thesis-projects issueshttps://gl.kwarc.info/kwarc/thesis-projects/-/issues2021-10-21T16:01:27Zhttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/11Theory Exploration2021-10-21T16:01:27ZFlorian RabeTheory ExplorationBy now MMT is strong enough to generate values of arbitrary types and compute with them.
This can be used for a generic theory exploration algorithm: Generate a formula, generate some values, test the formula.
If the formula is true for...By now MMT is strong enough to generate values of arbitrary types and compute with them.
This can be used for a generic theory exploration algorithm: Generate a formula, generate some values, test the formula.
If the formula is true for all values, it may be a theorem and is marked as such.
Later a human or a theorem can inspect the generated theorems and add the proofs.
----
For example, consider addition on the natural numbers with +, 0, and =.
MMT can generate formulas by
* using some (e.g., up to 3) free variables x,y,z
* systematically generate well-typed terms from them such as x, y, z, x+y, y+x, x+z, z+x, x+z, y+z, z+y, x+0, 0+x, and so on
* systematically generate values of N such as 0, 1, 3, 10, 13
* plug in all combinations of the values for x, y, z into the generated terms
* whenever two terms t(x,y,z) and t'(x,y,z) that agree on all values and conjecture the theorem forall xy.t(x,y,z)=t'(x,y,z)
Many important theorems can be guessed in this way, and for not-well-understood-yet theories, it's plausible that will include some new theorems.
----
There is already prior work on conjecture generation, historically by Simon Colton (look it up) and more recently by Josef Urban, Cezary Kaliszyck. The latter is quite compatible with this idea but is on different corpora.Florian RabeFlorian Rabehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/46Representing Game Levels in MMT (Project or Bachelor Thesis Topic)2021-09-02T12:10:45ZRichard MarcusRepresenting Game Levels in MMT (Project or Bachelor Thesis Topic)In the FrameIT project, we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
We visualize problems or puzzles in the form o...In the FrameIT project, we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
We visualize problems or puzzles in the form of interactive *scrolls*, where players can input their gathered knowledge to obtain more information and, eventually, solve the problem.
We want to have the conditions to solve the game level within MMT!
[WIP, description needs update]
das Konzept von Levels in MMT irgendwie repräsentieren (ähnlich wie wir Scrolls in MMT auch in einer strukturierten Art und Weise repräsentieren)
in Metainformationen sowas repräsentieren wie "the level is solved iff dist(C, A) evaluates to a numeric value"
Insb. könnte mensch sich überlegen, dass es durchaus Levelziele geben wird, die nur Unity überprüfen kann, die MMT-extern sein müssen.
Die könnte mensch trotzdem in MMT repräsentieren a la "level is solved iff [some condition mmt can check internally] and [some condition Unity needs to check]"
Geht in Richtung "flexiformale Repräsentation von Levelzielen in MMT"
For more information, please take a look at the [project issue](https://github.com/UFrameIT/UFrameIT/issues/18) and #1.https://gl.kwarc.info/kwarc/thesis-projects/-/issues/45Meta-Data in MathHub2021-03-31T12:55:24ZFlorian RabeMeta-Data in MathHub@mkohlhase @twiesing
MathHub is a huge library of mathematical libraries.
The underlying content format OMDoc supports metadata, which are currently not used in MathHub for display, cross-referencing, etc.
The infrastructure currently...@mkohlhase @twiesing
MathHub is a huge library of mathematical libraries.
The underlying content format OMDoc supports metadata, which are currently not used in MathHub for display, cross-referencing, etc.
The infrastructure currently also does not generate good cumulative statistics about the data that is valuable for display and other user interactions.
This topic will design how such metadata should be efficiently computed and stored and how it can be used to enhance the user experience.https://gl.kwarc.info/kwarc/thesis-projects/-/issues/43FrameIT in the Real World2021-01-29T13:30:22ZRichard MarcusFrameIT in the Real WorldIn the FrameIT project (#1 and https://uframeit.github.io), we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
For even ...In the FrameIT project (#1 and https://uframeit.github.io), we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
For even better application to real world problems, we want to add support for augmented reality versions with Mobile AR or devices like the hololens.
