Connection and Dispersion of Computation

CONNECTION AND DISPERSION OF COMPUT A TION KOBA Y ASHI, KOJI Abstract. In this pap er, we describe the impact on the computational com- plexit y of Connection and D isp ersion of CNF. In previous pap er [1], we told about str uctural differences in the P-complete pro blems and NP-complete problems. In this paper, we clarify the CNF’s disp ersion and HornCNF’s connect ion, and sho ws the difference betw een CNFSA T HornSA T. First we fo- cus on t he MUC de cision problem. W e clar ify the relat ionship b et w een MUC and th e classi fying of the truth v al ue assignmen t. Next, we clarify the clauses correlat ion and orthogonal by using the tw o inner product of clauses. Because HornMUC has highe r orthogonal, its orthogonal MUC is p olynomial si ze. Be- cause M UC has higher correlation, i ts orthogonal MUC is not p olynomial size. And, HornMUC whereas on ly b e a large p olynomial is at most its size ev en if ort hogonal than ort hogonal high, MUC wi ll be fit to size pol ynomial in th e size and orthogonalized using HornCNF m ore highly correlated shown. So D P 6 = P , and N P 6 = P . 1. CNF’s classifica tion and CNFSA T W e show the rela tionship b etw een CNFSA T and CNF’s classification. W e show the relatio nship be tw een MUC decision problem and CNFSA T. And W e show the relationship that determined by the CNF. And W e sho w the r elationship betw een CNF’s cla ssification and MUC dicition problem. 1.1. MUC decisi on problem. Describ es the MUC decision pro blem. MUC de- cision problem is the pr oblem to decide the CNF is MUC (Minim um Unsatisfiable Core) or not. MUC is the unsatisfiable CNF. And it changes MUC to satisfiable CNF that remov e one of the MUC’s claus e . MUC decisio n pro blem is combina- tion problem of c oNP-complete and P-complete pr oblem. And HornCNF’s MUC dicision problem is P-co mplete bec a use of P = coP . The relationship betw een the DP-complete and P-complete is; Theorem 1. If P 6 = D P t hen P 6 = N P . S o if we c an not r e duc e MUC dicision pr oblem to HornMUC dic ision pr oblem i n p olynomial time, then P 6 = N P . Pr o of. If P = N P then N P = coN P and P = DP . So MUC dicision pro blem can reduce HornMUC dicisition problem in p olyno mial time. So take the cont rap osi- tiv e, if we can not reduce MUC dicision problem to Ho rnMUC dicision problem in po lynomial time, then P 6 = N P .  1.2. CNF classification. Describ e the relationship that define the CNF . CNF clauses v alue corresp o nds to either true or false. Clauses are the rules that maps each truth v a lue ass ignment to truth v a lue. This is equiv ale nce rela tion that classify each truth v alues to equiv alent cla ss. 1 CONNECTION AND DIS PERSION OF COMPUT A TION 2 Definition 2. Cla uses equiv alence relation is the truth v alue assig nmen ts relation that equal the clause s v alue. Similarly , CNF equiv a lence relation is the truth v alue assignments relation that equal the all cla uses v alue set. I will use the term “CNF classification” to the truth v a lue as signments equiv a - lence class of CNF, and “CNF equiv alence class” to the equiv alence class of CNF classification. And “L o gical v alue assignment” to the set of the cla uses v alues. And “Cyclic v alue ass ignment” to the log ical v alue assig nment that ha ve only one false v alue c la use, and “All true assig nmen t” to the log ical v alue assignment that hav e no false v alue. Num b er of cyclic v alue assignment matches the n um be r of the cla uses. All true assignment is o nly one. The co m bined truth v alue as signment and logic a l v alue assignment is the truth v alue table of the clauses. Esp ecially , I will use the term “Logical v alue table” to the truth v alue table. 