XSAT of linear CNF formulas

Transcription

1 XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D Kön, Germany; eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open questions with respect to the computationa compexity of inear CN formuas in connection with reguarity and uniformity are addressed. In particuar it is proven that any -reguar monotone CN formua is XSAT-unsatisfiabe if its number of causes m is not a mutipe of. or exact inear formuas one finds surprisingy that -reguarity impies -uniformity, with m = + (-)) and aowed -vaues obey (-) = 0 (mod ). Then the computationa compexity of the cass of monotone exact inear and -reguar CN formuas with respect to XSAT can be determined: XSAT-satisfiabiity is either n trivia, if m is not a mutipe of, or it can be decided in sub-exponentia time, namey On ( ). Subexponentia time behaviour for the wider cass of reguar and uniform inear CN formuas can be shown for certain subcasses ony. Introduction. The computationa compexity of inear formuas has been studied extensivey by Porschen et.a., see e.g. [,2]. NP-competeness of exact satisfiabiity (XSAT) and not-a-equa satisfiabiity (NAE-SAT) was proven for inear monotone formuae and extended to exact inear formua (without monotony). Both variants of SAT were identified as NP-compete aso for -reguar and even -uniform subcasses of CN formuas by these authors. Open questions remained concerning the NP-competeness of monotone -reguar and -uniform exact inear CN instances. Porschen et. a. conjectured that XSAT is NP-compete for this CN cass, and aso in monotone -uniform and -reguar inear CN casses. This communication coects some properties of inear and exact inear CN instances which might be of reevance to the cited conjectures. The paper is an enarged version of earier papers [3, 3 ], in particuar theorems,2,7, and 8.

2 2 Throughout this paper I wi adopt the notation used in [,2] which is shorty repeated here. A Booean formua in Conjunctive norma form (CN) by definition is a conjunction of causes, where each cause is a disjunction of iteras. A itera is an occurrence of a Booean variabe or its negation. In a inear CN formua any two causes have at most one variabe in common. The cass of such formuas is denoted by LCN. In an exact inear formua any two causes have exacty one variabe in common. The cass is denoted by XLCN. A monotone formua contains positive iteras ony. Monotony is denoted by a subscript +, e.g. LXCN. -reguarity is the property that each variabe occurs times. It is denoted by a superscript, e.g. CN. inay, uniformity means that every cause contains the same number of iteras. If this number is, one writes e.g. - CN. XSAT is the probem of deciding whether for a given CN formua there is a truth assignment (aso caed mode) that evauates exacty one itera in each cause of to true. If there is at east one such assignment is said to be x-sat, otherwise x-unsat, for short. If the context is cear we denote the number of causes of some formua, i.e. by m, its number of variabes, V( ), by n. The paper is organized as foows. irst we consider exact inearity in genera. This property imposes severe imitations on the structure of such formuas. Quite generay it can be shown that the majority of causes in an exact inear CN formua is onger than the argest occurrence x ( ) of any variabe x, i.e. xv ( ) V ( C) max ( x) for xv ( ) m max ( x) many causes C. This is a generaization of the statement that -uniform and - reguar XLCN do not exist uness (as stated in theorem 3 in [2]). Some genera reations for XLCN formuas are then derived. When reguarity is added further severe restrictions in exact inear formuas arise. In particuar it is shown that an exact inear -reguar CN formua necessariy is - uniform, with certain restrictions on the aowed -vaues. As a by-product we get the number of causes and the number of variabes for fixed and as m ( ) and 2 n ( ) /. Then we turn to the question of XSAT-satisfiabiity of monotone formuas in genera and introduce a simpe method using some straightforward considerations concerning the number of true iteras. Then we consider the impications of -reguarity for the x-satisfiabiity of genera monotone CN formuas. In particuar it is proven that any monotone -reguar CN formua is x-unsat if m is not a mutipe of. or monotone -reguar exact inear CN formuas we get the interesting resut that either is x-unsat (if (mod ), i.e. - is not divisibe by ), or x-satisfiabiity is decidabe in time n ess than (2 n n O ) up to poynomia corrections, with some depending on. or monotone inear (not exact inear) formuas which are -uniform and -reguar a simiar type of sub-exponentia

