The Geometrician: a Computer Prototype of Problem Solving in Geometry Construction
Bibliographic record
Abstract
The Geometrician: a Computer Prototype of Problem Solving in Geometry Construction Edgar R. Acosta Villase˜ nor (avillase@connect.carleton.ca) Institute of Cognitive Science; 1125 Colonel By Drive Ottawa, ON K1S 5B6 Canada Rafael P´ erez y P´ erez (rpyp@servidor.unam.mx) Instituto de Investigaci´on en Matem´aticas Aplicadas y en Sistemas UNAM, Mexico, DF 04510 Mexico Keywords: creativity; problem solving; geometry. Introduction Most theories of creativity assume the interaction of two kinds of cognitive processes: the generation and evalua- tion of possible ideas (Sternberg & Lubart, 1999). P´erez y P´erez and Sharples (2001) described in great detail both processes in their Engagement-Reflection computer model of creativity (E&R model). Originally the model was developed with the aim of describing in detail a cog- nitive account of creative writing. As a way to improve the model, and to evaluate its potential for problem solving, a computer program based on the E&R model known as the Geometrician was implemented. The Geometrician The Geometrician solves geometry construction prob- lems in which, given some initial geometric objects (e. g. points, lines), new geometric objects are constructed employing only a straightedge and a compass. The E&R model, and its implementation in the Geometrician are outlined in this document. Engagement & Reflection The E&R model establishes that all knowledge struc- tures in the system are created from a set of previous solved problems provided by the user. Once these struc- tures are created the system starts to solve the problem through a cycle between two processes: engagement and reflection. Engagement is the generative process in the E&R model. During engagement, the system employs the problem’s context as a cue to probe memory and re- trieve a set of possible actions to perform in order to solve the problem. After a number of actions are produced, or if the system is unable to retrieve more actions from memory (i. e. if an impasse is declared), the reflection process takes control. During reflection, the system evaluates the actions generated so far and eliminates those that are not use- ful to solve the problem, checks the coherence of the sequence of actions generated during engagement, tries to break impasses, determines whether the problem has been solved, and generates a set of guidelines that drive the production of material during engagement. Then the system switches back to engagement. In this way, the outputs of the system are the result of the interaction between engagement and reflection. The cycle ends when the problem is solved or when it is impossible to break an impasse. Each time a problem is solved, the solution is added to the system’s knowledge base. Implementation The actual implementation of the Geometrician does not embody the whole E&R model (e. g. the func- tion to eliminate useless actions has not been finished yet). However, the Geometrician contributes with some characteristics not present in the original E&R model, as for example the capacity to execute the E&R cycle recursively to solve sub-problems of the current prob- lem. A sub-problem is created each time the sequence of produced actions lacks coherence. Discussion The prototype provides some insights on how useful the E&R model is for problem solving in geometry. In an ex- periment the Geometrician was provided with an initial knowledge base consisting of 3 solved problems. With this information the system was able to solve four new and more complex problems. Another interesting feature of the model is that dif- ferent solutions were produced on different runs. This occurred because the search on memory could retrieve more than one candidate action, and the engagement procedure selected only one. Thus, the decisions made by the system influenced the way in which the problem was solved. Conclusion Although the Geometrician is just a prototype subject to further development, the interaction between the en- gagement and reflection procedures proved to be useful on problem solving. References P´erez y P´erez, R., & Sharples, M. (2001). Mexica: A computer model of a cognitive account of creative writ- ing. J. Expt. Theor. Artif. Intell., 13, 119–139. Sternberg, R. J., & Lubart, T. I. (1999). The concept of creativity: Prospects and paradigms. In R. J. Stern- berg (Ed.), Handbook of creativity. Cambridge Univer- sity Press.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.004 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".