Computer Science – Computational Complexity
Scientific paper
2012-02-06
Computer Science
Computational Complexity
6 pages, English and Japanese (see Other formats - Source)
Scientific paper
This paper talks about difference between P and NP by using topological space that mean resolution principle. I focus restrictions of antecedent and consequent in resolution, and show the influence of restrictions to computation complexity. First, I introduce RCNF that mean the topology of resolution principle to formula of CNF. RCNF is HornCNF that mean the resolution principle of CNF. Variable values of RCNF formulas are presence of restrictions of CNF formula clauses. Second, I show the restrictions of inference rule of resolution principle. Resolution have restriction that antecedents adjoin each other. Therefore, one resolution cannot infer many restrictions, but consequent of resolution become single clause. As a result, RCNF become HornCNF. And we can make antecedents product as consequent by using some resolutions which have same joint variable. But we must combine some independent antecedents to make such product. Third, I show the P-Completeness of RCNF. By using unit resolution, we can reduce HornCNF to RCNF with logarithm space. And RCNF is HornCNF. Therefore, RCNF is P-Complete. Last, I show the size of RCNF that reduce CNF. We can construct CNF that some truth value assignment which make one clause false (or that formula true) become totally separated and simply independent. And we can divide the totally separated and simply independent truth value assignment over polynomial scale. Therefore, RCNF formulas which change from these formulas exceed polynomial size. And CNF is not P.
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