Towards Characterizing Strict Outerconfluent Graphs
Bibliographic record
Abstract
CDEG, "Computerized Diagrammatic Euclidean Geometry," is a computerized formal system for giving diagrammatic proofs in Euclidean geometry which uses planar graphs in drawing its diagrams.Here we discuss some of the graphtheoretic problems that arise in this context.This computer proof system implements a diagrammatic formal system for giving diagram-based proofs of theorems of Euclidean geometry that are similar to the informal proofs found in Euclid's Elements.The theoretical ideas underlying this system and the original version of CDEG are described in detail in the book Euclid and his Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry [4].A much updated version of CDEG is now publicly available at [1].Interested readers are encouraged to download CDEG and to try it out for themselves.When we say that CDEG is a diagrammatic computer proof system, this means that it allows its user to give geometric proofs using diagrams.Internally, CDEG represents a diagram abstractly as a planar graph along with some additional information about how elements of the graph relate to the geometric objects they represent.The nodes in the graph represent points in the plane, while the edges represent line segments and arcs of circles.In general, one diagram drawn by CDEG can actually represent many different possible collections of lines and circles in the plane.What these collections all share, and share with the diagram that represents them, is that they all have the same planar topology.This means that any one can be stretched into any other.So, for example, a diagram containing a single line segment represents all possible single line segments in the plane, since any such line segment can be stretched into any other.Two sample CDEG diagrams are shown in Fig. 1.The first diagram represents a circle drawn with a point inside of it and two points on its circumference.The circle is drawn in a way that appears to be rectangular rather than circular, but recall that all we care about here is the topology of the diagram.The second diagram, which occurs in the proof of Euclid's Proposition 1, shows an equilateral triangle inscribed in the intersection of two circles.In CDEG diagrams, dotted lines represent circles while solid lines represent straight lines, and all nodes, edges, and regions of the underlying graph are labeled for reference by numbers drawn in boxes.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".