Algorithmic complexity of pair cleaning method for k-satisfiability problem. (draft version)

Algorithmic complexit y of pair cleaning metho d for k-sati s fiabilit y problem. (draft v ersion) Sergey Kardash April 18, 2012 Abstract It’s known that 3-satisfiabilit y problem is NP-complete. Here p olynomial algorithm for solving k-satisfiability ( k ≥ 2) problem is assumed. In case theoretical points are righ t, sets P ans NP are equal. 1 In tro duction Definition 1. F ormulae A(x) is c al le d k-CNF if A ( x ) = n \ i =1 k [ j =1 x σ ij u ij , σ ij ∈ { 0 , 1 } , u ij ∈ { 1 , · · · , m } , ∀ i ∈ { 1 , · · · , n } , ∀ j ∈ { 1 , · · · , k } T - c onjuntion op er ation, S - disjuntion op er ation, m - numb er of variables in formulae, n - numb er of clauses, k - numb er of variables in e ach disjunction, n t - numb er of clause gr oups. x σ = ( x, σ = 0 ¯ x, σ = 1 Example 1. 3 -CNF A ( x ) = ( x 1 ∪ x 2 ∪ x 3 ) ∩ ( ¯ x 1 ∪ x 3 ∪ ¯ x 4 ) . Her e m = 4 , n = 2 , k = 3 , n t = 2 . Definition 2. L et formulae A ( x ) is k-CNF. Pr oblem of defining whether e quation A ( x ) = 1 has solution or not is c al le d k-satisfiability pr oblem of formulae A(or k-SA T(A)). Example 2. k- satisfiability pr oblem of formulae A describ e d in Example 1 (k-SA T(A)) is defining whether ∃ x ∈ B m (b o ole an ve ctor of size m): A ( x ) = 1 . It’s evident that x 0 = (1 , 1 , 1 , 1) makes A ( x 0 ) = 1 . A ( x 0 ) is satisfiable. k-CNF B ( x ) = ( x 1 ∪ x 2 ) ∩ ( ¯ x 1 ∪ x 2 ) ∩ ( x 1 ∪ ¯ x 2 ) ∩ ( ¯ x 1 ∪ ¯ x 2 ) is an example of n ot satisfiable task. Ther e is no x 0 : A ( x 0 ) = 1 . On the c ont r ary A ( x ) = 0 , ∀ x . It was pr oved that 2-sa tisfiability problem has poly nomial s olution (by Kr om [2]). W e a re g oing to show po ly nomial a lgo- rithm(from n ) for an y k − S AT . By the wa y we describe metho d of getting 1 explicit solution of corresp onding e q uation A ( x ) = 1 in case s ource task is s a tisfiable which is po ly nomial from n and metho d of so lving equation A ( x ) = 1 whic h is p olynomia l from nu mber of such solutions. 2 Metho d description Initially new mathematic ob jects and op era tions for them are int ro duced. After description of method in pure mathematic w ay algorithmic presentation which is more readable is giv en. Almost each structure has 2 common s tr uctures asso ciated with it: 1)v aria ble set asso ciated with this structure and 2)some v alue sets of these v ariables. Thoug h they will be defined s eparately it’s easy to see common logic o f their in tro duction. Let x s 1 s 2 ··· s k = ( x s 1 , x s 2 , · · · , x s k ). F urther in order to av oid enumeration of v ariables w hich are no t related to describ ed structure we list impor tant v ariables using such notation. 