Cognitive Logic
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Cognitive Logic Enables P=NP

Cognitive logic  is a set of innate functions of the human brain, that is different from mathematical logic. Mathematical logic is wrestling within a paradox, and is a subset of cognitive logic. When mathematicians are looking for the foundations of mathematics, biologists may offer cognitive logic as the model. The knowledge learning and reasoning (KLAR) algorithm simulates the cognitive logic of the human brain, and validates this foundation. Formally KLAR is an NP algorithm that learns relations as knowledge, and retrieves relations deductively and reductively. In computer science P is a class of deterministic Turing algorithm that processes recursion functions efficiently. NP is a class of nondeterministic Turing alorithm that recognizes relations efficiently. Whether we can program a P algorithm as an NP algorithm is an open question denoted by P=NP. The knowledge learning and reasoning algorithm applies a mirrored memory structure as its knowledge system and employs two inverse functions to retrieve member-class relations learned in the knowledge system. Therefore, the knowledge learning and reasoning algorithm is able to  perform the NP task efficiently. Our work indicates that Boolean logic satisfiability problems can be solved in linear time by KLAR.

 

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The breakthrough came from the biological discovery that the human memory system is a mirrored perceptual and conceptual memory system. Within this memory system there exists three cognitive logic functions: induction, deduction, and reduction. Induction is the function that learns relations and stores them into the knowledge system. Deduction is the function that retrieves conceptual information mapped from perceptual information. Reduction is the function that retrives perceptual information mapped from conceptual information. This biological model can be converted into a mathematical model of axiomatic iterative set theory. In computer science, the knowledge learning and reasoning algorithm is a computable Oracle Turing machine, which includes two parts: a knowledge structure (KS) and its embedded logic functions of induction, deduction and reduction. KLAR can solve NP class  problems in polynomial time because of its iterative set data structure and inverse functions of deduction and reduction. The foundation of KLAR is the perceptual-conceptual iterative architecture of the human brain. The perceptual-conceptual iterative relation is a model of axiomatic set theory, the concept of which was first introduced by Gödel in 1947. Based on this conception of iterative set, in 1956 Gödel was able to foresee that P=NP was completely within the realm of possibility.

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Deterministic Turing Algorithm (P)

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Nondeterministic Turing Algorithm (NP)

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