


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 memberclass 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.
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 perceptualconceptual iterative architecture
of the human brain. The perceptualconceptual 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.
Deterministic Turing Algorithm (P)
Nondeterministic Turing Algorithm (NP)


