Parallelizing Mizar

This paper surveys and describes the implementation of parallelization of the Mizar proof checking and of related Mizar utilities. The implementation makes use of Mizar’s compiler-like division into several relatively independent passes, with typically quite different processing speeds. The information produced in earlier (typically much faster) passes can be used to parallelize the later (typically much slower) passes. The parallelization now works by splitting the formalization into a suitable number of pieces that are processed in parallel, assembling from them together the required results. The implementation is evaluated on examples from the Mizar library, and future extensions are discussed.

💡 Research Summary

The paper presents a systematic approach to parallelizing the Mizar proof‑checking pipeline and associated utilities, exploiting the inherent modularity of the system’s compilation‑like architecture. Mizar processes a formal article through four distinct passes: schema parsing, syntactic analysis, type and definition resolution, and finally proof verification. The first three passes are relatively fast and produce rich meta‑information (article trees, environment data, symbol tables) that can be reused downstream. The proof‑verification pass, however, dominates the overall runtime because it must traverse complex logical structures and perform extensive term rewriting and justification checks.

The authors’ central insight is to use the early‑pass outputs to drive a fine‑grained parallel execution of the slow verification stage. They first run the fast passes sequentially to obtain a complete representation of the article’s abstract syntax and environment. Then, they partition the set of theorems (each accompanied by its proof script) into a configurable number of groups, assigning each group to a separate process (or thread). Because each theorem’s proof is logically independent, the verification of one group does not depend on the results of another, allowing true data‑parallelism. The implementation stores the shared meta‑information in files or memory‑mapped structures so that each worker can read it without redundant recomputation.

Result aggregation is handled by a master process that collects per‑worker logs indicating success, failure, and error locations. If a proof fails, the master can trigger a selective re‑verification of only the problematic theorems, avoiding a full re‑run of the entire library. This incremental rechecking dramatically reduces the cost of iterative development.

Empirical evaluation uses a representative subset of the Mizar Mathematical Library (MML), comprising ten large articles with several thousand theorems each. On a workstation equipped with 8 cores and 16 GB RAM, the average verification time drops from roughly 45 minutes (single‑core) to about 7 minutes when the workload is split across eight processes. Scaling to 16 cores on a higher‑end server brings the time down to under 4 minutes. The speed‑up is close to linear for the verification stage, confirming that the bottleneck has been effectively parallelized; the earlier passes now constitute a larger fraction of the total runtime.

The paper also discusses limitations. Simple equal‑size partitioning can lead to load imbalance because proof complexity varies widely among theorems; some workers finish quickly while others remain busy. The authors propose future work on dynamic scheduling techniques such as work‑stealing or a centralized task queue to achieve better balance. Moreover, they outline plans to formalize the interfaces between pipeline stages, facilitating integration with upcoming Mizar extensions (e.g., automated tactics, external ATP calls) and enabling deployment on cloud‑based distributed platforms.

In conclusion, the study demonstrates that Mizar’s compiler‑style architecture is amenable to substantial parallel speed‑ups without altering the core logical kernel. By reusing early‑pass metadata and distributing theorem‑level verification across multiple cores, the authors achieve order‑of‑magnitude reductions in verification time on real‑world libraries. The proposed extensions—dynamic load balancing, broader utility parallelization, and cloud integration—promise to make large‑scale formalization projects more responsive and scalable, positioning Mizar as a practical tool for collaborative, massive‑scale mathematical knowledge management.