1.3. CNF cl ass ification and MUC di cision problem. MUC decision pro blem is a matter to determine the truth as an input CNF, can b e though t as the pro blems dealing with CNF classification. The problem is the decision pro blem that the logical v alue ass ignment includes all the cyclic v alue assignment and excludes all tru as signment. So MUC decision pro blem can b e divided int o tw o calculatio ns , CNF classification and dec is ion of the logical v alue assignments. CNF classification includes the difference of MUC decision problem and Horn- MUC decision problem. In decision of logical v alue assig nment , There is no dif- ference b etw een MUC decision problem a nd Hor nMUC decision pr oblem. And these can b e determined in p o lynomial time either. Espe cially , all logical v alue assignments is cyclic v alue ass ig nment , the decision of log ic a l v alue as signment can determined in polynomial time of the num b er of the clauses. Theorem 3. De cision of lo gic al value assignment c an b e do ne in p olynomia l time either MUC de cision pr oblem or H ornMUC de cision pr oblem. Thus, the differ enc e in c omputational c omplexity o f the MUC de cision pr oblem and HornMUC de cision pr oblem will app e ar i n size of the lo gic value assignments of CNF classific ation. Pr o of. The decision of log ical v alue assignment of MUC decision pr oblem a nd Horn- MUC decision problem is the computation that the lo gical v alue assignment includes all cyclic v alue ass ig nment and excludes all truth assignment. W e can determine the decisio n by determine the a ll lo gical v a lue a ssignment. Th us, W e can determine the decision only the polyno mial time of the lo gical v alue assignment. And we can handle in p o lynomial time o f the log ical v alue assi g nment that reduce from MUC decision problem a nd HornMUC decisio n problem. So, if they hav e the coputation c omplexity difference b etw een MUC decision problem and HornMUC decision problem, the difference included in CNF cla ssification.  Therefore, What has to be noticed is the CNF classifica tion. 2. MUC as Periodic function In view of p erio dic function, let us then co nsider MUC. CNF claus es classificatio n hav e the p erio dicity in truth v alue ta ble. Th us , we can deal with CNF as p erio dic function. The truth table grows lar g er the t yp e o f v ariables included in the clauses, and truth v alue assignment is changed and the cla us e is false by v a riable’s p os itive and negative. Thus, in p er io dic function of clauses, v ariable is cycle and v ariable’s CONNECTION AND DIS PERSION OF COMPUT A TION 3 po sitive/negativ e is phase. And CNF is the perio dic function that put many clauses per io dic function. Clauses is the notation in the frequency do main, logical v alue table is the notation in the time doma in. Definition 4. I will use the term “Clause cycle” to the num ber of the v aria bl e s in clauses. The num ber of Clause cycle is equal to th e n um ber of all cases of c hange the clauses v ariable’s po s itive/negativ e. I will use the term “Cla use phas e” to the p os itive/negative co nfiguration o f the v aria bles in clauses. Clause phase is the position of the truth v alue assignment that the clause make fals e in truth v a lue table. The clause pha se difference of tw o clauses is equal the minim um Hamming distance of these truth v alue assignment that ma ke these clauses false. In MUC, there is a equiv alence that ma kes only each clause fal se, and the truth v alue assignment do es not b elong to the false equiv alence of other clauses. In other words, the equiv alence class is not a combination o f other clauses of MUC. Thus, betw een the clause and the other clauses, there is the orthog o nal that could no t save the lo gical v alue in transpo sition. T o put it the other w ay round, a ny false clause in the