# Mikrokosmos User's guide (v0.8.0)

## 1 Mikrokosmos

Mikrokosmos is an untyped and simply typed λ-calculus interpreter, borrowing its name from the series of progressive piano études Mikrokosmos written by Bela Bartok. It aims to provide students with a tool to learn and understand the λ-calculus.

## 2 Installation

Quick start: under GNU/Linux, it should be possible to install the interpreter and the Jupyter kernel only with the two following commands, provided that cabal and pip are installed.

cabal install mikrokosmos
sudo pip install imikrokosmos


If you want to add custom libraries, they should be placed under ~/.mikrokosmos/. The following sections explain in more detail each step.

### 2.1 Mikrokosmos

Mikrokosmos is installable from Hackage. The Haskell tools cabal and stack can be used to install it. For instance, it can be installed using cabal with the following commands.

cabal update
cabal install mikrokosmos


Binaries are also available from the GitHub repository. Manual installation is also available by cloning the git repository and using cabal or stack as follows.

git clone https://github.com/mroman42/mikrokosmos.git
cd mikrokosmos
cabal install

git clone https://github.com/mroman42/mikrokosmos.git
cd mikrokosmos
stack install


The ghc compiler can be also directly used from its version 8. In that case, you must check that all the Haskell libraries on the following list are available first.

git clone https://github.com/mroman42/mikrokosmos.git
cd mikrokosmos
ghc Main.hs


Please refer to the mikrokosmos.cabal file for an updated list of libraries.

Library Version
base >=4.7 && <5,
mtl >=2.2,
containers >= 0.5,
parsec >=3,
ansi-terminal any
multimap any
HUnit >=1.0
options any
tasty any
tasty-hunit any
tasty-quickcheck any
tasty-golden any
directory 1.0

### 2.2 Libraries

Mikrokosmos comes bundled with a collection of lambda calculus libraries. They will be directly loaded into the interpreter unless you specify the flag --no-libs.

Apart from these, you can write your own libraries and load them onto the interpreter. A library consists on a file with extension .mkr and containing definitions. The following is an example of the contents of a file ski.mkr.

# SKI combinators
S = \x.\y.\z. x z (y z)
K = \x.\y.x
I = S K K

# Schönfinkel combinators
C = \f.\x.\y.f y x
B = \f.\g.\x.f (g x)
W = \x.\y.(y y)

# Y combinator
Y := \f.(\x.f (x x))(\x.f (x x))


To load a library into the interpreter, it must be placed under your home directory in a folder called ~/.mikrokosmos, or, more easily, in the current working directory. Mikrokosmos will automatically look for libraries on the following list of directories. The :load <filename> command must then be used to load the library <filename>.mkr.

~/
~/.mikrokosmos/
./
./lib/


### 2.3 Jupyter kernel

Please note that the Jupyter kernel does not correctly displays Unicode characters on Windows. This is a known bug, see Issue #49.

Using pip, the Jupyter kernel can be installed with the following command. Use the flag -U if you want to update a previous install to the latest version.

pip install imikrokosmos


Development versions of the kernel can be installed from its GitHub repository. For instance, you can use the following command if you want to download the source from the master branch of the repository.

pip install git+https://github.com/mroman42/jupyter-mikrokosmos.git@master


#### 2.3.1 Manual installation

It is possible to install the kernel by simply copying the files under jupyter-mikrokosmos on their respective directories. Three files are needed for it to work.

• The files kernel.js and kernel.json must be placed in one of the following paths.

~/.local/share/jupyter/kernels/jupyter-mikrokosmos/
/usr/share/jupyter/kernels/jupyter-mikrokosmos/


This can be achieved automatically with the jupyter kernelspec command.

cd mikrokosmos
sudo jupyter kernelspec install jupyter-mikrokosmos

• The mikrokosmoskernel.py file must available for Python on the working directory, or, more generally, on the sys-path. This can be achieved simply by installing it with pip as

sudo pip install imikrokosmos


You can check the value of sys.path on your Python interpreter and copy manually the script to one of those locations.

## 3 First steps

### 3.1 First steps on the command line

Once installed, the interpreter can be opened by executing mikrokosmos in a terminal. It will open a prompt where lambda expressions can be evaluated.

You can write expressions using \x. to denote a lambda abstraction on the x variable and you can bind names to expressions using =. As you can see in the image, whenever the interpreter finds a known constant, it labels the expression with its name. You can type :help into the prompt to get a summary of the available interpreter commands.

Commands available from the prompt:
<expression>		 evaluates the expression
:quit       		 quits the interpreter
:restart    		 restarts the interpreter
:verbose <on/off> 	 sets verbose mode on/off
:color <on/off> 	 sets color mode on/off
:ski <on/off> 		 sets ski mode on/off
:types <on/off> 	 untyped/simply typed lambda calculus
:help       		 shows this help


### 3.2 First steps with the Jupyter kernel

If Jupyter and the Mikrokosmos Jupyter kernel are installed, a Jupyter Notebook server can be started by executing jupyter notebook. You can see more information in how to use a Jupyter notebook server here.

Once a notebook is opened, the mikrokosmos kernel can be selected in the menu Kernel > Change kernel > IMikrokosmos. Multiple expressions can be evaluated in one cell but each line will be evaluated separately.

## 4 The Mikrokosmos interpreter

In both cases, the interpreter will evaluate commands and expressions line-by-line. In the follwing sections we describe the options that are available both to the command line interpreter and to the Jupyter kernel. Remember that the :help option will show all the available commands.

### 4.1 Verbose mode

The verbose mode makes the interpreter show every step on the beta reductions it performs internally using De Bruijn notation, coloring the next substitution at each step. It can be activated and deactivated by executing :verbose on and :verbose off.