Therefore, one of the main tasks will be to capture the geometry of the environment and make it possible to place points and measure information based on this.https://gl.kwarc.info/kwarc/thesis-projects/-/issues/41Linking Formalized Situations to specific Game Objects/Situations2020-12-20T09:26:44ZRichard MarcusLinking Formalized Situations to specific Game Objects/SituationsIn the FrameIT project (#1), we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
Based on parametric situation theories, ...In the FrameIT project (#1), we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
Based on parametric situation theories, which are instantiated by "pretty small Unity scripts", we want to link the parameters to specific points in the world modeled in Unity.
Our goal here is to create a toolkit for game developers that allows them to integrate the formalized knowledge into the games in the form of modular elements that can linked to game objects.
Work Areas:
- 1. Making the formalized knowledge available to the Unity Editor itself (https://learn.unity.com/tutorial/editor-scripting#), which could enable interactions like this:
- list available fact types so that developers can select which gadgets can create these facts
- list available scrolls so that developers can build levels based on the problems defined in the scrolls
- 2. Scripting the generation of semantic game objects
- we could define certain object attributes on the knowledge side like "base_tree"
- if the developer assigns this attribute to an object in the game it should then be enhanced automatically: place it in a 90 degree angle to the ground, generate snapzones for root and top, configure colliders
- 3. Advanced Toolkit Interactions based on 1 and 2, e.g., if a Fact is connected to a Gadget, we know which Gadget should be used to create a certain fact for a solution
- 4: Building a graphical tool for this. Unity recently acquired [Bolt](https://assetstore.unity.com/packages/tools/visual-scripting/bolt-163802). Maybe it could be integrated into this, maybe we want/need our own editor, needs investigation.Michael Kohlhasemichael.kohlhase@fau.deMichael Kohlhasemichael.kohlhase@fau.dehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/27OEIS importer and extension for MathDataHub2020-12-08T09:05:50ZMichael Kohlhasemichael.kohlhase@fau.deOEIS importer and extension for MathDataHub[MathDataHub](http://data.mathhub.info) (MDH) is a deep FAIR system for mathematical data sets. Currently, it has mostly been used for systematic data sets like the [Small Groups Library](https://data.mathhub.info/collection/smallgp/).
...[MathDataHub](http://data.mathhub.info) (MDH) is a deep FAIR system for mathematical data sets. Currently, it has mostly been used for systematic data sets like the [Small Groups Library](https://data.mathhub.info/collection/smallgp/).
Having the [OEIS](http://oeis.org) (Encyclopedia of Integer Sequences) in [MathDataHub](http://data.mathhub.info) would be a major undertaking and a major success. Some of this we have already done, see [E. Luzhnica and M. Kohlhase (2016) Formula semantification and automated relation finding in the OEIS. ICMS 2016](http://kwarc.info/kohlhase/papers/icms16-oeis.pdf). This needs to be made sustainable and much more complete. [MathDataHub](http://data.mathhub.info) and [MathHub](http://mathhub.info) are the right tools/targets for this. We have to
* get our hands on the OEIS dumps - KWARC has the necessary contacts - and build an infrastructure to keep the content synced.
* build a parser for the OEIS that is as lossless as possible
* formalize the meta-theory of OEIS in MMT.
* Build the MDDL Schema theories for the OEIS
* (optionally) Build specialized User Interfaces on top of MDH.
* (optionally) build a programmatic API for the OEIS.
Clearly this is a project for a whole group, and the later points are optional.