1 Definition 3. Clause gr ou p signe d T s 1 s 2 ··· s k ( A ) is a set of al l clauses S k j =1 x σ t j s j wher e u i 1 u i 2 · · · u ik = s 1 s 2 · · · s k . V ariable set asso ciate d with T u s 1 u s 2 ··· s k ( A ) (or X ( T s 1 s 2 ··· s k ( A )) ) is x s 1 s 2 ··· s k . V alue of clause gr oup T s 1 s 2 ··· s k ( A ) is a value of x s 1 s 2 ··· s k such that k-CNF c onsist e d of al l clauses fr om clause gr oup T s 1 s 2 ··· s k is e qual to 1. V alue set induc e d by clause gr oup T s 1 s 2 ··· s k ( A ) (or V ( T s 1 s 2 ··· s k ( A ) ) is a set of al l values of this clause gr oup. Example 3. Though cla uses x 1 ∪ x 2 ∪ x 3 and ¯ x 1 ∪ x 2 ∪ ¯ x 3 have differ ent de gr e es they b elong to the same clause gr oup T 123 in c ase they pr esent in formulae A. Example 4. F or example clause gr oup T 123 c onsists of clauses x 1 ∪ x 2 ∪ x 3 and ¯ x 1 ∪ x 2 ∪ ¯ x 3 . V alue set induc e d by this clause gr oup c an b e pr esente d us ing table b elow: x 1 x 2 x 3 0 0 1 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 Each r ow c orr esp onds to one value of x 123 . We have e xclude d fr om this list o nly sets wh ich make 3-CNF ( x 1 ∪ x 2 ∪ x 3 ) ∩ ( ¯ x 1 ∪ x 2 ∪ ¯ x 3 ) e qual to 0 ( x 123 = (0 , 0 , 0) and x 123 = (1 , 0 , 1) ). Definition 4 . k-CNF A(x) al l cla uses of that c an b e classifie d into n t clause gr oups is c al le d k-CNF of de gr e e n t . It also c an b e signe d as A n k ( x ) or A k ( x ) or A n ( x ) . Example 5. 2-SA T A ( x ) = ( x 1 ∪ x 2 ) ∩ ( ¯ x 1 ∪ x 2 ) ∩ ( x 2 ∪ x 3 ) ∩ ( ¯ x 2 ∪ ¯ x 3 ) has 2 clause gr oups T 12 and T 23 , so it’s de gr e e is 2 and it c an b e signe d as A 2 2 ( x ) or A 2 ( x ) or A 2 ( x ) . Definition 5. Cla use c ombination F for fo rmulae A ( x ) c onsiste d fr om clause gr oups T u i 1 1 u i 1 2 ··· u i 1 k ( A ) , T u i 2 1 u i 2 2 , ··· ,u i 2 k ( A ) , · · · , T u i l 1 u i l 2 ··· u i l k ( A ) (or F ( T u i 1 1 u i 1 2 ··· u i 1 k , T u i 2 1 u i 2 2 , ··· ,u i 2 k , · · · , T u i l 1 u i l 2 ··· u i l k , A ) ) is a set of liste d clause gr oups. V ariable set asso ci- ate d with it is x h 1 h 2 ··· h r wher e e ach va riable index fr om set of cla use gr oups is pr esente d only onc e. W e’ll deal with different v alue sets of v ar ia bles asso ciated with clause combination and in order not to confuse them let’s write them out s e pa rately . Definition 6. V alue of clause c ombination F ( T u i 1 1 u i 1 2 ··· u i 1 k , T u i 2 1 u i 2 2 , ··· ,u i 2 k , · · · , T u i l 1 u i l 2 ··· u i l k , A ) is a value of x h 1 h 2 ··· h r - vari- able set asso ciate d with it such that k-CNF c onsiste d of al l clauses asso ciate d with liste d clause gr oups e qual t o 1. Definition 7 . V alue set of cla use c ombination F ( T u i 1 1 u i 1 2 ··· u i 1 k , T u i 2 1 u i 2 2 , ··· ,u i 2 k , · · · , T u i l 1 u i l 2 ··· u i l k , A ) b ase d on A ( x ) is a set of values of this clause c ombination. Definition 8. V alue set of clause c ombination F ( T u i 1 1 u i 1 2 ··· u i 1 k , T u i 2 1 u i 2 2 , ··· ,u i 2 k , · · · , T u i l 1 u i l 2 ··· u i l k , A ) induc e d by A (x) is a set of al l values of this clause c ombination. It’s easy to see that v alue se t induced by clause com bination F ( T u i 1 1 u i 1 2 ··· u i 1 k , T u i 2 1 u i 2 2 , ··· ,u i 2 k , · · · , T u i l 1 u i l 2 ··· u i l k , A ) is a v alue set based on this clause com bination. Example 6. L et we have 2 clause gr oups: T 12 ( A ) which has clauses x 1 ∪ x 2 and ¯ x 1 ∪ x 2 in formulae A and T 23 ( A ) which has clauses x 2 ∪ x 3 and ¯ x 2 ∪ ¯ x 3 . Then value set induc e d by clause c ombination F ( T 12 , T 