### 4.2 SKI mode

Every lambda expression can be written in terms of SKI combinators, and Mikrokosmos implements a translation algorithm. The SKI mode shows every result also in terms of SKI combinators. It can be activated and deactivated by executing :ski on and :ski off.

### 4.3 Color mode

Colored output will be activated by default. It can be deactivated however by executing :color off in the interpreter; :color on will activate the color again.

### 4.4 Simply typed λ-calculus

Mikrokosmos uses untyped λ-calculus by default, but you can activate the simply-typed λ-calculus interpreter using :types on. The syntax is the same on both; but there are some differences with the untyped version.

• Every term will be displayed along with its type.
• Non-typeable expressions will not be computed. Instead, an error message will be shown if a type cannot be found for the expression.
• Every computation terminates. Non-terminating computation cannot be expressed on simply-typed lambda calculus.

In particular, Mikrokosmos can be used to check proofs on intuitionistic propositional logic.

To this end, implementation of simply typed λ-calculus is presents the following type constructors.

Types Description
A → B Function type, from elements of type A to elements of type B.
A × B Product type, pairs of elements of type A and B.
A + B Union type, disjoint union of A and B.
Unit type, with only one element.
Empty type.

You can directly use these types via the built-in term constructors INL, INR, ABORT, UNIT and (-,-). However, it is recommended to load the standard library and work with the type constructors already available on the library.

## 5 The Mikrokosmos language

The Mikrokosmos language aims to be a minimal implementation of pure lambda calculus. A tutorial on the language in Jupyter notebook format can be found on tutorial.ipynb, but the online tutorial, already covers most of the features and it requires no installation. The following sections briefly describe the language.

### 5.1 Lambda expressions

A lambda abstraction $$(\lambda x. x)$$ can be written as \x.x. Combinations of alphanumeric characters are valid variable names. Application of lambda expressions is written by yuxtaposition. For example, the following code applies the identity function to itself.

(\a.a)(\a.a)


### 5.2 Bindings

A expression can be evaluated and the results of the evaluation bound to a name using the = operator. Sometimes, we do not want to evaluate the expression before the binding, maybe because it is a non-terminating computation. In these cases, we can use the := operator to assign a value to a variable without evaluating it.

plus = \a.\b.\c.\d.((a c) ((b c) d))
fix != \f.(\x.f (x x))(\x.f (x x))


Comments must always start with a # and they always expand for one line. This rule applies to the interpreter, the libraries and the Jupyter notebook.

### 5.4 Type constructors

When the :types option is on, and the standard library is loaded, a predefined set of type constructors and destructors are allowed on λ-expressions.

Constructor Type
(-,-) A → B → A × B
inl A → A + B
inr B → A + B
unit ⊤
caseof (A + B) → (A → C) → (B → C) → C
abort ⊥ → A
absurd ⊥ → ⊥

## 6 Standard libraries

Apart from the standard libraries. You can write your own libraries and place them under the ./lib folder. The installation of libraries has been discussed here.

The reserved keyword :load is used to load libraries. If the library name is foo.mkr, it will be loaded using :load foo. Note that the standard libraries are loaded by default if the flag --no-libs is not activated. You can write your own libraries as *.mkr files. Each line of a library is interpreted in the same way it would be interpreted on the command line interpreter.

You can check the definitions of the standard library and examples on this repository. There are representations of structures such as linked lists or trees in the standard library.

## 7 Mikrokosmos in the classroom

The recommended route for educators wanting to use Mikrokosmos in the classroom is to deploy the Mikrokosmos online interpreter on a web page accessible by the students. This avoids the need for a manual installation of the Haskell building tools for each one of the students. An alternative route is to install the Mikrokosmos Jupyter kernel into a Jupyter server accessible by the students.

In order to embed Mikrokosmos into your web page, you need the following code in your HTML header.

<script src="https://cdnjs.cloudflare.com/ajax/libs/codemirror/5.29.0/codemirror.min.js"></script>
<script src="https://mroman42.github.io/mikrokosmos-js/codemirrormkr.js"></script>
<script language="javascript" src="https://mroman42.github.io/mikrokosmos-js/rts.js"></script>
<script language="javascript" src="https://mroman42.github.io/mikrokosmos-js/lib.js"></script>
<script language="javascript" src="https://mroman42.github.io/mikrokosmos-js/out.js"></script>
<script language="javascript" src="https://mroman42.github.io/mikrokosmos-js/runmain.js"></script>
<script language="javascript" src="https://mroman42.github.io/mikrokosmos-js/mikrobox.js" defer></script>


Once this code is added, the mikrokosmos Javascript function will be available. You can create a Mikrokosmos code pad on your document as follows. As an example, assume we are preparing a lesson on the SKI combinators, and we want the students to describe the natural numbers in terms of SKI combinators. We could add the following pad into our web page.

<div class="mikrojs-console">
<script type="text/mikrokosmos">
:ski on
S = λa.λb.λc.(a c) (b c)
K = λa.λb.a
I = λa.a

# Exercise. Write the natural numbers in terms of lambda
# expressions using Church encoding. The interpreter will
# translate them into SKI combinators. Describe the pattern the
# natural numbers follow in terms of SKI combinators.

</div>


An example of this usage can be found in the tutorial.

## 8 Support and contributions

If you are an user seeking support, please use the GitHub issue tracker of the main repository. Any bug, failure, or unintended behaviour of the program, specially if it differs from the behaviour described on the user's guide, should be reported on the issue tracker. When doing so, please try to include all the relevant information, the steps you may take to reproduce the error, and the platform and version of Mikrokosmos you were using.