A Critique of a Polynomial-time SAT Solver Devised by Sergey Gubin

This paper refutes the validity of the polynomial-time algorithm for solving satisfiability proposed by Sergey Gubin. Gubin introduces the algorithm using 3-SAT and eventually expands it to accept a broad range of forms of the Boolean satisfiability problem. Because 3-SAT is NP-complete, the algorithm would have implied P = NP, had it been correct. Additionally, this paper refutes the correctness of his polynomial-time reduction of SAT to 2-SAT.

💡 Research Summary

The paper provides a thorough critique of Sergey Gubin’s claimed polynomial‑time algorithm for solving 3‑SAT and his subsequent reduction of SAT to 2‑SAT. Gubin’s method begins by converting a 3‑SAT instance, expressed as a conjunction of clauses (c_1 \land c_2 \land \dots \land c_m), into a triangular array of “compatibility matrices” (C_{i,j}). Each matrix is an (8 \times 8) Boolean table that records, for every pair of truth‑table rows of clauses (c_i) and (c_j), whether the two rows are simultaneously true and agree on any shared variables. The algorithm then proceeds iteratively: for each clause index (i) (starting from 2), it “depletes” all matrices in rows (i) through (m) using clause (c_{i-1}). Depletion means that an entry (e_{a,b}) in a matrix (C_{i,j}) is set to zero if the corresponding columns in (C_{i-1,i}) and (C_{i-1,j}) have no common non‑zero entry in the same row. After all depletion steps, Gubin asserts that the bottom‑right matrix (the one involving the last two clauses) being all zeros indicates unsatisfiability, while any remaining 1 indicates satisfiability. He also claims that the same framework can be adapted to recover a satisfying assignment.

The critique identifies two fundamental flaws. First, the depletion process discards essential meta‑information: while a matrix entry may survive, the algorithm never records which specific truth‑table rows of the involved clauses correspond to that entry. Consequently, the algorithm cannot guarantee that a surviving entry actually extends to a global assignment satisfying all clauses. Second, the size of each compatibility matrix is insufficient to capture the combinatorial explosion that occurs when more than two clauses interact. An (8 \times 8) matrix encodes only the (2^6) possible assignments for the six literals appearing in two 3‑literal clauses. When a third clause is considered, the relevant space expands to (2^9) assignments, which cannot be faithfully represented in the same matrix without additional dimensions. The paper shows that this compression inevitably leads to loss of information, causing the algorithm to make incorrect decisions.

To demonstrate the failure concretely, the authors present several counter‑examples. The simplest is the unsatisfiable formula \