3 3 behavior can ony be found in restricted, rather artificia subcasses, namey those LCN for which each cause is not connected to ( 2) other causes, and a subcass we termed entanged inear chains. Impications for NP-competeness of XSAT of inear and -reguar formuas is shorty discussed in a concuding section. Theorems and proofs. Throughout this section formuas and causes are considered to be non-empty. Aso a positive and negative itera of the same variabe in one cause is excuded. Theorem : Let XLCN and et L : max ( x) be the argest occurrence of a variabe in. Then xv ( ) there are at most L causes C in with C L. PROO: Let C be a cause with ( x ) L for some x V( C). Denote by C2, C3,..., C L the other L- causes that have x in common with C. We assume x ( ) in particuar for the other variabes fromvc ( ) \{ x }. Then there exist V( C ) ( x ) ( x )... ( x ) ( x ) L ( V( C ) ) : madd many additiona causes 2 3 V( C ) i i which have exacty one xi V ( C ) in common with C, but no variabe from VC ( ) in common with each other due to exact inearity. None of these causes C, C 2,..., C can consist of ess than L L L L m add iteras because they must have one variabe in common with each of the L causes C, C2, C3,..., C L. Therefore, for a these causes V ( Cadd ) L. Now assume that there exists a further cause. It shoud have a variabe in common with C due to exact inearity, but cannot have such a connection because a occurrences of the variabes of C are competed aready. Thus no further cause exists. Theorem 2: or XLCN the foowing reations hod (i) The number of causes m is given by m V( C) ( x), for arbitrary C. xv ( C) (ii) ( x) V ( C) ( x) V ( C ') for any two causes C and C of. xv ( C) xv ( C ')

4 4 PROO: or XLCN any cause C is connected to a other causes by exacty one variabe. Since every variabe in C occurs times in the tota number of connections of C to other causes is x ( ). Since this must be the tota number of causes (the cause C itsef) one gets the xv ( C) stated resut for m. Aternative Proof: rom the considerations of the proof of theorem it foows that the tota number of causes is given by m L m V ( C ) ( x). But a considerations stay vaid, if one add xv ( C ) starts with an arbitrary C. This proves (i). (ii) foows from (i), since (i) shows that ( x) V( C) m is an invariant independent of C. xv ( C) Coroary : or -uniform XLCN one has: The sum of occurrences of a variabes beonging to a given cause C is a constant independent of C. This invariant is given by m. PROO: oows from theorem 2, (i) and (ii) since V ( C) V ( C ') for -uniform formuas by definition. Together with theorem this impies that must be arger or equa to the argest occurrence in the formua. Otherwise exact inearity woud be vioated. Note that this is a generaization of the observation made in [2], that no -uniform, -reguar XLCN with < exist. Now we turn to exact inear formuas which in addition are -reguar, i.e. each variabe occurs exacty times. The centra resut is the observation that -reguarity impies -uniformity in exact inear formuas. Theorem 3: The cass XLCN consists soey of -uniform formuas with either (mod ) or m 0 (mod ). urthermore the number of causes and variabes of is given by, respectivey: m ( ), 2 n ( ) /. PROO: The first part of theorem 3 is a specia case of theorem 2, with a occurrences being equa. Thus m V ( C) V ( C) ( ). But m must be independent of C, so we have V ( C) xv ( C) independent of C, or. or the second part of the theorem we observe that the