sa me equiv alence can transp ose ano ther clause ea ch other. Thus, b etw een the clause a nd the ano ther, there is the corr e la tion that can save the lo gical v a lue in tr a nsp osition. Theorem 5. Each clauses of the MUC have the ortho gonality that c an not r epla c e with a c ombinatio n of other clauses in the MUC. If MUC have the e quivalenc e class that have some false cla uses, the clauses ha ve the c orr elatio n that c an r eplac e e ach other. Pr o of. W e prov e the clauses orthog onality o f MUC using pro of by contradiction. W e assume Cla us es C 1 without o rthogonality is included in the MUC. Because C 1 do es not ha ve the o rthogona lit y , the truth v alue assignments that C 1 is false will be false in another clauses. But this is inco nsistent with the terms of the MUC (there ar e truth v alue assignment clause that clause is false). Th us F ro m the pr o of b y contradiction, clauses of MUC hav e orthog onality . It is clear that the clauses hav e the cor relation if th e cla uses have the same equiv alence class.  Logic v alue assignment o f MUC and ortho g onality/correlation ha s a deep rela- tionship. Log ical v a lue Assignment represents the relationship b etw een the some clauses which is false in truth v alue assignment. In other w ords, the same v alue clauses in same log ical v alue ass ig nment hav e cor relation. T o put it the o ther wa y round, the different v alue clauses in same logical v a lue as signment ha ve orthogo- nality . In addition, there are the c y cric v a lue assignment to ea ch clauses in MUC, there must also b e orthog onal of the each clauses. Ther efore, to increa se the or- thogonality b et ween MUC claus es, it is necessar y that all logical v a lue assignmen t is cyclic v alue assignment in MUC. Theorem 6. Al l clauses in MUC that al l lo gic al value assignment is cyclic value assignmet is ortho gonal. And any truth value assignment is fal se at only one clause. Pr o of. It is clea r from the definition of the cyc lic v a lue assignment.  In addition, MUC is the CNF that is false of all tr uth v alue assig nment, and there is always a clause corresp onding to the equiv alence classes. In other words, MUC is complete sy stem ba sed on the equiv a lence classes o f the tr uth v alue assig nment. CONNECTION AND DIS PERSION OF COMPUT A TION 4 Theorem 7. MUC is c omplete system b ase d on the e quivalenc e classes of t he truth value assignment. Pr o of. It is clear from the definition of the equiv a lence classes of the tr uth v alue assignment.  Thu s, MUC thats all logica l v alue as signments are cyclic v a lue assignments is complete orthogonal function. Theorem 8. MUC thats al l lo gic al value assignments ar e cyclic value assignments is c omplete ortho gonal function. And every clauses ar e ortho gonal b asis of the c omplete ortho gonal function. I wil l use the term “Ort ho gonal MUC” to the MUC. Pr o of. Judging from the ab ove6 7, this is clear .  Thu s, b y reducing logical v alue a ssignments to all cyclic v a lue assignments with keeping the orthogona lity , we ca n express the MUC size in the num ber of cla uses. 3. Or thogon aliza tion of MUC W e reduce MUC to orthogo nal MUC under the constraints of the HornMUC. First, we define the tw o inner pro duct of clauses, and define the or thogonality of clauses. Secondly , w e clarify the limitations HornMUC. An d we show how to reduce the MUC to the orthogonal MUC. And we show that we can reduce Ho rnMUC to o rthogona l MUC in polynomial time from its connectivit y , and can not reduce MUC. I will use the term “F a ct cla us e ” to the HornCNF’s clauses that include only po sitive v ar iable. “ Goal clause” to the HornCNF’s clauses that include only negative v aria ble. “Rule clause” to the Ho rnCNF’s clauses that is no t fact clause a nd goal clause. “Ca se clause” to the HorncNF’s clauses that is fact c la uses and r ule clauses. 