This is a prototype geometry solver based on a forward-chaining theorem prover. The basic forward-chaining design is analogous to the deductive database described in [Chou00], in which the prover operates by alternating steps: (1) fixed-point computation of logical inferences, and (2) choosing a valid geometric construction. However, our prover also contains an implementation of exploration-based forward-chaining, by which we choose a construction by maximizing an intrinsic exploration-based metric during the fixed-point stage, and selecting the construction that maximizes the metric. This form of exploration alone turns out to be sufficient to solve a hard reduction of an olympiad-level geometry problem, which even the deductive database plus geometry domain-specific algebraic reasoning ([Trinh24]) cannot solve. Please see the blog post for more details.

Instructions

Building for testing and development

Note: These build instructions assume a unix-like system.

1. Create a workspace directory, and clone this repository into the workspace:

   mkdir meringue-workspace
   cd meringue-workspace
   git clone https://github.com/peterhj/meringue
   

2. Run the bootstrap script in this repository to checkout the vendored dependencies into the workspace:

   cd meringue
   ./bootstrap.sh
   

3. The default Makefile target is for a release build:

   make
   

   or, you may target a debug build:

   make debug
   

Running on the reduced IMO 2009 P2 test cases

After following the above build instructions, running any of the three scripts run_reduced_imo_2009_p2_v{1,2,3}.sh will run the prover one-shot on the triple (theory, problem, random seed). The default theory is located at data/geometry.txt and contains Horn clause-style rules sufficient for proving our reduced versions of IMO 2009 P2. The reduced problem definitions are located at data/reduced_imo_2009_p2_v{1,2,3}.txt.

Note that the reduced problems are ordered by increasing difficulty. So, v1 should be very easy to solve, v2 should take some more iterations (often 30-40), while v3 may require restarting the solver with a different seed to get a result within a reasonable time.

We also experimented with running DD+AR only (from AlphaGeometry [Trinh24]) on our reduced versions of IMO 2009 P2. We found that DD+AR in fact fails to solve the v3 reduction. Please see our branch at https://github.com/peterhj/alphageo-experiments/tree/exp for our implementation of the experiment.

Related Work

• Newclid: Vladmir Sicca, Tianxiang Xia, Mathïs Fédérico, Philip John Gorinski, Simon Frieder, Shangling Jui [arXiv:2411.11938]

References

[Chou00] Shang-Ching Chou, Xiao-Shan Gao, and Jing-Zhong Shang. A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering. Journal of Automated Reasoning, 25. 2000.

[Trinh24] Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, He He, and Thang Luong. Solving olympiad geometry without human demonstrations. Nature, 625. 2024.