FGPS

Formal geometric problem solver (FGPS) based on FormalGeo.

This project utilizes FormalGeo as its problem solver and conduct experiments on datasets formalgeo7k and formalgeo-imo. Our paper introducing the functions and implementation of FGPS can be obtained at Symmetry 2024.

More information about FormalGeo will be found in homepage. FormalGeo is in its early stages and brimming with potential. We welcome anyone to join us in this exciting endeavor.

Installation

You can get the source by cloning the FGPS repository.
Create Python environment:

$ conda create -n FGPS python=3.10
$ conda activate FGPS

Clone project:

$ git clone --depth 1 https://github.com/BitSecret/FGPS.git

Install python dependencies:

$ cd FGPS
$ pip install -e .

Run

Enter the path:

$ cd src/fgps

Download datasets formalgeo7k_v1 and formalgeo-imo_v1:

$ python utils.py --func download_datasets --path_datasets ./user_datasets

Create logs dir:

$ python utils.py --func create_log_archi --path_logs ./user_logs

View help information for personalizing configuration:

$ python utils.py --help

Auto problem-solving based on search

Check our search results on formalgeo7k_v1 dataset:

$ python check_search.py --func check_search --path_datasets ./user_datasets

Run the search method on your own computer:

$ python search.py --path_datasets ./user_datasets --path_logs ./user_logs --dataset_name formalgeo7k_v1 --method fw --strategy bfs 

Interactive problem-solving

Notes: In the latest update, formalgeo7k_v1 has removed the expanded problem. This will cause functions in check_run.py with the expanded parameter to throw errors. Please access the project formalgeo7k to obtain the original formalgeo7k_v1 dataset. Additionally, versions of FormalGeo starting from 0.0.4 have also removed functions related to the expanded problem. Please revert back to version 0.0.3.

Check our results:

$ python check_run.py --func check_run --path_datasets ./user_datasets

Verify the annotated problems in the dataset:

$ python run.py --func auto_run --path_datasets ./user_datasets --path_logs ./user_logs --dataset_name formalgeo7k_v1
$ python run.py --func auto_run --path_datasets ./user_datasets --path_logs ./user_logs --dataset_name formalgeo-imo_v1

Run interactively and show detail solution:

$ python run.py --func run --path_datasets ./user_datasets --path_logs ./user_logs --dataset_name formalgeo7k_v1
$ python run.py --func run --path_datasets ./user_datasets --path_logs ./user_logs --dataset_name formalgeo-imo_v1

You can find detail solution output in ./user_logs/run/problems.

Others

Draw fig.8 and fig.9:

$ python check_run.py --func draw_run_results --path_datasets ./user_datasets
$ python check_search.py --func draw_search_results --path_datasets ./user_datasets

The image will be saved to ./231016/run/auto_logs.

Results

We conducted experiments on the formalgeo7k_v1, comparing different search methods and strategies in terms of problem-solving success rate, solution time, and the number of steps required for problem-solving.
Forward search (FW) starts from the known conditions of the problem and continuously apply theorems to derive new conditions until the goal is achieved. Backward search (BW), on the other hand, begins with the problem-solving goal, expands it into multiple sub-goals, and repeats this process until all sub-goals are resolved.
The search-based methods construct a search tree during the problem-solving process. We have the flexibility to choose various strategies to traverse the search tree and reach the goal. Breadth-first search (BFS) begins by expanding the top-level nodes of the search tree and then proceeds layer by layer into the depth. Depth-first search (DFS) recursively selects nodes from the search tree from shallow to deep and continues this process. Random search (RS) randomly selects an expandable node at each stage of expansion. Beam search (BS) selects $k$ nodes in each stage of expansion and can be viewed as a trade-off between BFS and RS.
We conducted experiments on 2 Intel i9-10900X, 1 AMD Ryzen 9 5900X, and 1 AMD Ryzen 9 7950X, running the search algorithms using multiple processes while maintaining a CPU utilization rate of 80%. The maximum search depth was set to 15, and the beam size was set to 20. The total duration of the experiments was approximately 3 days. When the timeout for each problem was 300 seconds, the best success rate for problem-solving was approximately 30%. When the timeout for each problem was increased to 600 seconds, the specific results are as follows.

