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A digital scientific notation

Leibniz is an attempt to define a digital scientific notation, i.e. a formal language for writing down scientific models in terms of equations and algorithms. Such models can be published, cited, and discussed, in addition to being manipulated by software.

The best way to get an impression of what Leibniz is and what you can do with it is to read the one-page introduction "Leibniz by example". Then you can move on to the other examples and to the manual. You can also watch my presentation at RacketCon 2020.

Readers interested in the studying the implementation (which needs a serious cleanup) should start by looking at the file for an overview of the code structure.

Leibniz is named after Gottfried Wilhelm Leibniz, who made important contributions to science, mathematics, formal logic, and computation, topics that are all relevant to this project. He invented a widely used notation for calculus, laid the foundation of equational logic by his definition of equality, and anticipated formal logic with his "calculus ratiocinator".


In a word: experimental. The major milestone that the implementation has reached is to play its role of a digital scientific notation: Leibniz specifications are embedded into the plain text discourse written for human readers, just like traditional semi-formal mathematical notation. This matters because it eliminates an important source of mistakes: the translation from human-readable and peer-reviewed descriptions of models and methods into computer-readable code.

However, many features that I have planned for the language are still missing: built-in collection types (lists/arrays, sets, ...) interfaces to databases and external datasets, support for workflows. Although in principle today's Leibniz can be used for everything (given that it's Turing-complete), it is still insufficient to express many aspects of computational science in a sufficiently concise and convenient form.

The master branch of this repository contains the version of Leibniz that has been described in the article Verifiability in computer-aided research: the role of digital scientific notations at the human-computer interface. Since then, I have been working on a major redesign on the branch pollen. As its name suggests, it uses the Pollen library rather than Scribble documentation system that the master branch builds on. The main reason is finer control over the generated HTML files, with in particular the embedding of the machine-readable XML code into the human-readable HTML file. A Leibniz document is thus now a single file that can be identified by a single URL or a single hash code.

A second implementation of Leibniz, incomplete so far, builds on the live object programming system Pharo, a descendant of Smalltalk. The main reason for this development is the explore the possibilities for authoring and exploring models formulated in Leibniz interactively. Pharo serves both as an implementation substrate (I expect to be able to extend or re-use the Pharo software development tools for Leibnis) and as a source of inspiration, given the long tradition in the Smalltalk community of eliminating the developer-user dichotomy that is, in my opinion, a major obstacle in computational science.

Required software, installation

This first implementation of Leibniz is written in Racket, whose support for implementing languages and language extensions is particularly useful for this project. In addition to Racket itself, Leibniz depends on the following libraries:

To install Leibniz and its dependencies, first install the Racket system on your computer, and then type, in a terminal window:

   raco pkg install git://\?path=leibniz

To run the Leibniz test suite, type

   raco test -c leibniz

You can then use Leibniz in two ways:

  • In Racket's IDE, called DrRacket. Any file starting with

    #lang leibniz

    is treated as a Leibniz document. Clicking on the Leibniz button creates a human-readable HTML version and a machine-readable XML version of the document, and opens the HTML file immediately in a browser for inspection.

  • Write your Leibniz documents using any text editor, and generate the HTML/XML files using the leibniz command line utility. It is part of installation process, but the location where it ends up is very platform-dependent. The good news is that the precise location is indicated near the end of the installation process, so have a careful look at the log output of raco pkg install ....

For more information, see the Leibniz manual:

  • In DrRacket, go to the "Help" menu and select "Racket Documentation". This will open the table of contents of the Racket documentation in a browser. Search for "Leibniz" and click the link.

  • From a terminal command line, run "raco docs leibniz"


I expect to properly document and release this code at some time, under a meaningful license. But for now, it is research code covered by the CRAPL license.


The following articles are helpful to understand the context in which Leibniz is developed:

Leibniz is based on equational logic and term rewriting. This seems an appropriate choice for scientific models that are traditionally written as mathematical equations. Algorithms are expressed by giving a direction to certain equations, indicating that the left-hand side is supposed to be replaced by the right-hand side in simplifying an expression. Term rewriting has been used for a long time in computer algebra, notably by Mathematica.

Leibniz differs from Mathematica and most other computer algebra systems in using an order-sorted term algebra, in which each term is assigned a sort, which is similar to what is called a type in programming languages. For a detailed discussion of order-sorted algebra, see

Term rewriting in order-sorted algebras has been implemented in the specification languages OBJ and its modern offshoot Maude. For readers familiar with these languages, a Leibniz "context" is roughly the same as an "object" in OBJ or a "functional module" in Maude. Reading the Maude documentation is currently the best preparation for understanding Leibniz.

However, Leibniz is much simpler than Maude, lacking both Maude's flexible syntax and its support for non-functional modules. This is due to a very different focus: Maude is a language for writing specifications for complex software, whereas Leibniz is a notation for scientific models. Scientific models are much simpler than most software, but they can be processed by a wide range of software. Leibniz must therefore be easy to implement in a wide range of software packages, whereas reimplementing Maude is of little interest, given that its source code is open.

Branch notes

Most branches of this repository contain experiments that test the utility and feasibility of ideas for improvements and new features. Each branch has a short note in this place that explains its reason for being.

Note that all branches except master may be rebased, or modified in other ways. If you want to fork this repository, please don't rely on any branch other than master.

This branch replaces Scribble by Pollen as the underlying document processing platform. The advantage of using Pollen is that all processing can be done at the xexpr level. This means in particular that the messy macro system for defining contexts can be replaced by much simpler plain functions. Another advantage is that Pollen allows precise control over the HTML output, which makes it possible to embed the XML representation as a script in the HTML file.


A Digital Scientific Notation




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