One of the high proints of this project is that there is a good chance that there will be funding (and possibly future employment options via the [NFDI](https://de.wikipedia.org/wiki/Nationale_Forschungsdateninfrastruktur) KWARC is part of the [MaRDI Consortium](http://wias-berlin.de/mardi/).Michael Kohlhasemichael.kohlhase@fau.deMichael Kohlhasemichael.kohlhase@fau.dehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/30SUMO to OMDoc/MMT exporter.2020-12-08T09:04:14ZMichael Kohlhasemichael.kohlhase@fau.deSUMO to OMDoc/MMT exporter.The Suggested Upper Merged Ontology [SUMO](http://www.adampease.org/OP/) is an upper ontology that lightly formalizes common sense situations. It woucd be very nice to have a translator of SUMO into MMT Theories. For that we need
* A me...The Suggested Upper Merged Ontology [SUMO](http://www.adampease.org/OP/) is an upper ontology that lightly formalizes common sense situations. It woucd be very nice to have a translator of SUMO into MMT Theories. For that we need
* A meta-theory for the KIF (Knowledge Interface Format)
* a translator for the [SUMO data](https://github.com/ontologyportal/sumo) to OMDoc/MMT.
SUMO seems licenced mostly under GPL, so we are allowed to do this and even republish it on MathHub (as long as we keep the GPL).
This would be a good basis for JLogic.Max RappMax Rapphttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/35FrameIT State Interface2020-12-03T17:45:48ZRichard MarcusFrameIT State InterfaceIn the FrameIT project (#1), we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
Since (formalized) knowledge is generate...In the FrameIT project (#1), we use the Unity game engine in combination with MMT to have access to mathematical knowledge management techniques and interact with the knowledge in the game world.
Since (formalized) knowledge is generated interactively during runtime, it is difficult to debug with existing MMT utilities.
Accordingly, we need adaptions to make use of these and new concepts to interact with the generated content on the programming language level.
Based on this, we would like to have an interface that helps developers to monitor the state of the underlying knowledge during gameplay.Michael Kohlhasemichael.kohlhase@fau.deMichael Kohlhasemichael.kohlhase@fau.dehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/40Configure WissKI instances from symbolic metadata2020-11-24T09:32:20ZMichael Kohlhasemichael.kohlhase@fau.deConfigure WissKI instances from symbolic metadataAt the moment, the [WissKI](https://wiss-ki.eu) instances created by the [WissKI Distillery](https://wisski.agfd.fau.de) are very elementary. It would be good if the user could configure them to their needs and then have them automatical...At the moment, the [WissKI](https://wiss-ki.eu) instances created by the [WissKI Distillery](https://wisski.agfd.fau.de) are very elementary. It would be good if the user could configure them to their needs and then have them automatically pre-configured. This project has the following parts:
* determine configuration options and their values; examples are
* desired modules,
* theming
* data for the DSGVO pages (impressum and such)
* admins, users (for automated account creation)
* data testament (what happens after the project is over; see https://gitlab.cs.fau.de/AGFD/wisski/-/issues/24) and its documentation page.
* develop a JSON schema for storing the choices.
* develop a configuration dialog that guides users through the data collection process, documenting the options and previewing the consequences of the choices.
* develop a configuration process that implements the configuration in a new WissKI instance.
* (possibly) allow to change and re-configure the instance.Tom WiesingTom Wiesinghttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/38Integrate Theorem provers into MMT2020-07-30T11:53:04ZMichael Kohlhasemichael.kohlhase@fau.deIntegrate Theorem provers into MMTOMDoc/MMT allows to represent a wide range of logics and reasoning calculi as (meta)-theories via the Curry/Howard isomoprphism. The MMT system supplies automated type- and this proof-checking for all of these out of the box. But it does...OMDoc/MMT allows to represent a wide range of logics and reasoning calculi as (meta)-theories via the Curry/Howard isomoprphism. The MMT system supplies automated type- and this proof-checking for all of these out of the box. But it does not offer any proof automation.
For some logics - e.g. first-order logic - we have very good proof automation - e.g. EProver, Spass, Vampire, the Hammers, ...