The number of problems for different levels.

total	l1	l2	l3	l4	l5	l6
6981	2409	1896	1248	824	313	291

Results of search-based problem-solving success rates

method	strategy	solved	unsolved	timeout
FW	BFS	38.86	7.42	53.72
FW	DFS	36.16	9.80	54.05
FW	RS	39.71	9.07	51.22
FW	BS	25.28	38.72	36.00
BW	BFS	35.44	2.68	61.88
BW	DFS	33.73	2.42	63.84
BW	RS	34.05	2.65	63.30
BW	BS	34.39	12.86	52.74

Detail results of search-based problem-solving success rates

method	strategy	total	l1	l2	l3	l4	l5	l6
FW	BFS	38.86	59.94	38.61	28.53	17.35	8.63	3.78
FW	DFS	36.16	55.75	40.03	22.92	12.38	7.03	4.12
FW	RS	39.71	59.24	40.03	33.65	16.38	5.43	4.81
FW	BS	25.28	46.12	22.57	13.46	5.83	2.88	0.34
BW	BFS	35.44	67.21	33.70	11.14	6.67	6.07	1.03
BW	DFS	33.73	65.88	30.85	8.89	6.55	5.11	0.69
BW	RS	34.05	66.58	31.70	8.65	5.83	4.47	0.69
BW	BS	34.39	67.08	31.33	9.46	6.31	5.75	1.03

Timing (solved)

method	strategy	total	l1	l2	l3	l4	l5	l6
FW	BFS	185.05	121.92	205.74	333.33	331.26	243.92	251.11
FW	DFS	132.35	84.07	159.07	254.29	196.02	230.39	218.05
FW	RS	92.21	57.34	89.67	177.34	189.74	166.96	199.31
FW	BS	58.76	35.09	81.66	106.97	207.55	191.40	121.36
BW	BFS	57.67	30.28	94.34	152.26	147.55	193.21	134.65
BW	DFS	46.45	26.34	81.17	115.83	94.85	133.23	0.85
BW	RS	65.82	42.79	101.71	156.27	145.50	202.37	3.37
BW	BS	75.55	47.44	120.53	172.50	165.80	207.28	146.26

Timing (unsolved)

method	strategy	total	l1	l2	l3	l4	l5	l6
FW	BFS	574.94	434.92	584.16	615.29	624.06	663.15	681.07
FW	DFS	548.81	418.40	546.35	594.46	605.72	641.69	655.53
FW	RS	542.46	394.11	546.29	591.92	606.37	632.35	649.81
FW	BS	355.74	277.04	345.92	358.74	429.92	446.75	452.67
BW	BFS	579.24	549.32	572.41	592.66	588.48	596.18	597.48
BW	DFS	581.50	550.26	577.09	593.69	592.71	595.34	598.38
BW	RS	580.74	554.28	578.62	589.06	588.93	592.29	597.23
BW	BS	513.86	338.82	508.33	566.67	574.09	585.96	578.35

Step size (solved)

method	strategy	total	l1	l2	l3	l4	l5	l6
FW	BFS	58.44	21.86	68.71	119.56	144.80	161.44	822.18
FW	DFS	87.16	33.17	95.99	225.57	232.25	257.59	727.17
FW	RS	41.89	19.14	47.66	64.32	108.26	143.41	611.00
FW	BS	18.64	11.01	25.95	37.86	46.19	48.44	541.00
BW	BFS	15.14	20.66	4.81	5.60	2.31	1.00	1.00
BW	DFS	5.17	4.96	5.83	5.86	3.94	1.00	1.00
BW	RS	4.34	5.02	3.13	2.81	1.27	1.00	1.00
BW	BS	1.67	1.49	1.95	3.08	1.25	1.00	1.00

Step size (unsolved)

method	strategy	total	l1	l2	l3	l4	l5	l6
FW	BFS	63.65	45.63	61.78	76.31	77.91	56.86	65.50
FW	DFS	154.25	89.50	148.21	154.27	152.39	204.89	378.16
FW	RS	66.38	65.17	68.91	75.37	56.14	62.68	62.81
FW	BS	37.44	27.70	36.13	41.83	42.55	40.06	54.81
BW	BFS	266.77	1147.53	195.47	21.51	10.16	4.49	59.40
BW	DFS	517.90	2649.89	154.48	6.99	2.18	1.62	17.26
BW	RS	181.74	828.78	113.62	15.26	2.14	1.34	10.19
BW	BS	7.17	22.14	6.81	3.02	2.48	1.73	2.01

Acknowledge

This project is maintained by FormalGeo Development Team and Supported by Geometric Cognitive Reasoning Group of Shanghai University (GCRG, SHU).
Please contact with the author (xiaokaizhang1999@163.com) if you encounter any issues.

Citation

A BibTeX entry for LaTeX users is:

@article{zhang2024fgeosss,
AUTHOR = {Zhang, Xiaokai and Zhu, Na and He, Yiming and Zou, Jia and Qin, Cheng and Li, Yang and Leng, Tuo},
TITLE = {FGeo-SSS: A Search-Based Symbolic Solver for Human-like Automated Geometric Reasoning},
JOURNAL = {Symmetry},
VOLUME = {16},
YEAR = {2024},
NUMBER = {4},
ARTICLE-NUMBER = {404},
URL = {https://www.mdpi.com/2073-8994/16/4/404},
ISSN = {2073-8994},
DOI = {10.3390/sym16040404}
}