3.1. Clauses inner pro duct and inner harmony . Let us star t with defining the inner pro duct and inner harmony , a nd considering the the these or tho g onality . How ever, considering the dualit y of the conjunction and disjunction, and consider the dual inner pr o duct. Definition 9. I will use the term “ Inner pro duct” as follow; h C 1 , C 2 i = h C 1 ⊥ C 2 i = W ( C 1 ( x i ) ∧ C 2 ( x i )) = ∃ x i ( C 1 ( x i ) ∧ C 2 ( x i )) I will use the term “Inner ha rmony” as the duality of the inner pro duct. h C 1 , C 2 i d = h C 1 ⊤ C 2 i = W  C 1 ( x i ) ∧ C 2 ( x i )  = V ( C 1 ( x i ) ∨ C 2 ( x i )) = ¬∃ x i  C 1 ( x i ) ∧ C 2 ( x i )  = ∀ x i ( C 1 ( x i ) ∨ C 2 ( x i )) W () , V () is the disjunction and conjunction of all truth v alue assign ment v alue. It also defines the inner pro duct and inner ha rmony of more than t w o. An d it also defines the inner pro duct a nd inner harmony of CNF. Also, we can define the orthog onality of inner pro duct a nd inner ha rmony . depending on whether tha t is false in the inner section and ca n be determined for e a ch of the orthogona lit y . Defined as follows: Note that the duality of inner pro duct replace true and false . Definition 10. When the inner pro duct is false, the clauses are orthog onal at inner pro duct. I will us e “ C 1 ⊥ C 2 ” to orthogonal at inner product. Orthogonal CNF at inner pro duct is unsatisfiable CNF. CONNECTION AND DIS PERSION OF COMPUT A TION 5 When the inner harmony is true, the clauses a re orthogonal a t inner harmony . I will use “ C 1 ⊤ C 2 ” to orthogonal at inner harmony . Or thogonal CNF at inner harmony is the CNF that all logical v alue assignment are cyclic v alue ass ignment. I will use ter m “ Clauses o rthogona lization” to reducing tw o cla uses to orthogona l clauses at inner harmony . When the inner harmony is orthognal, the CNF that include thes e clauses are orthogo na l at inner harmony . I will use “ C 1 ⊤ C 2 ” to orthogo nal at inner harmony . 3.2. HornMUC constrain ts. Describe s the Hor nMUC’s co nstraints. HornMUC is the CNF that ea ch clauses have at most one po sitive v ariables. This res trict the phase difference of clauses. Therefo r e, it is a lso restrict the inner harmony . First, we show the res triction of the phase difference in the Hor nMUC cla uses. Theorem 11. Phase differ enc e b etwe en the t wo clauses of HornMUC would b e at most one. In other wor ds, HornMUC cla uses ar e c onne ct e d to gether. Pr o of. W e assume that Hor nMUC hav e t wo clauses C 1 = ( x 1 ∨ x 2 · · · ) , C 2 = ( x 2 ∨ x 1 · · · ) . These clauses are true when T = ( x 1 , x 2 · · · ) = ( ⊥ , ⊥ · · · ) . Thus, HornMUC must include the cla use that is false at truth v alue assignment T . But in order to satisfy this condition, HornCNF include ( x 1 ∨ x 2 ) or b e false reg ardless ( x 1 , x 2 ) . This is contradicts the ass umption that HornMUC include C 1 , C 2 . Thus F rom the pro o f b y contradiction, HornMUC do not include the claus e s these phase differnet more than t wo.  W e ca n see from the Hor nMUC restrict what the structure of HornCNF is re- stricted. HornMUC’s phase difference is at most o ne, Ho rnMUC structure ha s b een connected not only a t who le but also each par t. Thus, HornMUC ca n not cons tr uct the structure w ith the non-co nnected part. Ho rnMUC constitutes a partial order of phases. And cla us e cycle do not affect to the HornMUC’s partial order . Theorem 12. HornMUC c an not c onstruct the structur e with the non- c onne cte d p art. And HornMUC c onstitut es a p artial o r der of phases. Pr o of. Sho ws the HornMUC’s connection. F ro m men tioned ab ove 11, a ll cla uses are connecting o r cross ing. W e ass ume that