23 ) is a set of al l p ossible values of x 123 which make 2-SA T ( x 1 ∪ x 2 ) ∩ ( ¯ x 1 ∪ x 2 ) ∩ ( x 2 ∪ x 3 ) ∩ ( ¯ x 2 ∪ ¯ x 3 ) e qual to 1. x 1 x 2 x 3 0 1 0 0 1 1 Each r ow of t he list is a value of clause c ombination F ( T 12 , T 23 ) , i. e. x 123 = (0 , 1 , 0) . Definition 9 . Rela tionship s tructur e for k-CNF A ( x ) ( R ( A ) ) is a set of al l p ossible clause c ombinations c onsiste d of ( k + 1 ) clause gr oups. Example 7 . F or 2-CNF A ( x ) = ( x 1 ∪ x 2 ) ∩ ( x 1 ∪ ¯ x 2 ) ∩ ( x 2 ∪ x 3 ) ∩ ( x 1 ∪ ¯ x 3 ) ∩ ( x 1 ∪ x 4 ) ∩ ( ¯ x 1 ∪ x 4 ) clause gr oups ar e: T 12 , T 23 , T 13 , T 14 . R ( A ) = { F ( T 12 , T 23 , T 13 ) , F ( T 12 , T 23 , T 14 ) , F ( T 12 , T 13 , T 14 ) , F ( T 23 , T 13 , T 14 ) } . 2 Definition 10. V alue set of r elationship structur e i nduc e d by k-CNF A ( x ) ( V i ( R ( A ))) is a set of value sets of clause c ombinations induc e d by A ( x ) involve d in r elationship structu r e b ase d on k-CNF A ( x ) . Example 8. F or Example 7 value s et of r elationship structu r e induc e d by k-CNF A ( x ) is a set of tables liste d b elow: V ( F ( T 12 , T 23 , T 13 , A )) : x 1 x 2 x 3 1 0 1 1 1 0 1 1 1 , V ( F ( T 12 , T 23 , T 14 , A )) : x 1 x 2 x 3 x 4 1 0 1 1 1 1 0 1 1 1 1 1 , V ( F ( T 12 , T 13 , T 14 , A )) : x 1 x 2 x 3 x 4 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 , V ( F ( T 23 , T 13 , T 14 , A )) : x 1 x 2 x 3 x 4 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 0 1 V i ( R ( A )) = { V ( F ( T 12 , T 23 , T 13 , A )) , V ( F ( T 12 , T 23 , T 14 , A )) , V ( F ( T 12 , T 13 , T 14 , A )) , V ( F ( T 23 , T 13 , T 14 , A )) } . Definition 11. V alue set of r elationship structur e b ase d on k-CNF A ( x ) ( V b ( R ( A )) ) is a set of value sets of clause c ombinations b ase d on A ( x ) invo lve d in r elationship struct u r e b ase d on k-CNF A ( x ) Example 9. F or Example 7 value set of r elationship structu r e b ase d on k-CNF A ( x ) is any s et V b ( R ( A )) = ( V 1 , V 2 , V 3 , V 4 ) wher e V 1 ⊆ V ( F ( T 12 , T 23 , T 13 )) , V 2 ⊆ V ( F ( T 12 , T 23 , T 14 )) , V 3 ⊆ V ( F ( T 12 , T 13 , T 14 )) , V 4 ⊆ V ( F ( T 23 , T 13 , T 14 )) . In example: V 1 : x 1 x 2 x 3 1 1 0 1 1 1 , V 2 : x 1 x 2 x 3 x 4 1 1 1 1 , V 3 : x 1 x 2 x 3 x 4 1 0 0 1 1 0 1 1 1 1 0 1 , V 4 : x 1 x 2 x 3 x 4 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 0 1 Definition 12. V alue set of r elationship struct u r e b ase d on k-CNF A ( x ) is c al le d empty ( V ( R ( A )) = ∅ ) if at le ast one value set of clause c ombination va lue set of r elationship structur e c onsists of is empty. Definition 13. L et R (A) - r elationship structur e f or k-CNF A ( x ) . V ( R ( A )) = { V 1 , V 2 , · · · , V t , } , G ( R ( A )) == { G 1 , G 2 , · · · , G t , } - 2 value sets of this rel ationship stru ctur es b ase d on A ( x ) . We c al l V ( R ( A )) include d in G ( R ( A )) ( or V ( R ( A )) ⊆ G ( R ( A )) ) if V i ⊆ G i , ∀ i ∈ { 1 , · · · , t } . Example 10. Le t V(R(A)) is a set describ e d in Example 9 and G(R(A)) is a set fr om example 8. V ⊆ G . Inde e d al l value sets of r elationship structur e b ase d on k-CNF A ( x ) ar e include d in the value set