5 5 tota number of iteras of can be counted in two different ways eading to V ( ) n m since is -reguar and -uniform (which basicay is a specia case of (iii) ). Thus 2 n m / ( ) / or. Since n must be an integer, (-) must be a mutipe of for to exist. If 0 (mod ) aso m 0 (mod ). A other aowed vaues of ead to m 0 (mod ). Coroary 2: or prime the cass XLCN spits into two subcasses with (mod ) and 0(mod ) respectivey. PROO: oows directy from theorem 3 which states that (-) must be a mutipe of. Note that the formua for m for boc designs ( =) is a specia case of a combinatoria consideration in a different context by Ryser [4]. Next I consider XSAT satisfiabiity of some monotone XLCN casses. The method used wi be as foows: or any XSAT-mode y of, i.e. a satisfying assignment which evauates to true with exacty one true itera per cause, the number of true iteras must be equa to m, the number of causes. Thus a XSAT-modes of are among the soutions of the equation ( y ) m, where ( y) G denotes the number of true iteras of any CN formua G under an assignment y. The soutions of the equation ( y ) m wi be caed pseudomodes, because a modes are among these soutions but not vice versa. By counting pseudomodes one has an upper bound on the number of modes. Each pseudomode can be tested on XSAT-satisfiabiity in poynomia time. Since a modes, if one exists, must be among the pseudomodes, the decidabiity of XSAT is bounded by the number of pseudomodes up to poynomia time corrections. Now, for monotone CN the tota number of true iteras is easy to cacuate since each variabe contributes either 0 or x ( ) to the tota sum: chosen assignment. We can now prove ( y) y ( x), where y {0,} specifies the xv ( ) x x Theorem 4: Let and 0 (mod ) CN. Then is x-unsat. PROO: Let (y) denote the tota number of true iteras in for an arbitrary assignment y. The coection. Since is - of pseudomodes is defined by M () : {0,} n pseudo y : ( y) yx( x) xv ( )

6 6 reguar a (x) are equa, and consequenty can tae vaues with n, depending on the assignment. (To be precise: { xv ( ) : y x } ). Thus pseudomodes are defined by ( y ). But the number of causes is not a mutipe of by assumption. So the equation has no soution, M ( ). There is no pseudomode and consequenty no mode. pseudo Note that inearity behod exact inearity was not needed for theorem 4. A specia case of theorem 4 is Theorem 5: Let XLCN and (mod ). Then is x-unsat. PROO: or ( ) as proven in theorem 3, we have 0(mod ) since - is not a mutipe of by assumption. Thus is x-unsat according to theorem 4. Thus in order to carify the computationa compexity of the cass of monotone -reguar exact inear CN formuas one ony needs to consider the subcass with (mod ). Note that Lemma in [2] stating that a members of are x-unsat, is a specia case of theorems 4 and 5 since XLCN m ( ) is not an integer mutipe of. Theorem 6: Any XLCN is either x-unsat or is decidabe in time exp ( )n(n) O f n or better for fixed and up to poynomia corrections, where n V ( ). PROO: irst we note that there is a and is -uniform with m ( ), as proven in theorem 3. Now, if - is not a mutipe of, m is not divisibe by and is x-unsat according to theorem 5. So, if by assumption, (mod ) the equation m does have an integer soution =m/. We can now proceed aong the ines of the proof of theorem 4. Since 0 for the assignment in which a variabes are set to 0 (fase) one can construct a pseudomodes corresponding to by fipping any of the n variabes from 0 to (true). Thus M pseudo n n n, where the ast equaity hods because is -reguar and -uniform. m / n / Since for and given n and m are uniquey determined by equations given in theorem 3, we can go to ever arger formuas, n, fixed, by choosing appropriatey arge (n). One finds