there are Ho rnMUC that can b e divided in to tw o subse ts of non- connected to each o ther. W e ass ume th a t we can split the HornMUC into t w o subsets that is not connected. The subsets h av e no common v aria bles or hav e only common neg ative v a riables. But if the subsets have no common v a r iables, HornMUC’s unsatisfia bilit y do not c hange to delete one o f these subset. So, This is co ntradicts the assumption of HornMUC. And if the subsets hav e only co mmon nega tive v ariables, Hor nMUC’s unsatisfia bilit y do not change to delete the claus e s that include that neg ative v a riables. So, from the pro o f by contradiction, w e can not divide HornMUC into t wo subsets of non-connected to each other. Shows the HornMUC’s structure of par tially o rdered. About HornCNF’s upp er clause C U and low er clause C L ; · · · ≥ C U ≥ C k ≥ C k − 1 ≥ · · · ≥ C 1 ≥ C L ≥ · · · ⇄ · · · ≥ ( x U ∨ x k · · · ) ≥ ( x k ∨ x k − 1 ∨ · · · ) ≥ · · · ≥ ( x 1 ∨ x L ∨ · · · ) ≥ ( x L ∨ · · · ) ≥ · · · It is clear that r eflexive is satisfied. It is clear that transitivity is satisfied from HornCNF’s constraints (clause include at mo st one po sitive v ar iable.) CONNECTION AND DIS PERSION OF COMPUT A TION 6 W e can see the antisymmetric from the following; ∀ C 1 , C 2 (( C 1 ≤ C 2 ) ∧ ( C 2 ≤ C 1 )) = ∀ F i =1 , 2 , 3 , 4 ( ∀ x 1 , x 2 (( x 1 ∨ F 1 ) ∧ ( x 2 ∨ x 1 ∨ F 2 ) ∧ ( x 2 ∨ F 3 ) ∧ ( x 1 ∨ x 2 ∨ F 4 ))) = ∀ x 1 , x 2 (( x 1 = x 2 ) ∧ ( x 2 ∨ x 1 ) ∧ ( x 1 ∨ x 2 )) = ∀ x 1 , x 2 ( x 1 = x 2 ) F i =1 , 2 , 3 , 4 : C N F So HornMUC clauses constitutes a partial order.  This constr a in HornMUC’s inner harmony . Hor nMUC co nstitutes a partial or der from empt y clause leading to fact clause, rule clause, goal clause, and finally empty clause. Thus, we wan t to split the HornMUC, we can only cut in to upper and low er part of this pa rtial order. Theorem 13. When we split the HornMUC, we c an only cut into upp er and lower p art of this p artial or der. Each p art is a p artial or der. Cutting clause exist only one p art. Pr o of. It is clea r that Hor nMUC is a partial order.  F or example, if you divide the ( x 0 ) ∧ ( x 0 ) b y MUC; ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 2 ∨ x 3 ) ∧ ( x 2 ∨ x 3 ) ( x 0 ∨ x 1 ) is divide to ( x 0 ∨ x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ∨ x 3 ) . But b ecause HornMUC ca n not b e greater than the dista nce of tw o c la uses, we can not divide an y clause s . 3.3. Clause O rthogonalization by using HornM UC. Descr ib es the way to Orthogona lize clause b y using HornMUC. F or men tio ned ab ov e 1213, we are con- strained when split MUC b y using HornMUC b eca us e o f Ho r nMUC’s partial or der. And w e must cut the clauses when we or thogonalize clauses to o rthogonal part and correla tio n part. F or example, if you or tho g onalize ( x 0 ∨ x 1 ∨ x 2 ) and ( x 0 ∨ x 3 ∨ x 4 ) , w e must cut follows; ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ) = ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ) = ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ∨ x 2 ) = ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ) ∧ ( x 0 ∨ x 3 ∨ x 4 ∨ x 1 ∨ x 2 ) rep eat that cutt ing target clause to orthogo na l part and correlation par t. And finally correlation clause is absorb ed to another clauses whic h matc h base clause. In addition, this reduction is a dded only HornMUC, so we can k eep CNF’s satisfies po ssibility . Definition 14. I will use term “Clause cut” to divide cla use b y using HornMUC. Theorem 15. By cutting c orr elation clauses to ortho gonal p art and c orr elation p art, we c an ortho gonalize the clauses. Pr o of. It is c le a r that w e can ach ieved by generalizing the ab ove pro cedur e. So I omit.  3.4. HornMUC Orthogonalization. Describes that w e can reduce