This page is part of the author's research project on Bongard problems.
The following table lists the problems solved by Phaeaco (specifically, its Bongard Problem Solving module), until 18-Nov-2006, the date when this page was last updated. It presents only a snapshot of a work in progress. I try to keep it updated, but I do not guarantee that the latest information has been incorporated.
| Problem | Date first solved by Phaeaco |
Times(*)
of Phaeaco |
Relative(**)
Times of Phaeaco |
Success Rate of Phaeaco |
Times(*)
of Human Subjects |
Relative(**)
Times of Human Subjects |
Success Rate
of Human Subjects |
| BP #3 | May 2003 | 3.0 sec +/- 0.3 sec | 1 | 50/50 | 5 sec +/- 2 sec | 1 | 8/8 |
| BP #23 | May 2003 | 7.7 sec +/- 0.9 sec | 2.57 | 40/50 | 12 sec +/- 9 sec | 2.4 | 12/12 |
| BP #1 | June 2003 | 6.5 sec +/- 0.3 sec | 2.18 | 48/50 | 7 sec +/- 5 sec | 1.4 | 11/11 |
| BP #5 | June 2003 | 9.2 sec +/- 0.8 sec | 3.07 | 24/50 | 20 sec +/- 28 sec | 4.0 | 11/11 |
| BP #6 | July 20, 2003 | 4.4 sec +/- 0.8 sec | 1.47 | 33/50 | 18 sec +/- 18 sec | 3.6 | 11/11 |
| BP #4 | Aug 10, 2003 | 6.8 sec +/- 0.3 sec | 2.27 | 10/50 | 15 sec +/- 8 sec | 3.0 | 8/10 |
| BP #2 | Oct 26, 2003 | 9.0 sec +/- 1.1 sec | 3.0 | 50/50 | 7 sec +/- 2 sec | 1.4 | 9/10 |
| BP #8 | Dec 9, 2003 | 7.9 sec +/- 0.3 sec | 2.63 | 14/50 | 8 sec +/- 2 sec | 1.6 | 3/3 |
| BP #15 | Dec 11, 2003 | 6.5 sec +/- 0.2 sec | 2.17 | 50/50 | 6 sec +/- 5 sec | 1.2 | 3/3 |
| BP #56 | Apr 25, 2004 | 33.6 sec +/- 9.4 sec | 11.20 | 7/50 | ? | ? | ? |
| BP #39 | Apr 28, 2004 | 3.7 sec +/- 0.3 sec | 1.23 | 32/50 | ? | ? | ? |
| BP #22 | Aug 8, 2004 | 30.2 sec +/- 17.9 sec | 10.07 | 3/50 | 1 min +/- 36 sec | 19.2 | 10/12 |
| BP #11 | Oct 22, 2004 | 16.2 sec +/- 1.7 sec | 5.40 | 40/50 | 47 sec +/- 46 sec | 9.4 | 8/10 |
| BP #21 | Oct 26, 2004 | 15.9 sec +/- 0.5 sec | 5.30 | 15/50 | 49 sec +/- 71 sec | 9.8 | 6/11 |
| BP #85 | Nov 6, 2004 | 10.6 sec +/- 1.3 sec | 3.53 | 24/50 | ? | ? | ? |
It should be noted that "solving a Bongard problem" is not an all-or-none event. People occasionally stumble, even on easy problems (except the trivial ones). The time required for solving a Bongard problem should be regarded not as a single number, but as a statistic, with a mean, a standard deviation, and also a success rate (for all non-trivial problems). Phaeaco's Bongard Problem Solver currently achieves slightly worse performance than people on some problems, while it marginally surpasses human performance on others. Of course, in comparing Phaeaco with human solvers I only refer to relative times (see columns for "relative times", and footnote (**), below), since Phaeaco's absolute times depend on the speed of the computer it is running on. Ideally, Phaeaco's relative times should match the corresponding relative times of human subjects.
(*) The times reported for Phaeaco are from a PC with a 1.5GHz Intel Celeron processor. The times of human subjects are so far informal (and hence, unpublishable), collected by asking up tp 12 friends of mine to take an experimental session. The +/- value defines the limits of the 95% confidence interval for the mean. Formal experimental collection of times for human subjects is scheduled to take place in the fall of 2004 - winter 2005.
(**) A relative time for a problem is the quotient of the time taken to solve that problem over the time taken to solve problem BP #3. The latter is taken to be the unit of measurement for relative timings in Bongard problems due to its simplicity and immediacy (probably the fastest-solved Bongard problem by both human subjects and Phaeaco).
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