7 7 n n / ( ) / n. Now n can be evauated for arge n ( fixed), e.g. by means of n/ Stirings formua, to give the stated resut, with f() given by f ( ) ( ) / ( ) / n( ( ) / ). inay, the worst agorithm to determine x- satisfiabiity of woud be to ist a pseudomodes and chec each of them for x-satisfiabiity (in time poynomia). Since any mode, if one exists, is bound to be among the pseudomodes, the process of determining x-satisfiabiity of is imited by their number, i.e. M p.. t many steps. pseudo -uniform and -reguar inear CN (not exact inear) are much ess restricted, and their behaviour with respect to XSAT can be expected to differ from their exact counterparts consideraby. Though a considerations concerning the pseudomodes with the resut M pseudo n n n stay m / n / vaid, this time n/ stays O(n) because and can be chosen independenty and need not be changed to attain ever arger formuas. Nonetheess one finds subcasses which sti exhibit sub-exponentia behaviour in the sense of theorem 6. To iustrate this point we construct an arbitrariy arge -reguar -uniform inear formua from its exact inear version. Definition: Let and G be two identica -reguar -uniform exact inear CN formuas except for their variabe sets : V ( ) V ( G). Then we ca G on V ( ) V ( G) a inear 2-chain and its generator. Obviousy G is an -reguar -uniform inear formua which is twice as arge, both in number of causes and number of variabes, as its generating formua. But since and G are competey independent x-satisfiabiity can be determined by doing this for aone and transferring the resut to the other variabe set. A ess trivia inear formua which cannot be divided into two non-overapping subsets (nonoverapping means: for any pair of causes of the two subsets their respective variabe sets have no eement in common) is an entanged inear 2-chain, defined as foows: Let G be a inear 2-chain and js be its iteras numbered with respect to the variabe set {a s},s {,...,n,n,...2n} and causes Cj, j{,...m,m,...,2m}. Then choose a cause number j {,... m} and a variabe index s {,... n}, move js from cause j in to the same cause but the corresponding variabe pace in G, i.e. j, s n. To eep G -reguar move j m, s n to the corresponding pace in, i.e. j m, s. The resuting formua is -reguar -uniform and inear by

8 8 construction, but can no onger be divided into two non-overapping subsets. We ca it an entanged inear 2-chain. By adding further identica formuas on different variabe sets and entange them with the precursor one can generate arbitrariy ong entanged inear chains, imited ony by the number of iteras of the generator. This way we have identified a non-trivia subcass of sub-exponentia with respect to XSAT: LCN which is Theorem 7: Let LCN be a monotone entanged inear p-chain, p {2,...,n}. Then XSAT can be decided in sub-exponentia time O(exp( n( n) n) at most. Proof: Consider the unentanged p-chain first. X-satisfiabiity of the generator of the p-chain can be determined in sub-exponentia time according to theorem 6. Since a assignments for the generator have the same effect on the other chain ins when transferred accordingy, i.e. y sn y for s s {,..., n} and {,..., p }, compexity of the compete chain is ony enhanced by a factor of O(p). Now consider a specific entangement between two chain ins. Let js be the itera that has been transferred to position j, s n, and j m, s n the one that is transferred bacwards to j m, s. If we set ys ys n cause j has the same number of true iteras as cause j+m, and the situation is unchanged with respect to the unentanged 2-chain. The same consideration can be made for a other pairs of ins in the chain. In the next theorem we introduce another subcass of which has sub-exponentia LCN behaviour with respect to XSAT. In preparation we prove the foowing Lemma: or LCN et {( C, C ') : V ( C) V ( C ') } / 2 denote the number of pairs of causes which have no variabe in common (doube counting is taen care of by the factor /2), i.e. are not connected at a. And et m ( ) denote the number of causes of the XL corresponding exact inear formua, and m the number of causes of. Then m( m m XL ) / 2. PROO: Since in a inear formua any two causes either have exacty one variabe in common or none we have m( m ) {( C, C ') : V ( C) V ( C ') {0,}} = {( C, C ') : V ( C) V ( C ') } {( C, C ') : V ( C) V ( C ') } 2 ( ) n. or a -uniform and -reguar formua n gives the stated formua. m additionay. Substituting m for n and soving for