So we could get proof automation in MMT by just translating to the input language of these and when they can find a proof obtain back a proof certificicate (translating that into the existing ND calculus we can leave to a later stage).
1. This idea should be implemented in MMT for one instance (e.g. FOLEQ and EProver) so that we can increase proof efficiency (and proofs are the most important check of adequacy of formalizations).
2. Building on this, we want to develop a general infrastructure for more logics and provers
3. Finally, we should extend this to even more logics by using views induced by conservative extensions. These translate "super-logics" back down to the "source logics". If there is a prover for the "source logic", then we can use it in the "super-logic".
This would one of the most important extensions of the OMDoc/MMT ecosystem.
@dmueller @frabeMichael Kohlhasemichael.kohlhase@fau.deMichael Kohlhasemichael.kohlhase@fau.dehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/28Build an integrated SPARQL index/endpoint for MathHub2020-07-30T11:25:15ZMichael Kohlhasemichael.kohlhase@fau.deBuild an integrated SPARQL index/endpoint for MathHubWe are slowly building up capacities for [tetrapodal search](https://kwarc.info/people/mkohlhase/papers/cicm20-search.pdf) in [MathHub](https://mathhub.info), i.e. search capacities across mathematical aspects.
For that we need an inde...We are slowly building up capacities for [tetrapodal search](https://kwarc.info/people/mkohlhase/papers/cicm20-search.pdf) in [MathHub](https://mathhub.info), i.e. search capacities across mathematical aspects.
For that we need an index for the *organizational aspect* which can mostly be expressed in terms of RDF graphs and queried via SPARQL. We already have quite a lot of data from our theory graphs and theorem prover libraries; see [the ULO paper](https://kwarc.info/kohlhase/papers/cicm19-ulo.pdf).
Now we need a dedicated index and query engine for them integrated into [MathHub](https://mathhub.info). I am thinking of [GraphDB](https://www.ontotext.com/products/graphdb/) a powerful system that is free in the basic tier. And give it a good test-drive with our data.Tom WiesingTom Wiesinghttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/8Theory Graph Visualization and Interaction Front-End2020-05-01T23:34:39ZMichael Kohlhasemichael.kohlhase@fau.deTheory Graph Visualization and Interaction Front-EndThe central data structure of the OMDoc/MMT format and system is that of a theory graph, where nodes are theories and the edges are theory morphisms, (or alignments, see #7). We extensively use this structure in all of our services, but ...The central data structure of the OMDoc/MMT format and system is that of a theory graph, where nodes are theories and the edges are theory morphisms, (or alignments, see #7). We extensively use this structure in all of our services, but only have very limited means of actually seeing them and interacting with them (we currently generate static SVG presentations via graphviz.
The topic here is to build a client-side (i.e. in-browser) graph interface based on a javascript ligrary for graph layout and interaction (most probably [vis.js network](http://visjs.org). This interface should support
* interactive graph visualization: showing/hiding parts of a graph, highlighting certain features, clustering, ...
* inspection of theories and edges (by clicking on them)
* addition/deletion/update of nodes and edges (e.g. for alignments)
* graph export for didactic purposes.
* ... I am sure more ideas will pop up when we play with this ...Dennis MüllerDennis Müllerhttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/363D Theory Graph Visualization2020-05-01T23:34:36ZRichard Marcus3D Theory Graph VisualizationAs a follow up to #8, we have developed an interactive [3D graph viewer](https://github.com/UniFormal/tgview3d).
Based on this, there are several possibilities for further research:
- extending the virtual reality prototype with state-of...As a follow up to #8, we have developed an interactive [3D graph viewer](https://github.com/UniFormal/tgview3d).
