Relational Lysis and Computational Efficiency II: The Resolution of 3-SAT by Boundary Assertion

Abstract

Relation Lysis is used to resolve a 3-SAT instance by asserting boundaries, and what is stored afterwards is the boundaries rather than the assignments they admit. The act of distinction establishes an envelope in which all 2n addresses stand at genesis; each clause contributes one boundary whose excluded sector is the clause’s own k contents at the opposite orientation, forced rather than asserted, so the whole instance is asserted in n+1+km structural acts. Those boundaries are the complete record: membership, the excluding boundary, the count of satisfying assignments and the ith of them are each read from them, and no list of assignments is created at any point. On a threshold instance over 70 contents the assertion occupies 965 symbols and admits 597,681 assignments, which written out are 41,837,670; none is formed and discarded. The assertion is carried to n = 10,000 at 137,801 acts, and on Tseitin formulas, where resolution is provably exponential, it grows from 109 to 325 acts while the classical work grows from 24,702 to 784,575,041 elementary operations. The boundaries compose, by one mediation per depth and with no address formed, into one boundary at depth whose strata are lighthouses: every address of a lighthouse stands, so the list is read with nothing to reject and no test to perform. A boundary’s two termini are a conjugate pair, so writing one writes both, and the terminus that excludes is never presented. How far a boundary reaches at the coarsest depth, and the order of mediation in which its contents stand, are fixed by the assertion. One instance is worked in full from the assertion to the answer. Eighteen experimental protocols carry 799 checks, all passing. The findings suggest that structure precedes enumeration and using boundaries to store information may result in faster compute and savings in data storage.