9 9 Theorem 8: or members of the cass LCN for which each cause is unconnected to exacty ( 2) other causes XSAT is decidabe in time O exp n( n) n or better up to poynomia time corrections. PROO: If fufis the assumptions of the theorem one can infer m( 2) / 2, with m. rom the emma we then have m ( 2) ( ) where m has been substituted for n because is -reguar and -uniform. The reation for m- can be rewritten as m ( )( ). This is the formua for an exact inear formua which is -reguar and -uniform and has m+ causes. Indeed we can add causes to with the foowing properties: they have + iteras each and have one variabe in common. So a in a there are ( ) new variabes. Since each cause of the origina is unconnected to ( 2) causes by assumption we can add iteras corresponding to of the new variabes to each of the origina causes in such a way that now each cause has exacty one variabe in common with each other cause, the new causes incuded. Thus the newy constructed is an eement of ( ) XLCN. or the rest of the proof we must assume that is aso monotone. Now we need to show that the newy constructed ' is x-sat iff is. Let x r be the variabe that the added causes have in common. Assume that an assignment y exists which maes ' x-sat. This assignment must have y' r and y' s 0 for a other new variabes, otherwise the added causes woud not be x-sat. Now remains, but since y maes x-sat, aso must be x-sat. If on the other hand y is an assignment that x-satisfies, then y' t y t for the od variabes and y' and y' s 0 for the newy added ones is an assignment that x-satisfies. According to theorem 6 x-satisfiabiity for can be determined in sub-exponentia time. So the same hods for. r Concuding remars. Theorem 3 states a somewhat surprising resut: -reguarity impies -uniformity for exact inear formuas, and one cannot choose and arbitrariy. A tacit assumption in the proof of theorem 3 is that a formua XLCN with at east causes exists at a. But it is not difficut to construct such formuas, at east for ow vaues of and, e.g. 3, 7. Porschen et. a. conjectured in [2] that NP-competeness of XSAT hods for XLCN and raised the question whether this property coud even be transferred to. Theorems 5 and 6 of XLCN

10 0 this study give hints on how this question is to be setted. irst of a, a arge fraction of this cass, namey those formuas where - is not a mutipe of, turns out to be unsatisfiabe with respect to XSAT, according to theorem 5. The rest, as shown in theorem 6, does not dispay the exponentia behaviour of running times when tested for XSAT, as is usuay expected for NP-cass probems, 2 n, but a faster efficiency at most of order n n. The resuts do not excude the possibiity that XSAT satisfiabiity of this cass even is decidabe in poynomia time. It woud be interesting to find an appropriate agorithm. Porschen et. a. aso conjectured that is NP-compete. In theorems 7 and 8 we have LCN named subcasses of which show the same sub-exponentia behaviour as exact inear LCN ones, i.e. a XSAT-decidabiity in ess than 2 cn steps. On the other hand, these subsets are rather artificia and probaby do not cover a substantia fraction of LCN. So our study does not excude the correctness of the cited conjecture. inay I woud ie to mention, that the method using pseudomodes (see proof of theorems 4 and 6) was appied to the monotone case here ony. or non-monotone formuas a more genera theorem [5] can be used and the resuts are more compicated and depend on the detaied structure of the distribution of positive and negative iteras. E.g. one negative itera per variabe is enough to destroy XSAT-satisfiabiity of instances from CN competey, because necessary condition min is raised to n > m, so the m cannot be achieved by any. Thining of -reguar exact inear HORN formuas as a further exampe, one has to dea with inhomogeneous distributions of positive iteras in otherwise negative XLCN. This eads to more compicated situations which are difficut to judge with respect to their asymptotic running times. An exception is a cass of exact HORN formuas (XL and -reguar and exacty one positive itera per cause) with the additiona condition that there is at most one positive itera per variabe. Then min m, and there is exacty one pseudomode to test, impying poynomia time behaviour, a situation which has been described in [6]. References. [] T. Schmidt, Computationa compexity of SAT, XSAT and NAE-SAT for inear and mixed Horn CN formuas, Ph.D. thesis, Institut für Informati, Univ. Kön, Germany (200).

11 [2] S. Porschen; T. Schmidt, E. Specenmeyer, A. Wotzaw, XSAT and NAE-SAT of inear CN casses, Discrete App. Math. 67 (204) -4. [3] B.R. Schuh, XSAT of exact inear CN Casses, [3 ] B.R. Schuh, Exact Satisfiabiity of Linear CN ormuas, submitted to Discrete Appied Mathematics. [4] H. J. Ryser, A note on a combinatoria probem, Proc. American Math. Soc., (950), [5] B.R. Schuh, unpubished. [6] B.R. Schuh, A Criterion for Easiness of Certain SAT Probems,