Euler Reasoning Tool

General commands

Note: Run the new command line interface starts with euler2 while the old command line interface start with euler. To see all EulerX commands, please use

• euler -h
• euler2 -h

The basic commands (run taxonomy alignment problem) are

• euler -i INPUT.TXT [-e ENCODINGS] [--reasoner=REASONER] [OTHER_OPTIONS]
• euler2 align INPUT.TXT [-e ENCODINGS] [-r REASONER] [OTHER_OPTIONS]

Consistent example: abstract4

• Input visualization

• euler -i abstract4.txt --iv

• euler2 show abstract4.txt iv

• Get all possible worlds (individual pw view and summary view) and polynomial encoding:

• Using DLV:

  • euler -i abstract4.txt -e mnpw
  • euler2 align abstract4.txt -e mnpw

• Using Potassco:

  • euler -i abstract4.txt -e mnpw --reasoner=gringo
  • euler2 align abstract4.txt -e mnpw -r gringo

• Support combined concept:

• euler -i abstract4.txt -e mncb

• euler2 align abstract4.txt -e mncb

Inconsistent example: coverage

• Black-box inconsistency repair

• euler -i coverage.txt -e mnpw --repair=topdown

• euler -i coverage.txt -e mnpw --repair=HST, this will give you MIS

• euler2 align coverage.txt -e mnpw --fouineone, this will give you MIS, MCS, MUS, MAA

• Mixed black-box/white-box inconsistency repair, this will give you MIS

• euler -i coverage.txt -e mnpw --ie

• euler2 align coverage.txt -e mnpw --ie

Use-cases of examples

Input visualizations (Examples abstract4):

The input txt file:

taxonomy 1 Taxonomy1
(A B C D E)

taxonomy 2 Taxonomy2
(A B D F G)

articulation abs abstract
[1.A {is_included_in equals} 2.A]
[1.B equals 2.B]
[1.C {is_included_in overlaps} 2.F]
[1.D equals 2.D]
[1.E {equals overlaps} 2.G]

The input visualization:

The command line:

• euler -i abstract4.txt --iv
• euler2 show abstrac4.txt iv

Inconsistency analysis (Example [coverage]

Here is an example of two inconsistent taxonomies (the set of articulations have conflict(s)). (https://github.com/EulerProject/EulerX/blob/main/example/coverage/coverage.txt))

By performing a model-based diagnosis, we can find the minimal inconsistent subset (MICS), which shows the smallest subset of articulations that are inconsistent, and maximal consistent subset (MCS), which directs us to how to solve the inconsistency. The command line

• lattice.sh coverage.txt

The lattice visualization

In the above figure, solid red octagons and solid green boxes, in order, denote MIS and MCS. Similarly, dotted red octagons and green boxes show inconsistent and consistent sets, respectively. The solid blue lines show the transition from a consistent set to an inconsistent one, and the dotted green/blue arrows show the transition between consistent and inconsistent sets of articulations. In the current example, {1,2,3} are the minimal inconsistent set of articulations. To resolve the inconsistency, following the solid blue lines, we can choose either of the three solid green rectangles denoting sets {1,2}, {1,3}, and {2,3}.

In the new command line interface, we can use the "fourinone" command:

• euler2 align coverage.txt --fourinone

The four-in-one lattice visualization

The four-in-one lattice will give you a powerset lattice which shows four special knowledge products here, including:

• MIS: Minimal Inconsistent Subsets (red box)
• MCS: maximal Consistent Subsets (green box)
• MUS: Minimal Unique (articulation) Subsets (green octagon)
• MAA: Maximal Ambiguous Articulation (sets) (cyan diamond)

Computing single MIS is necessary to compute all MIS, then consider about a lattice approaches for a use-case. If there are n articulations, the lattice will have 2^n nodes, each node is a combination of articulations while each edge is a subset relation between the two sets. This lattice we called it Diagnostic Lattice. We color each consistent set of node as green, while color each inconsistent set of node as red. Based on propagation rule, we know any subsets of a green node is green and any supersets of a red node is red. So, each MIS is a red node whose offsprings are green and each MCS is a green node whose ancestors are red. Not only inconsistency analysis, we can use lattice approach in ambiguous analysis approach. Given n articulations, instead of looking whether the result is logically consistent, we look at the number of possible worlds output. We assert the result is unique when there is only one possible world while it is ambiguous when there is two or more. This is still a diagnostic lattice and is conceptually similar to inconsistency analysis lattice. The MIS and MCS in inconsistency lattice can be regarded as MUS, short for Minimal Unique (articulation) Subsets, and MAA, short for Maximal Ambiguous Articulations (sets).

The consistent example (Example abstract4)

Individual / summary-view possible worlds visualization The command line:

• euler -i abstract4.txt -e mnpw
• euler2 show pw after running alignment problem

The visualization of summary-view of “all worlds”

Possible worlds cluster The command line:

• euler -i abstract4.txt -e mnpw --pwcluster --simpcluster
• euler2 show sv showing summary view, after running alignment problem and showing possible worlds