions are always welcome – especially in the form of pull requests.

The main references are listed below.

Objectives

The course presents classical mathematical topics from a computing science perspective: giving specifications of the concepts introduced, paying attention to syntax and types, and ultimately constructing DSLs of some mathematical areas mentioned below.

Learning outcomes as in the course syllabus.

• Knowledge and understanding
  • design and implement a DSL (Domain-Specific Language) for a new domain
  • organize areas of mathematics in DSL terms
  • explain main concepts of elementary real and complex analysis, algebra, and linear algebra
• Skills and abilities
  • develop adequate notation for mathematical concepts
  • perform calculational proofs
  • use power series for solving differential equations
  • use Laplace transforms for solving differential equations
• Judgement and approach
  • discuss and compare different software implementations of mathematical concepts

The course is elective for both computer science and mathematics students at both Chalmers and GU.

Course setup

• Lectures
  • Introduction: Haskell, complex numbers, syntax, semantics, evaluation, approximation
  • Basic concepts of analysis: sequences, limits, convergence, ...
  • Types and mathematics: logic, quantifiers, proofs and programs, Curry–Howard, ...
    • Type classes, derivatives, differentiation, calculational proofs
  • Domain-Specific Languages and algebraic structures, algebras, homomorphisms
  • Polynomials, series, power series
  • Power series and differential equations, exp, sin, log, Taylor series, ...
  • Linear algebra: vectors, matrices, functions, bases, dynamical systems as matrices and graphs
  • Laplace transform: exp, powers series cont., solving PDEs with Laplace
• Weekly exercise sessions
  • Half time helping students solve problems in small groups
  • Half time joint problem solving at the whiteboard

Lectures

The latest PDF snapshot of the book can be found in L/snapshots but please also consider buying the "real thing".

The "source code" for the chapters are in subdirectories of L/: L/01/, L/02/, etc. where chapter N is approximately course week N.

Exercises

Most chapters end with weekly exercises.

In L/RecEx.md you will find a list of recommended exercises for each chapter of the lecture notes.

Exams

The exams + solutions are available under the Exam/ subdirectory: for example 2016-Practice/, 2016-03/, 2016-08/, 2017-03/, 2017-08/, 2018-03/, 2018-08/, 2019-03/, 2019-08/, 2020-03/, 2020-08/, 2021-03/, 2021-08/.

References

Some important references:

Functional programming

• Thinking Functionally with Haskell, Richard Bird, Cambridge University Press, 2014 URL
• Introduction to Functional Programming Using Haskell, Richard Bird, Prentice Hall, 1998. A previous (but clearly different) version of the above.
• An Introduction to Functional Programming, Richard Bird and Phil Wadler, Prentice Hall, 1988. A previous (but clearly different) version of both of the above.

DSLs

• Functional Programming for Domain-Specific Languages, Jeremy Gibbons. In Central European Functional Programming School 2015, LNCS 8606, 2015. URL

  This is currently the standard reference to DSLs for the functional programmer.

• Folding Domain-Specific Languages: Deep and Shallow Embeddings, Jeremy Gibbons and Nicolas Wu, ICFP 2014. URL

  Available at the same link: a highly recommended short version and the two videos of Jeremy presenting the most important ideas of DSLs in a very accessible way.

• Programming Languages, Mike Spivey. Lecture notes of a course given at the CS Department in Oxford. Useful material for understanding the design and implementation of embedded DSLs. URL

• Domain-Specific Languages, Martin Fowler, 2011. URL

  The view from the object-oriented programming perspective.

The computer science perspective

• Communicating Mathematics: Useful Ideas from Computer Science, Charles Wells, American Mathematical Monthly, May 1995. URL

  This article was one of the main triggers of this course.

  Short summary of the recommendations

Mathematics

• The Language of First-Order Logic, 3rd Edition, Jon Barwise and John Etchemendy, 1993. Out of print, but you can get it for one penny from Amazon UK. A vast improvement over its successors (as Tony Hoare said about Algol 60).

• Mathematics: Form and Function, Saunders Mac Lane, Springer 1986. An overview of the relationships between the many mathematical domains. Entertaining, challenging, rewarding. Fulltext from the library

• Functional Differential Geometry, Gerald Jay Sussman and Jack Wisdom, 2013, MIT. A book about using programming as a means of understanding differential geometry. Similar in spirit to the course, but more advanced and very different in form. An earlier version appeared as an AIM report.

License: CC-BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.