**Authors:** Koji KOBAYASHI

This paper describes about P vs NP by using topological approach. We modify computation history as “Problem forest”, and define special problem family “Wildcard problem” and “Maximal complement Wildcard problem” to simplify relations between every input. “Problem forest” is directed graph with transition functions edges and computational configuration nodes with effective range of tape. Problem forest of DTM is two tree graph which root are accepting & rejecting configuration, which leaves are inputs, trunks are computational configuration with effective range of tape. This tree shows TM's interpretation of symmetry and asymmetry of each input. From the view of problem forest, some NTM inputs are marged partly, and all DTM inputs are separated totally. Therefore NTM can compute implicitly some type of partial (symmetry) overrap, and DTM have to compute explicitly. “WILDCARD (Wildcard problem family)” and “MAXCARD (Maximal complement Wildcard problem family)” is special problem families that push NTM branches variations into inputs. If “CONCRETE (Concrete Problem)” that generate MAXCARD is in P-Complete, then MAXCARD is in PH, and these inputs have many overrap. DTM cannot compute these overrap conditions implicitly, and these conditions are necesarry to compute MAXCARD input, so DTM have to compute these conditions explicitly. These conditions are over polynomial size and DTM take over polynomial steps to compute these conditions explicitly. That is, PH is not P, and NP is not P.

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[v1] 2017-01-03 01:45:00

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