Hor nMUC to orthogonalize MUC in p olynomial time. F or mentioned above 15, we can reduce HornMUC to ortho g onalize MUC b y cutting all co rrelation parts. And we can re- duce more easier HornMUC to orthogonalize MUC by using HornCNF’s c o nstraint. CONNECTION AND DIS PERSION OF COMPUT A TION 7 Specifica lly , we can reduce that we cut the cla use a t the negative v aria ble corr e- sp o nd to the p o sitive v ar iable of low er clause that is near er from fact clause. T hus, we can absorb every cutting. F or exa mple to O rthogona lize ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 1 ∨ x 2 ) ∧ ( x 2 ) , cutting ( x 1 ∨ x 2 ) at x 0 , c utting ( x 2 ) at x 0 , and cutting ( x 0 ∨ x 2 ) at x 1 . ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 1 ∨ x 2 ) ∧ ( x 2 ) = ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 2 ) = ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 2 ) = ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 2 ) ∧ ( x 0 ∨ x 2 ) = ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 2 ) = ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) = ( x 0 ) ∧ ( x 0 ∨ x 1 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) Because the num ber of claus e s do not increa se and the num ber of v a riables in each clauses increa se at most p olyno mial s ize , so this reductio n increase at most po lynomial size. And HornMUC clauses constitutes a partial or der and it is enough to o r thogonalize each clause a t low er clauses. So the r e duction Hor nMUC to or- thogonal MUC takes at most polyno mial time. Theorem 16. W e c an r e duc e HornMUC to ortho gonal MUC at most p olynomial size and time. Pr o of. It is c le a r that w e can ach ieved by generalizing the ab ove pro cedur e. So I omit.  Theorem 17. The clauses and variables of the ortho gonal MUC that r e duc e d fr om Horn MUC c onstitute total or der. Pr o of. F rom the ab ove pro cedure, the clause of orthogonal MUC include the nega- tiv e v ar ia ble that’s p ositive v a r iable is included in the low er clauses. So, all clauses hav e order, and orthog o nal MUC constitute total order.  Theorem 1 8. If we c an n ot r e duc e MUC to ortho gonal MUC in p olynomial time by u sing clause cutting, then P 6 = N P . Pr o of. F rom the ab ov e 17, if MUC is Hor nMUC, then w e ca n reduce the MUC to o r thogonal MUC by using cla use cutting in po ly nomial time and s ize. And if P = N P , then we can r e duce MUC to HornMUC in po ly nomial time and s ize . So , if P = N P and w e can reduce MUC to HornMUC in p olyno mial time, then we ca n reduce the MUC to o rthogona l MUC in po lynomial time and size. T aking the cont rap ositive, if we can not reduce MUC to orthogonal MUC in po lynomial time and size, then P 6 = N P o r w e can not reduce MUC to HornMUC in polyno mial time and size. So, from the ab ov e 1, if we can not reduce MUC to orthogona l MUC in p olynomial time and size b y using clause cutting, then P 6 = N P .  3.5. MUC Orthogonalization. Describes that we ca n not reduce MUC to o r - thogonalize MUC in polyno mia l time. U nlike HornMUC, MUC can be set to any phase difference be tw een the clauses . So MUC cla uses hav e high dispartion and correla tio n. But for men tioned above 12, Hor nMUC connected each clauses. So w e m ust use many HornMUC each other to cut every dispart areas to orthogonal part and correlation part. And b ecause of HornMUC’s connection, we can not use s a me HornMUC to cut different areas. CONNECTION AND DIS PERSION OF COMPUT A TION 8 F or example to this, we think MUC include follow clauses; O 3 ( x 0 , x 1 , x 2 ) ∧ E 3 ( x 3 , x 4 , x 5 ) O 3 ( x 0 , x 1 , x 2 ) = ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) E 3 ( x 3 , x 4 , x 5 ) = ( x 3 ∨ x 4 ∨ x 5 ) ∧ ( x 3 ∨ x 4 ∨ x 5 ) ∧ ( x 3 ∨ x 4 ∨ x 5 ) ∧ ( x 3 ∨ x 4 ∨ x 