# The MMT Language and System

In this tutorial, we build an implementation of first-order logic (FOL) in MMT.

It assumes that

• MMT has been installed,
• archives are placed in some folder, which is refered to as CONTENT,The MMT installer should take care of this
• jEdit is used for editing mmt files.Other editors will work but might make editing awkward.

### Define a Meta-Logic

Actually, we will skip this step for now because only advanced users need to define their own meta-logic.

Instead, we can choose one of the meta-logics already defined in MMT. The most important example is the logical framework LF, whose MMT URI is http://cds.omdoc.org/urtheories?LF.

This theory is defined in the archive MMT/urtheories, which is automatically cloned into the folder CONTENT when setting up MMT.

Other important meta-logics defined in this archive include

• http://cds.omdoc.org/urtheories?PLF extends LF with shallow polymorphism
• http://cds.omdoc.org/urtheories?LFModulo, which extends LF with a rewrite system

These can also be combined.

For FOL, LF is sufficient as a meta-logic.

### Define the Formal System in an Appropriate Meta-Logic

We create a new file fol.mmt, in which we will define FOL.

#### Creating the File

It does not matter where this file is located. But to allow for running MMT build targets over it later, it is convenient to place it in a new MMT archive:

• create a folder CONTENT/myarchive
• create a file MANIFEST.MF in it containing (at least) the lines

id: myarchive
narration-base: http://mydomain.org/myarchive

• and create a file source/fol.mmt in it

The id is some string that is used to refer to the archive. The narration base is some URI that is used as the default namespace for forming unique identifiers for the content of the archive.

#### Create a Theory for FOL

Open the file in jEdit and create an empty theory:

namespace http://mydomain.org/mmt-example [GS]

theory FOL : http://cds.omdoc.org/urtheories?LF =

[GS]


Here

• The namespace declaration defines a unique namespace (a URI) for our example.The URI does not have to be a URL, i.e., it does not have to point to a physical location. It only acts as a unique identifier.
• The theory introduces an MMT theory called FOL with meta-theory http://cds.omdoc.org/urtheories?LF.Alternatively, we can write ur:?LF because the namespace prefix definition import ur http://cds.omdoc.org/urtheories [GS] is implicitly present.
• [GS] refers to ASCII 29, the toplevel delimiter used by MMT. In jEdit, it can be inserted via the symbol button for it or by typing jGS .

#### Define the Syntax and Semantics of FOL

From this point on forwards, defining a language proceeds according to LF.

The details of doing so are given in the self-documenting mmt files in the archive CONTENT/MMT/examples, which has been cloned automatically when setting up MMT.

It contains the full definition of FOL in the file tutorial/1-sfol.mmt. To complete this step of the tutorial, copy over the declarations in this file step by step. (Note that this tutorial makes you use a different namespace than the that file. That’s important to make sure all declarations have unique URIs.)

To insert Unicode symbols, you can use LaTeX commands with j instead of . For example, jrightarrow inserts \rightarrow. And because that one is so important, jra is a shortcut for it. The complete list of preconfigured abbreviations can be found in the section Abbreviations of the jEdit settings.

#### View the Logic in the Browser

We can start the MMT web server from within jEdit to view our definition in the browser:

#### Build and Serve the Archive

To build your archive, use the command build myarchive mmt-omdoc fol.mmt. This will build all mmt files in your archive and convert them into omdoc (which is the XML format that MMT uses internally, corresponding to the binary files produced by a compiler). The omdoc files are put into the folders narration (one omdoc file per source file) and content (one omdoc file per module in those source files). Additionally, index files for fast querying are placed in the folder relational.

You can build the archive from within jEdit (by using the jEdit console as above).

Alternatively, you can build it directly from the MMT shell without involving jEdit. Start the shell by running mmt.jar. Then type

build myarchive mmt-omdoc fol.mmt
server on 8081


The shell is completely independent from jEdit (but works with the same archives). Therefore, we use a different port number above.

Eventually, you can leave the shell by typing exit.

### Use FOL to Build a Small Algebra Library

We can now apply our defined system to formalize some knowledge. We pick algebra as an example.

#### Formalize the Library

Create a file algebra.mmt in your archive and put

namespace http://mydomain.org/mmt-example [GS]

theory Semigroup : ?FOL =

[GS]


This creates a new theory for semigroups, this time using our new theory FOL as the meta-theory. Because both theories are in the same namespace, we can use a relative URI to refer to http://mydomain.org/mmt-example.

As for FOL, we do not give the details of the formalization here. Instead, we refer to the self-documenting file tutorial/2-algebra.mmt in the archive CONTENT/MMT/examples.

#### Build and Serve the Library

Like before, you can now

• build the library using build myarchive mmt-omdoc algebra.mmt
• serve it using server on 8081.