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PreprintCC BY 4.0arXiv

A Study in Solution Models for SAT using the PBNF-transform

  • Pelle Brooke Borgeke
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Abstract

In this article, we use the PBNF-transform, or Polynomial Boolean Normal Form, described earlier in two articles and now extended to higher arity, to solve Boolean satisfiability problems (SAT), which are part of the P versus NP problem. SAT consists of a statement of conjunctions of clauses, and in our SAT-solving models, we dualize the logical equations, inverting the problem by \emph{multiplying} letters (literals) in the transferred clauses, now phrases: pqrs⋯ sℓ, and using a binary iterated composition based on \emph{adding polynomials} for the conjunctions in our solution formula. We obtain Polynomial Boolean Equations, PBE=0, by also renaming the truth value 1↦ 0(T). To find a solution, we look for repeating literals in the clauses to factor out. With reversed literal repetition, we can even reduce the CNF to the sum of the transformed clauses.

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Cite

BibTeXRIS
BibTeX
@misc{borgeke2026study,
  title = {{A Study in Solution Models for SAT using the PBNF-transform}},
  author = {Pelle Brooke Borgeke},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30523},
  eprint = {2610.30523},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30523},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Study in Solution Models for SAT using the PBNF-transform
AU  - Pelle Brooke Borgeke
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30523
UR  - https://arxiv.org/abs/2610.30523
AB  - In this article, we use the PBNF-transform, or Polynomial Boolean Normal Form, described earlier in two articles and now extended to higher arity, to solve Boolean satisfiability problems (SAT), which are part of the P versus NP problem. SAT consists of a statement of conjunctions of clauses, and in our SAT-solving models, we dualize the logical equations, inverting the problem by \emph{multiplying} letters (literals) in the transferred clauses, now phrases: pqrs⋯ s_ℓ, and using a binary iterated composition based on \emph{adding polynomials} for the conjunctions in our solution formula. We obtain Polynomial Boolean Equations, PBE=0, by also renaming the truth value 1↦ 0(T). To find a solution, we look for repeating literals in the clauses to factor out. With reversed literal repetition, we can even reduce the CNF to the sum of the transformed clauses.
ER  -