# Complexity Considerations, cSAT Lower Bound

## TL;DR

This paper explores the complexity of Boolean algebra, establishes exponential lower bounds for certain transformations, and presents an algorithm that suggests P does not equal NP, also discussing relativization and oracle implications.

## Contribution

It introduces a complexity lower bound for transformations in Boolean algebra and proposes an algorithm placing the problem in NP, supporting P ≠ NP.

## Key findings

- Exponential lower bound for transformation complexity outside P
- Algorithm solving the problem in polynomial time on NDTMs
- Relativization results distinguishing deterministic and non-deterministic models

## 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.

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Source: https://tomesphere.com/paper/0704.0514