of r elationship structu r e induc e d by k-CNF A ( x ) . Definition 14. L et we have 2 clause c ombinations F ( T i 1 , T i 2 , · · · , T i s , A ) and F ( T j 1 , T j 2 , · · · , T j r , A ) . L et they have c ommon variables x i 1 , x i 2 , · · · , x i s - those variabl es which pr esent in b oth clause c ombinations. Cle aring of given p air of value sets V 1 and V 2 of clause c ombinations F ( T i 1 , T i 2 , · · · , T i s , A ) and F ( T j 1 , T j 2 , · · · , T j r , A ) c orr esp ondingly b ase d on k-CNF A(x ) is a pr o c ess of deleting x 1 a 1 a 1 ··· a z ∈ V 1 for which ∄ x 2 b 1 b 2 ··· b u ∈ V 2 : x 1 i 1 i 2 ··· i s = x 2 i 1 i 2 ··· i s and de leting x 2 b 1 b 2 ··· b u ∈ V 2 for which ∄ x 1 a 1 a 1 ··· a z ∈ V 1 : x 1 i 1 i 2 ··· i s = x 2 i 1 i 2 ··· i s . Cle aring pr o c e dur e is bri efly marke d as C ( V 1 , V 2 ) . Example 11. L et’s take 2 values of clause c ombinations fr om Example 8: V ( F ( T 12 , T 23 , T 13 , A )) : x 1 x 2 x 3 1 0 1 1 1 0 1 1 1 and V ( F ( T 23 , T 13 , T 14 , A )) : x 1 x 2 x 3 x 4 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 0 1 . Common variables ar e x 123 = ( x 1 , x 2 , x 3 ) . L et’s explo r e table which c orr esp onds to V ( F ( T 12 , T 23 , T 13 )) . x 1 123 (1) = (1 , 0 , 1) has c orr esp onding x 2 1234 (3) = (1 , 0 , 1 , 1 ) (in br ackets x 2 1234 (3) , 3 is a numb er of r ow in the t able) and it should b e save d. x 1 123 (2) has even 2 c orr esp onding r ows: x 2 1234 (4) and x 2 1234 (5) . But for last one, x 1 123 (3) , we c an ’t find c orr esp onding values fr om se c ond table with the same c ommon variables and it should b e delete d fr om values b ase d on V ( F ( T 12 , T 23 , T 13 )) . The same should b e done with x 2 1234 (1) and x 2 1234 (2) . After cle aring 3 V 1 : x 1 x 2 x 3 1 0 1 1 1 0 and V 2 : x 1 x 2 x 3 x 4 1 0 1 1 1 1 0 1 1 1 0 1 . It c an b e briefly marke d as C ( V ( F ( T 12 , T 23 , T 13 , A )) , V ( F ( T 23 , T 13 , T 14 , A ))) = ( V 1 , V 2 ) . Definition 15. Cle aring of value set of r elationship str u ctur e ( V r ) b ase d on k- CNF A ( x ) (p air cle aning metho d for formulae A ( x ) ) is a pr o c ess of cle aring of al l p ossible p airs of value sets of clause c ombination b ase d on k-CNF A ( x ) c ontaine d in V r until cle aring is imp ossible. We’l l note r esult of cle aning as C(V(R(A))). Pair cleaning metho d in alg orithmic form V new ← V sourc e ( R ( A )) rep eat V old ← V new for i = 1 → d − 1 do for j = i + 1 → d do ( V i new , V j new ) ← C ( V i new , V i new ) end for end for un til V new = V old where d - n umber of clause com binations in relationship str ucture, V sourc e ( R ( A )) - v alue set of relationship struc tur e induced b y A ( x ). Definition 16. L et V = V ( R ( A )) - value set of r elationship struct ur e b ase d on k-CNF A ( x ) . V is c al le d uncle ar able if V = C ( V ) . Lemma 1. L et V = V ( R ( A )) - value set of r elationship st ructur e induc e d by k-CNF A ( x ) . V r es = C ( V ) . V r es 6 = ∅ ⇔ ∃ V 1 ⊆ V r es wher e V 1 - uncle ar able value set of r elationship structur e b ase d on k-CNF A ( x ) wher e e ach value set of clause c ombination c onsists of 1 value of