5 ) In other words, O 3 is true when it contains an odd trues in truth v alue a ssign- men t, E 3 is true when it contains ev e n trues in truth v alue assig nmen t. O 3 and E 3 divide the truth v alue assignment. So if we w ant to orthog onalize the MUC that include O 3 and E 3 , we must cut every ar ea by using every o ther HornCNF. This orthogo na lize MUC is the MUC that expanded to CNF; O 3 ( x 0 , x 1 , x 2 ) ∧ ( E 3 ( x 3 , x 4 , x 5 ) ∨ (( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 ) ∧ ( x 0 ∨ x 1 ∨ x 2 )) Theorem 19 . When we ortho gonalize CNF that disp arte d some ar e a, W e must cut e ach ar e a by using every another HornMUC. Pr o of. T o as sume that we can orthogonalize t wo a rea by using one HornMUC. This time, there is other clos ure ar ea b etw e en the divided a rea. F rom assuming, w e can orthogo na lize these area by us ing one HornMUC. But w e must cro ss the HornMUC at the clo sure that divided a rea. This is contradicts the assumption that we can orthogo na lize by using one HornMUC. Th us, F rom the pr o of by c ontradiction, w e can not o rthogona lize t wo area by using o ne HornMUC.  As a example, let us co nstruct the MUC that hav e man y divided a rea amoun t of non-po lynomial size of MUC. W e should notice that the truth v a lue assignments having same even-odd of true v alue do not connect each other. By combining prop er O 3 and E 3 ab ov e, it is p ossible to co nfigure the CNF that is true when the truth v alue assig nment having same ev en-o dd of true. F o r example; O 4 ( x 0 , x 1 , x 2 , x 3 ) = O 3 ( x, x 0 , x 1 ) ∧ E 3 ( x, x 2 , x 3 ) This CNF is true only if the truth v alue having o dd of true v alue. This can be extended ea s ily for any num ber of v ariables. O n +1 ( x 0 , · · · , x n ) = O n ( x, x 0 , · · · , x n − 2 ) ∧ E 3 ( x, x n − 1 , x n ) Nun ber of O n clause is p o lynomial size of v ar iables. And truth v alue assignment having odd of true v a lue is amount of half of all truth v alue a ssignment, and every truth v a lue as signment ha ving same even-o dd of true do not co nnect each other. So O n hav e many divided area amount of non-p o lynomial siz e . And there is MUC that we can not reduce to orthogonal MUC in p olynomia l size. Theorem 20. ther e is MUC t hat we c an not r e duc e to ortho gonal MUC in p oly- nomial size by u sing HornMUC. Pr o of. F or mentioned ab ov e ex a mples, ther e is CNF that divide the truth v a lue assignment in to the sa me even-o dd of true v alue. And these CNF can b e a part of the MUC. So there is the MUC that divided area a mount of no n-p o lynomial size . F or mentioned ab ove 19, w e m ust cut each a rea to orthog onalize MUC. So there is MUC that we can not reduce to orthogonal MUC in p olynomia l size.  4. DP is not P and NP is not P Describ es the difference b etw een MUC decision problem and HornMUC dec is ion problem. F or mentioned a bove 3, the difference of the MUC decisio n problem and HornMUC decis io n problem will app ear in CNF classificatio n. And o rthogona l ba sis of inner harmony is different b etw e e n MUC and Hor nMUC. F or mentiond ab ov e CONNECTION AND DIS PERSION OF COMPUT A TION 9 16, all Hor nMUC hav e polyno mial size or thogonal basis. But F or mentiond ab ove 20, there is MUC that hav e non-po lynomial size ortho g onal basis. All orthogona l MUC’s clause is orthogonal basis, and corresp ond to the equiv alence class . So we can not r educe MUC to Hor nMUC in p o ly nomial size. So DP 6 = P . And for men tiond above 18, N P 6 = P . References [1] Koj i KOBA Y ASHI, Symmetry and Uncoun tabilit y of Computa- tion. 2010 , 2010ar Xiv1008.2247K [2] C. H. P apadimitriou and M. Y annak akis. The complexity of f acets (and some facets of com- plexit y), Journal of Computer and System Sciences 28 :244-259, 1984.