Artificial Intelligence


On the Dual Nature of Logical Variables and Clause-Sets

Authors: Elnaserledinellah Mahmood Abdelwahab

This paper describes the conceptual approach behind the proposed solution of the 3SAT problem recently published in [Abdelwahab 2016]. It is intended for interested readers providing a step-by-step, mostly informal explanation of the new paradigm proposed there completing the picture from an epistemological point of view with the concept of duality on center-stage. After a brief introduction discussing the importance of duality in both, physics and mathematics as well as past efforts to solve the P vs. NP problem, a theorem is proven showing that true randomness of input-variables is a property of algorithms which has to be given up when discrete, finite domains are considered. This insight has an already known side effect on computation paradigms, namely: The ability to de-randomize probabilistic algorithms. The theorem uses a canonical type of de-randomization which reveals dual properties of logical variables and Clause-Sets. A distinction is made between what we call the syntactical Container Expression (CE) and the semantic Pattern Expression (PE). A single sided approach is presumed to be insufficient to solve anyone of the dual problems of efficiently finding an assignment validating a 3CNF Clause-Set and finding a 3CNF-representation for a given semantic pattern. The deeply rooted reason, hereafter referred to as The Inefficiency Principle, is conjectured to be the inherent difficulty of translating one expression into the other based on a single-sided perspective. It expresses our inability to perceive and efficiently calculate complementary properties of a logical formula applying one view only. It is proposed as an alternative to the commonly accepted P≠NP conjecture. On the other hand, the idea of algorithmically using information deduced from PE to guide the instantiation of variables in a resolution procedure applied on a CE is as per [Abdelwahab 2016] able to provide an efficient solution to the 3SAT-problem. Finally, linking de-randomization to this positive solution has various well-established and important consequences for probabilistic complexity classes which are shown to hold.

Comments: © 2016 Journal Academica Foundation. All rights reserved. With perpetual, non-exclusive license for - Originally received 08-04-2016 - accepted 09-12-2016 - published 09-15-2016 J.Acad.(N.Y.)6,3:202-239 (38 pages) - ISSN 2161-3338

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[v1] 2017-09-05 04:26:06

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