this clause c ombination. Pr o of. ⇒ This can easily b e prov ed using induction. W e’ll ta ke induction not for clauses but for claus e groups. In this pro o f n t - num ber of clause groups. It’s eviden t that n t ≤ n . In case n t ≤ k + 1 statement is eviden t because cleaning of v alues of relationship structure is reduced to clear ing the only clause com bination. Let the case n 0 t = k + 1 be the basis of induction. Let’s assume statemen t is right fo r n t > k + 1. W e need to prov e ( n t + 1) case. Let A n t +1 ( x ) - so urce k-CNF (see Definition 4). R = R ( A ) - rela tionship s tr ucture for it. V - v a lue set of relationship structure induced by k-CNF A ( x ). Let V C = C ( V ) - re sult o f pair clear ing metho d which is not empty ( V C 6 = ∅ ). After clearing relationship structure induced by k-CNF with ( n t + 1) clause groups we hav e not empty v a lue set of relationship structure . Let’s choose an y clause group T n t +1 (w e’ll use b oth t yp es of notation - T i 1 i 2 ...i k which shows v ariables in volv ed in clause g r oup building and T j , j ∈ { 1 , · · · , n t + 1 } ) - a serial num b er of clause group from for m ulae A n t +1 ( x ). Let’s lo ok at B n t ( x ) - formulae whic h has the same cla us e gro ups as A n t +1 ( x ) exc luding T n t +1 . Let R B - relatio nship structure ba sed on B n t ( x ), V B - v alue set of this r elationship str uc tur e. It’s evident that all clause com binations of R B are clause combinations of R . Beside them R has clause combinations which contain T n t +1 with all p ossible combination without rep etition of k clause groups which are common for A n t +1 ( x ) and B n t ( x ) (i. e. F ( T n t +1 , T 1 , T 2 , · · · , T k )). Let’s V B has v alue s ets of claus e com binations the same a s v alue sets of corresp onding clause co m binations of V C . It’s evident that C ( V B ) = V B . V C 6 = ∅ ⇒ V B 6 = ∅ ⇒ exists V 1 B ⊆ V B where V 1 B - unclear able v alue set o f relationship struc tur e based on k-CNF B n t ( x ) wher e each v alue set o f clause co mbination consists of 1 v a lue. (a ccording to induction step). Now w e need show that ∃ V 1 A ⊆ V A - unclearable v a lue set of relationship structur e ba sed on k-CNF A n t +1 ( x ) where each v alue set of clause combination consists of 1 v alue. This pr o of is very trivial. Indeed, let’s lo ok a t T n t +1 (another notation for this clause group is T l ( n t +1)1 l ( n t +1)2 ··· l ( n t +1) k ). In this clause g roup there are 2 t yp es o f v aria bles: those that present at least in o ne clause g roup T j , j ∈ { 1 , · · · , n t } (common v ar iables) and those that absent in this set. Let’s explore first group (present). W e can say that exists such clause co m bination F ( T n t +1 , T i 1 , T i 2 , · · · , T i k , A ) from relationship structure R wher e all common v a r iables fro m T n t +1 can b e found at lea st in one of other mem b ers of this clause combination: T i 1 , T i 2 , · · · , T i k . This statemen t can e a sily b e proved b y building this clause combination. Number of common v ariables can’t be grea ter than k . So w e ca n find corres po nding clause gro up for each common v aria