Based on this, there are several possibilities for further research:
- extending the virtual reality prototype with state-of-the-art interaction concepts
- developing library-specific, specialized builds
- comparing different layout algorithms
- optimizing layout computationMichael Kohlhasemichael.kohlhase@fau.deMichael Kohlhasemichael.kohlhase@fau.dehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/3FrameIT for mathematical models.2020-04-27T14:44:15ZMichael Kohlhasemichael.kohlhase@fau.deFrameIT for mathematical models.Mathematical models (e.g. differential equations, ...) are used in many branches of social and natural sciences. There is talk (e.g. by [Karsten Tabelow](http://www.wias-berlin.de/people/tabelow)) of [building collections of math models...Mathematical models (e.g. differential equations, ...) are used in many branches of social and natural sciences. There is talk (e.g. by [Karsten Tabelow](http://www.wias-berlin.de/people/tabelow)) of [building collections of math models as a resource](
http://www.wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=2016&number=2267). I believe that the [FrameIT method](https://www.cicm-conference.org/2016/ceur-ws/W50.pdf) (see also #1) developed at KWARC is exactly the right framework for reprsesenting and applying such models, only that we do not need the game part. The main difference is that we want to explore the real world, not the game world; a main contribution of this thesis might be to design that kind of interaction interface that allows this. Maybe even a "search engine" that suggests math models given a "situation theory".
This application needs to be developed further, and may become a project proposal if successful.
cc: @dmuellerMichael Kohlhasemichael.kohlhase@fau.deMichael Kohlhasemichael.kohlhase@fau.dehttps://gl.kwarc.info/kwarc/thesis-projects/-/issues/31Lexicon Management in GLIF2020-04-20T14:32:19ZFrederik SchaeferLexicon Management in GLIF[**GLIF**](https://github.com/kwarc/GLIF) is a framework for describing the translation of natural language into logical expressions.
This requires the specification of a grammar, a target logic, a domain theory in that logic, and a sema...[**GLIF**](https://github.com/kwarc/GLIF) is a framework for describing the translation of natural language into logical expressions.
This requires the specification of a grammar, a target logic, a domain theory in that logic, and a semantics construction (mapping of parse trees into the domain theory).
If you participated in the LBS lecture, you should be familiar with the setup.
Adding a new word (like "woman") to a GLIF pipeline requires the following additions:
* abstract syntax: `woman_N : N;`
* concrete syntax: `woman_N = mkN "woman" "women";` (if e.g. German is also supported: `woman_N = mkN "Frau" feminine;`)
* domain theory: `woman : i -> o`
* semantics construction: `woman_N = woman`
There clearly is a lot of repetition here.
Your task would be to improve this by designing a lexicon format from which the necessary files can be generated automatically.
A naive attempt to write a lexicon entry could look like this:
```
woman
noun
eng: "woman" "women"
ger: "Frau" feminine
```
We probably want customization in different places. For example, the semantics construction for the name John might be `john_PN = john` or `john_PN = [P] P john` depending on the context. Also, not every project uses the resource grammar library, so the operations used in the concrete syntax might vary.
The lexicon management should also be supported in GLIF's Jupyter front-end.
It would also be interesting to take existing lexica and generate a generic lexicon from that, which can be imported into any projects (with the required customization to make it fit in).https://gl.kwarc.info/kwarc/thesis-projects/-/issues/26More MathWebSearch Instances2020-03-20T07:45:59ZMichael Kohlhasemichael.kohlhase@fau.deMore MathWebSearch InstancesThere are numerous mathematical resources that need [formula search engines](https://search.mathweb.org). Given Christoph Alt's thesis and MWS front-end this is mostly a matter of writing a good harvester. Some examples that come to mind...There are numerous mathematical resources that need [formula search engines](https://search.mathweb.org). Given Christoph Alt's thesis and MWS front-end this is mostly a matter of writing a good harvester. Some examples that come to mind
* The [Stacks Project](https://stacks.math.columbia.edu/) 7000 pages of algebra in LaTeX; see https://github.com/MathWebSearch/frontend/issues/18
* the [OEIS](https://oeis.org), We had a search engine for the formulae at one point, we just need to resurrect and modernize it (see #27)
* ...
