Authors: Koji KOBAYASHI
This paper describes about complexity of NP problems by using minimal circuit, and divide class P and NP. Inputs of uniform circuit family that compute P problem have some symme- try that indicated curcit structure. To clarify this symmetry, we define “Min- imal circuit” as subgraph of circuit which are necessary to compute subset of inputs. Minimal circuit divide problem to some symmetric partial problems. The other hand, inputs of NTM that compute NP problem have extra im- plicit symmetry that indicated nondeterministic transition functions. To clar- ify this implicit symmetry, we define special DTM “Concrete DTM Di”which index i correspond to selection of nondeterministic transition functions. That is, NTM split many different asymmetry DTM Di and compute all Di in same time. Consider Di and minimal circuit family, uniform circuit family N that solve NP problem have to include minimal circuit family that correspond to Di. These minimal circuit family have unique circuit gate and N must include these minimal circuit family and gates. Number of such minimal circuit is over polynomial size of input. Therefore, N is over polynomial size, and P is not NP.
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