ble which also contains this v ariable. Number of such clause groups is less or equa l k and if it’s les s we add a rbitrary cla us e groups in order to get clause combination whic h co n tains k + 1 clause gr oups. And no w let’s build another cla use combination F ( T i k +1 , T i 1 , T i 2 , · · · , T i k , A ) 4 which has k common clause groups with F ( T n t +1 , T i 1 , T i 2 , · · · , T i k , A ) and T i k +1 is a clause g roup from B n t ( x ) (this clause group can b e found b ecause n t > k + 1). By the wa y we nee d pro ve that each v ariable of clause combination in unclearable v alue set of relatio nship str ucture where each v alue set of cla us e combination consists of 1 v a lue has the s a me v alue in all cla use combinations of that v alue of rela tionship structure. This result will also be used in next le mma. That’s easy to be shown. Let x i - arbitrar y v ar iable presen ted in relationship structure. Let F 1 = F ( T x i x j ··· ) and F 2 = F ( T x i x z ··· ) - 2 differen t clause combinations whic h are parts of relationship structur e R . V 1 - unclear able v a lues of relationship structure wher e eac h v alue s et of clause com bination consists of 1 v alue. Let v alue V F 1 1 of clause com bination from V 1 which co r resp onds F 1 and v a lue V F 2 1 of clause combination fr om V 1 which cor r esp onds F 2 hav e different v a lue of v ariable x i . Then op eration C( V F 1 1 , V F 2 1 ) will give empt y sets to both v alues. But that’s con tradiction b ecause v a lues of relationship structure is unclearable. The fact that V C is not empty and V 1 B ⊆ V B means that v alue of cla use com bination F ( T i k +1 , T i 1 , T i 2 , · · · , T i k , A ) from V 1 B is also a v alue of the same clause combination from V B and from V C . The fact that it ca n’t b e deleted during clearing means that exists v alue V B T n of clause combin ation F ( T n t +1 , T i 1 , T i 2 , · · · , T i k , A ) fro m V C which has the sa me v alues o f common v ar iables as v alue of F ( T i k +1 , T i 1 , T i 2 , · · · , T i k , A ) from V 1 B . The only thing we need to prov e no w is tha t all clause c o m binations from V C which contain T n t +1 hav e v alue which can be added to V 1 B and V B T n to create new v alue of r elationship structure V 1 C which is unclearable. Let’s notice that these clause comb inations don’t give any new v ariables to clause combinations of R B and F ( T n t +1 , T i 1 , T i 2 , · · · , T i k , A ) . This fact and the fact that in V 1 B all v a lues of the same v a riables in different clause combinations are the same can give us a hint that v a lue of each clause combination which con tains T n t +1 consisted of the same v ariable v a lues as they presen ted in V 1 B and v alue of clause com bination F ( T n t +1 , T i 1 , T i 2 , · · · , T i k , A ) discuss ed in previous pa r agra ph. ⇐ This side is eviden t: the fact that ∃ V 1 ⊆ V r es means that V r es 6 = ∅ . Lemma is prov ed. Lemma 2. L et V 1 - value set of r elationship st r u ctur e b ase d on k-CNF A ( x ) wher e e ach value set of clause c ombination c onsists of 1 value of this clause c ombination. V 1 is uncle ar able ⇔ k-CNF A ( x ) is e qual t o 1 on this value set. Pr o of. ⇒ It was proved in Lemma 1 tha t cor resp onding v ar iables have the s ame v alues in different clause combinations. Let’s hav e a glance at k-CNF which v ariable s v alues are the same as in the s tr ucture. It’s eviden t that such k-CNF is equa l to 1 . Indeed for each clause exis ts claus e combination that in volv es this clause. Clause com bination is equal to 1 o n this set ⇒ clause itself is equal to 1. All clauses o n this set ar e equal 1 ⇒ k- CNF v a lue on this se t is equa l 1. ⇐ This pro of is trivia l. W e take v ar iable v alues x 12 ··· m that make k -CNF equal 1. It’s ev iden t that in v a lue set of relations hip structure V 1 based on A ( x ) ea c h v alue set of clause combination whic h is a mem b er o f V 1 and has the same v ariable v a lues as x 12 ··· m is unclear able. Lemma is prov ed. Theorem 1. Result of p air cle aning metho d applie d to sour c e k-CNF is not empty ⇔ ∃ solution of e qu ation k − C N F = 1 . Pr o of. Cons e c utiv e usa ge of Lemma 1 and Lemma 2 proves the theorem. Theorem 2. L et V - val ue set of r elationship structure b ase d on k-CNF A ( x ) , V C = C ( V ) - cle ar e d value set of re lationship stru ctur e, V 1 C - uncle ar able value set of r elationship structure b ase d on k-CNF A ( x ) wher e e ach value set of clause c ombination b ase d on k-CNF c onsists of 1 value, V F i - value set of clause c ombination F i , V C F i - values of clause c ombination F C i , V 0 F i - value of clause c ombination F i . Then V 0 F i ∈ V F i - memb er of V C = C ( V ) ⇔ ∃ V 1 C : V 0 F i ∈ V C F i - memb er of V 1 C Pr o of. Scheme of proo f is the same as for Lemma 1, it’s full description will be given a bit later. So we ha ve no t o nly algor ithm for solving k -satisfiability problem but also algorithm for solving equa tion A ( x ) = 1. Of cour s e in co mmon case it’s not p oly no mial (beca use n umber o f solutions is O (2 n )). But pro cess of getting each ro ot of equation is po lynomial. W e’ll descr ib e it in full preprint version of this pap er. 5 3 Complexit y Num b er of v alues clause g r oup can ta ke is less than 2 k . Num b er of v alues clause co m bination can take is less than 2 k ( k +1) . Num b er of clause co m binations in relatio nship structure is C k +1 n t . Num b er of compar isons during one iteration pa s s is less than 2 2 k ( k +1) ( C k +1 n t ) 2 . Num b er of iteratio ns is less than 2 k ( k +1) C k +1 n t . That means that n umber of o per ations for algorithm is less than 2 3 k ( k +1) ( C k +1 n t ) 3 . Therefore complexit y o f k − S AT is O ( n 3( k +1) t ). F or 3 - SA T it’s O ( n 12 t ). 2 − k n ≤ n t ≤ n ⇒ method’s co mplexity is O ( n 3( k +1) ). F or 3 -SA T it’s O ( n 12 ). That means that pa ir cleaning metho d is po lynomial a nd P = NP . References [1] Coo k, Stephen (April 20 00). The P versus NP Problem . C lay Mathematics Institute. Retrie ved 2006-10 -18. [2] Krom, Melven R. (1967), ” The Decision P roblem for a Clas s of First-Or der F ormulas in Whic h all Disjunctions are Binary”, Zeitschrift fur Mathematische L o gik und Grund lagen der Mathematik , 13 , pp. 15-20. 6