Complexity Considerations, cSAT Lower Bound

Abstract

This article discusses completeness of Boolean Algebra as First Order Theory in Goedel's meaning. If Theory is complete then any possible transformation is equivalent to some transformation using axioms, predicates etc. defined for this theory. If formula is to be proved (or disproved) then it has to be reduced to axioms. If every transformation is deducible then also optimal transformation is deducible. If every transformation is exponential then optimal one is too, what allows to define lower bound for discussed problem to be exponential (outside P). Then we show algorithm for NDTM solving the same problem in O(n^c) (so problem is in NP), what proves that P \neq NP. Article proves also that result of relativisation of P=NP question and oracle shown by Baker-Gill-Solovay distinguish between deterministic and non-deterministic calculation models. If there exists oracle A for which P^A=NP^A then A consists of infinite number of algorithms, DTMs, axioms and predicates, or like NDTM infinite number of simultaneous states.