Connection and Dispersion of Computation

📝 Abstract

This paper talk about the influence of Connection and Dispersion on Computational Complexity. And talk about the HornCNF’s connection and CNF’s dispersion, and show the difference between CNFSAT and HornSAT. First, I talk the relation between MUC decision problem and classifying the truth value assignment. Second, I define the two inner products (“inner product” and “inner harmony”) and talk about the influence of orthogonal and correlation to MUC. And we can not reduce MUC to Orthogonalization MUC by using HornMUC in polynomial size because HornMUC have high orthogonal of inner harmony and MUC do not. So DP is not P, and NP is not P.

💡 Analysis

This paper talk about the influence of Connection and Dispersion on Computational Complexity. And talk about the HornCNF’s connection and CNF’s dispersion, and show the difference between CNFSAT and HornSAT. First, I talk the relation between MUC decision problem and classifying the truth value assignment. Second, I define the two inner products (“inner product” and “inner harmony”) and talk about the influence of orthogonal and correlation to MUC. And we can not reduce MUC to Orthogonalization MUC by using HornMUC in polynomial size because HornMUC have high orthogonal of inner harmony and MUC do not. So DP is not P, and NP is not P.

📄 Content

arXiv:1106.5470v9 [cs.LO] 9 Aug 2011 CONNECTION AND DISPERSION OF COMPUTATION KOBAYASHI, KOJI Abstract. In this paper, we describe the impact on the computational com- plexity of Connection and Dispersion of CNF. In previous paper [1], we told about structural differences in the P-complete problems and NP-complete problems. In this paper, we clarify the CNF’s dispersion and HornCNF’s connection, and shows the difference between CNFSAT HornSAT. First we fo- cus on the MUC decision problem. We clarify the relationship between MUC and the classifying of the truth value assignment. Next, we clarify the clauses correlation and orthogonal by using the two inner product of clauses. Because HornMUC has higher orthogonal, its orthogonal MUC is polynomial size. Be- cause MUC has higher correlation, its orthogonal MUC is not polynomial size. And, HornMUC whereas only be a large polynomial is at most its size even if orthogonal than orthogonal high, MUC will be fit to size polynomial in the size and orthogonalized using HornCNF more highly correlated shown. So DP ̸= P , and NP ̸= P .

1. CNF’s classification and CNFSAT We show the relationship between CNFSAT and CNF’s classification. We show the relationship between MUC decision problem and CNFSAT. And We show the relationship that determined by the CNF. And We show the relationship between CNF’s classification and MUC dicition problem. 1.1. MUC decision problem. Describes the MUC decision problem. MUC de- cision problem is the problem to decide the CNF is MUC (Minimum Unsatisfiable Core) or not. MUC is the unsatisfiable CNF. And it changes MUC to satisfiable CNF that remove one of the MUC’s clause. MUC decision problem is combina- tion problem of coNP-complete and P-complete problem. And HornCNF’s MUC dicision problem is P-complete because of P = coP. The relationship between the DP-complete and P-complete is; Theorem 1. If P ̸= DP then P ̸= NP. So if we can not reduce MUC dicision problem to HornMUC dicision problem in polynomial time, then P ̸= NP. Proof. If P = NP then NP = coNP and P = DP. So MUC dicision problem can reduce HornMUC dicisition problem in polynomial time. So take the contraposi- tive, if we can not reduce MUC dicision problem to HornMUC dicision problem in polynomial time, then P ̸= NP. □ 1.2. CNF classification. Describe the relationship that define the CNF. CNF clauses value corresponds to either true or false. Clauses are the rules that maps each truth value assignment to truth value. This is equivalence relation that classify each truth values to equivalent class. 1 CONNECTION AND DISPERSION OF COMPUTATION 2 Definition 2. Clauses equivalence relation is the truth value assignments relation that equal the clauses value. Similarly, CNF equivalence relation is the truth value assignments relation that equal the all clauses value set. I will use the term “CNF classification” to the truth value assignments equiva- lence class of CNF, and “CNF equivalence class” to the equivalence class of CNF classification. And “Logical value assignment” to the set of the clauses values. And “Cyclic value assignment” to the logical value assignment that have only one false value clause, and “All true assignment” to the logical value assignment that have no false value. Number of cyclic value assignment matches the number of the clauses. All true assignment is only one. The combined truth value assignment and logical value assignment is the truth value table of the clauses. Especially, I will use the term “Logical value table” to the truth value table. 1.3. CNF classification and MUC dicision problem. MUC decision problem is a matter to determine the truth as an input CNF, can be thought as the problems dealing with CNF classification. The problem is the decision problem that the logical value assignment includes all the cyclic value assignment and excludes all tru assignment. So MUC decision problem can be divided into two calculations, CNF classification and decision of the logical value assignments. CNF classification includes the difference of MUC decision problem and Horn- MUC decision problem. In decision of logical value assignment, There is no dif- ference between MUC decision problem and HornMUC decision problem. And these can be determined in polynomial time either. Especially, all logical value assignments is cyclic value assignment, the decision of logical value assignment can determined in polynomial time of the number of the clauses. Theorem 3. Decision of logical value assignment can be done in polynomial time either MUC decision problem or HornMUC decision problem. Thus, the difference in computational complexity of the MUC decision problem and HornMUC decision problem will appear in size of the logic value assignments of CNF classification. Proof. The decision of logical value assignment of MUC decision problem and Horn- MUC decision problem is the computation that the logical value assignment includes all cyclic value assignment and excludes all truth assignment. We c

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