And the best would be to create a joint index where MWS searches all of these together. https://gl.kwarc.info/kwarc/thesis-projects/-/issues/25Intellij support for MMT-jupyter-notebooks2020-01-03T09:11:22ZDennis MüllerIntellij support for MMT-jupyter-notebooksSee https://github.com/UniFormal/IntelliJ-MMT/issues/28See https://github.com/UniFormal/IntelliJ-MMT/issues/28https://gl.kwarc.info/kwarc/thesis-projects/-/issues/20Representing Relational Databases in a Logical Framework2019-10-08T11:08:32ZFlorian RabeRepresenting Relational Databases in a Logical Framework@kohlhase @twiesing
There is a very elegant embedding of relational databases into MMT.
* We write a theory D that declares all the basic datatypes (integers, string, products, etc.) of the database language.
* A table schema is repres...@kohlhase @twiesing
There is a very elegant embedding of relational databases into MMT.
* We write a theory D that declares all the basic datatypes (integers, string, products, etc.) of the database language.
* A table schema is represented as a theory S with meta-theory D. The constants of S correspond to the columns of the table.
* Local consistency conditions (which are often omitted in relational databases) are represented as axioms in S.
* A table entry in S is represented as a morphism S -> D, i.e., entries are records whose fields are given by S.
* A join is represented as a pushout.
For example, the join of S with S' where the field f of S should be equal to the field f' of S', is the pushout of {f} -include-> S and {f} -{f=f'}-> S'.
The entries of the join are the set of universal morphisms out of the pushout for every pair (e,e') of entries for which the respective diagram commutes.
* A database view providing a table S backed by a table T are MMT morphisms m is represented as a morphism S -> T.
As a special case, include morphisms represent the selection of some columns from a table.
The entries of the view are the morphisms m;e for all entries e of T.
* A writable view providing S backed by T allows modifying an entry of T if the change can be propagated back to the corresponding entry of T.
If T is a declared theory, this is possible if m is a renaming/inclusion.
If m is a view or join, this is more complicated.
The goal of this topic is to
* work out the details of the above
* investigate the propagation of changes along writable views in general
* determine how these representations can be used to provide added value to database applications
In particular, we could represent the schema of a database in MMT (as sets of theories and views).
The entries would be stored in an actual database.
The theory D can use high-level datatypes, which MMT's codecs translate to the concrete types of the database.
Queries (like joins, selections, and views) can be formulated in MMT and translated to relational queries; conversely, entries can be translated back to MMT views.https://gl.kwarc.info/kwarc/thesis-projects/-/issues/24Identifying Mathematical Objects up to Canonical Isomorphpism2019-06-04T22:57:39ZFlorian RabeIdentifying Mathematical Objects up to Canonical Isomorphpism@mkohlhase @dmueller
# Motivation
Mathematics pervasively identifies objects if they are canonically isomorphic.
Examples:
* define the rationals as fractions (pairs of integers, quotiented by cancelling), then identify the integer z...@mkohlhase @dmueller
# Motivation
Mathematics pervasively identifies objects if they are canonically isomorphic.
Examples:
* define the rationals as fractions (pairs of integers, quotiented by cancelling), then identify the integer z with the fraction z/1
* the entire number hierarchy N <: Z <: Q etc. is built like above
* elements of ring R are identified with constant polynomials over R
* elements of ring R are identified with elements in localizations over R
* polynomials are identified with the respective elements in the field of fractions
* generators of a group are identified with elements of the generated group (which are technically equivalence classes)
It is very common that
* the isomorphism is from a smaller to a subset of a bigger structure
* the bigger structure is built from the smaller one (i.e., we cannot simply *define* the smaller one as a subset of the bigger one because we need it to build the bigger one)
* the embedding of the smaller into the bigger structure preserves some operations (e.g., the embedding N -> Z is a semiring morphism, the embedding of R into R[X] is a ring moprhism) and can therefore not simply be represented as just a function
* the set preserved properties is extended later on as additional properties are considered (e.g., the embedding N -> Z is also an order morphism)
Virtually all tools for formalized mathematics cannot handle this at all, let alone elegantly.
