Algebraic Indexing to Parallelize Atom Comparison in the ATP Saturation Loop

18 pages•Published: October 24, 2026

Abstract

This paper introduces a scheme to compare atoms and extract variable substitution bindings, as a part of unification, within the core loop of an automated theorem prover. The presented algorithm uses only linear algebraic transformations and no pointer operations and is designed to scale with parallelized hardware acceleration. Atoms are converted to a single 64-bit value using a fingerprint function, under which identity comparison is a single XOR and unifiability is dictated by membership in a linear span. Two mechanisms follow, blackout matching (reduced-basis membership) and power unification (recovering variable bindings by solving a linear system in GF(2^64)). Although the comparison is approximate, it serves as a prefilter whose candidate matches can be verified syntactically before inference. When correctly tuned, it achieves substantial candidate reduction during a simulated unification pass on randomly selected axioms with zero observed false negatives in targeted ground/pattern and pattern/pattern validation.

Keyphrases: automated theorem proving, bch codes, parallelization, saturation, secure computing, unification

In: Stephan Schulz, Konstantin Korovin and Michael Rawson (editors). Proceedings of 16th International Workshop on the Implementation on Logics. Proceedings of the 26th Conference on Logic for Programming, Artificial Intelligence, and Reasoning (short papers), vol 23, pages 72-89.

Links:	https://easychair.org/publications/paper/wxnd
	https://doi.org/10.29007/5n67

BibTeX entry

@inproceedings{IWIL-2026:Algebraic_Indexing_Parallelize_Atom,
  author    = {Theodore Kim and Adam Pease},
  title     = {Algebraic Indexing to Parallelize Atom Comparison in the ATP Saturation Loop},
  booktitle = {Proceedings of 16th International Workshop on the Implementation on Logics. Proceedings of the 26th Conference on Logic for Programming, Artificial Intelligence, and Reasoning (short papers)},
  editor    = {Stephan Schulz and Konstantin Korovin and Michael Rawson},
  series    = {Kalpa Publications in Computing},
  volume    = {23},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2515-1762},
  url       = {/publications/paper/wxnd},
  doi       = {10.29007/5n67},
  pages     = {72-89},
  year      = {2026}}

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