  Geometry problem solving (GPS) requires capacities of multi-modal
understanding, multi-hop reasoning and theorem knowledge application. In this
paper, we propose a neural-symbolic model for plane geometry problem solving
(PGPS), named PGPSNet-v2, with three key steps: modal fusion, reasoning process
and knowledge verification. In modal fusion, we leverage textual clauses to
express fine-grained structural and semantic content of geometry diagram, and
fuse diagram with textual problem efficiently through structural-semantic
pre-training. For reasoning, we design an explicable solution program to
describe the geometric reasoning process, and employ a self-limited decoder to
generate solution program autoregressively. To reduce solution errors, a
multi-level theorem verifier is proposed to eliminate solutions that do not
match geometric principles, alleviating the hallucination of the neural model.
We also construct a large-scale geometry problem dataset called PGPS9K,
containing fine-grained annotations of textual clauses, solution program and
involved knowledge tuples. Extensive experiments on datasets Geometry3K and
PGPS9K show that our PGPSNet solver outperforms existing symbolic and neural
solvers in GPS performance, while maintaining good explainability and
reliability, and the solver components (fusion, reasoning, verification) are
all justified effective.