It requires fundamentally different formal systems that we have not designed yet.
# Idea
For a certain special case, theory morphisms may be a solution:
* We can define the structures as theories and the embedding as a theory morphism, e.g.,
* N={n: type, z: n, s: n->n}
* Z={i: type, z:i, s:i->i, p:i->i, s(p(x)=x, p(s(x)=x, leq:i->i->prop}
* emb: N->Z = {n={x:i|z leq x}, z=z, s=s}
In particular, the theory morphism would capture which operations are preserved.
In this topic, you build a case study formalizing canonical isomorphisms in this way.
You will identify and possibly solve any theoretical or practical problems that come up along the way.
# Technical Problems
## Weak Embeddings
Often the embedding function is not a theory morphism because it does not preserve all properties of a structure s.
Such embeddings can only be expressed as theory morphism out of S if S is an inadequately weak formalization of s.
Often the theory S amounts to specifying a category in which s is the initial object.
More generally, we can think of s as the structure that is freely generated by the syntax of S.
An important special case are non-injective embeddings.
For example, the embedding of the natural numbers into the theory of rings is not necessarily injective.
This can only be expressed as a theory morphism if the natural numbers are formulated without the injectivity of successor.
Similarly, we may have to allow for non-surjective embeddings if we cannot capture the image of the embedding as a predicate subtype (e.g., when embedding natural numbers into real numbers, where the logic cannot express the needed predicate).
In the case of natural numbers, this means the induction axiom should be optional as well.
## Models as Theories
There are multiple ways to represent mathematical structures (here called models) in a formal system.
*Concrete* models of theory T are given as tuples of their universes and operations, e.g., (N,0,1+,*) for the semiring of natural numbers.
(Technically, this tuple also includes proofs of all axioms.)
Concrete models can be represented in two different way:
* internal models: models are represented as record values of a record type corresponding to T
* external models: models are represented as theory morphisms out of T
*Abstract* models of theory T are represented as theories.
These theories contain undefined constants and axioms. Examples are the models for N and Z above.
Typically, these theories M extend T or admit very simpl morphisms, usually injective renamings, T -> M.
The critical idea of this case study is to use abstract models.
This is the only way that allow using theory morphisms for the embeddings.
It remains open how this relates to other, concrete, representations of models.https://gl.kwarc.info/kwarc/thesis-projects/-/issues/23Formalization of legal content2019-05-02T10:12:22ZMax RappFormalization of legal contentAs part of the ALMANAC-project, we are currently developing new MMT-modules to deal with conflict-laden content. These include context graphs (theory graphs with attack relations), formalizations of nonmonotonic logics and argumentation ...As part of the ALMANAC-project, we are currently developing new MMT-modules to deal with conflict-laden content. These include context graphs (theory graphs with attack relations), formalizations of nonmonotonic logics and argumentation theories (as context graphs or in classical MMT) and computation and visualization components (Embedding of argumentation semantic solvers, TGView graph viewer).
The goal of this project is a first application of these developments in an effort to formalize paradigmatic documents (rulings, example cases) in law. Legal content involves a plethora of interesting formalization challenges such as defeasible reasoning, ambibuity, precedence and argumentation. There for it provides an ideal arena to gain a sense of the current capabilities of our systems as well as the open requirements.
The project would entail choosing a suitable document (or collection of documents) and flexiformalizing it. The final document will include content of various formalization levels which can be visualized and processed by ALMANAC's graphical and computational tools. The undertaking would be closely supervised jointly by members of the ALMANAC-project and our Legal-Tech cooperation partner [Prof. Dr. Axel Adrian](https://www.str2.rw.fau.de/lehrstuhl/honorarprofessor/).