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  • 8/11/2019 p216 Schaefer


  • 8/11/2019 p216 Schaefer


    (e) Every relation in S is definable by a CNFformula having at most 2 liter al s in eachconjunct.

    (f ) Every relation in S is the set of solutionsof a system of linear equation over the two-element fiel d {0,1}.

    Sections 2-4 are devoted to the statement andproof of thi s Dichotomy Theorem. (Although we usethe word "dichotomy" to describe thi s resul t, i t

    should be borne in mind that the dichotomy holdsonly i f P#NP; i f P=NP, the dichotomy would col-lapse.)

    A variat ion of the problem consists of allow-the constants 0 and l to occur in input formulas(e.g. a clause R(x,O,y) is allowed). Wedenotethis "satisf iabil ity-with-constants" problem bySATc(S)~ Our results for SATe(S) are sharper thanfor SAT(S): we obtain a compTete characterizationup to log-space equivalence. For any fini te set Sof logical relations, SATc(S) li es in one of sevenlog-space equivalence classes, described as follows:

    I. SATc(S) is decidable deterministical ly in logspace.

    2. The complement of SATe(S) is log-equivalent to

    the graph reachabili t~ problem (given a graph Gand nodes s,t of G, do s and t lie in the sameconnected component of G?).

    3. The complement of SATe(S) is log-equivalent tothe digraph reachabil~ty problem (given a direc-ted graph G and nodes s, t of G, is there a di-rected path from s to t?). In this case,SATc(S) is log-complete in co-NSPACE(Iog n).

    4. SATc(S) is log-equivalent tothe problem of deciding whether a graph isbipartite.

    5. SATc(S) is log-equivalent to the problem ofwhether an arbi trary system of li near equationsover the fi eld {0, ]} is consistent.

    6. SATc(S) is log-complete in P.

    7. SATc(S is log-complete in NP.

    This result is presented in Section 5. For "most"sets S, SATc(S is essentially ident ical to, andhas the same complexity as, SAT(S). (See Lemma 4.2.)Of course, i t is not known that the above sevenclasses are distinct.

    In Section 6, we present a polynomial-spaceanalogue of the Dichotomy Theorem, involving quan-ti fied formulas. Wedefine QFc(S) to be the analogof SATc(S) in which formulas contain universal andexistential quantifiers quantifying the proposition-al variables. The main theorem of this sectionstates that for any fi ni te set S of logical rela-tions, QFc(S is ei ther polynomial-time decidable

    or log-complete in polynomial space. For bothQFc(S ) and SATc(S) the polynomial-time decidableca~es are just- the cases (c) -( f) li sted above;cases (a) and (b) are excluded.

    Wemen~ion here a few par ticular completenessresults whic~fol low from these general theorems.Problems NPI,NP2 and NP3 are NP-complete.


    Given sets S . . . . . S_ each having at most 3 mem-bers, is there a subset T of the members suchthat for each i, ITnSiJ = l ?


    Given sets S. , . .. ,S m each having at most 3members, canlthe meJiibers be colored with twocolors so that no set is all one color?


    Given a graph G, can the nodes of G be coloredwith two colors so that each node has exactlyone neighbor the same color as itsel f? (G maybe restri cted to be planar and cubic.)(Theorem 7.1)

    Problems PI and P2 are log-complete in P.



    SAT3W(Weakly Positive Sati sf iabili ty )

    Given a CNF formula having at most 3 l iteralsin each clause, and having at most one negatedvariable in each clause, is i t sati sfiable?(Corollary 5.2)


    Given sets S .. . . . S m each having at most 3members, and'a distTnguished member s, can onechoose a subset of the members, containing s,so that no set has exactly one member chosen?(Corollary 5.2)

    This paper contains a ful l proof of theDichotomy Theorem. The other resul ts are, for themost part, stated without proof.

    Technical Note. The def init ion of "logical relation"given above is deficient in that i t fai ls to dif fe r-entiate between empty relat ions of diff er ing ranks.Therefore, we formally define a logical relation tobe a pair (k,R) with R~{O,1}k; but informally weshall continue to regard R i t self as being therelation.


    This section states and discusses the mainresult of this paper, the following theorem.

    Theorem 2.1. (Dichotomy Theorem for Sa tisf iab ility) .Let S be a fi ni te set of logical relations. I f Ssat isf ies one of the conditions ( a) -( f) below, thenSAT(S) is polynomial-time decidable. Otherwise,SAT(S) is log-complete in NP. (See below fordefinitions).

    (a) Every relation in S

    (b) Every relation in S

    (c) Every relation in S

    (d) Every relation in S

    (e) Every relation in S(f) Every relat ion in S

    is O-valid.

    is l-valid.

    is weakly positive.

    is weakly negative.

    is affine.is bijunctive.

    Definitions.(The following defini tions were all invented forthis paper and should not be assumed to agree withterminology used elsewhere.)

    The logical relation R is O-valid i f (0 ... .. O)E R. The logical relat ion R is l- val id i f(I .. .. . I)~R.

    2 1 7

  • 8/11/2019 p216 Schaefer


    The logic al rel at ion R is weakly pos iti ve(resp. weakly negative) i f R(x .. . . ) is l ogi ca ll yequ iva len t to some CNF f o r ~ l a having at mo st onenegated ( resp. unnegated) va ri ab le in each con-junct.

    The logical relat ion R is b ijunc tive ifR(x I .. . . ) is lo gi ca ll y equivalent to some CNF for -~ul~ having at most 2 l it er a ls in any conjunct.

    The logical relat ion R is af fine if R(x .. . . )is logically equivalent to somesy--~m of-linearequations over the two-element fi el d {0 , I} ; that

    is, if R(x~ .. . ) is log ic all y equivalent to a con-jun ct io~ o~ formulas of the forms ~I ~ ~Cn = 0and ~l ~' '' ~n =l , where ~ denotes additi onmodulo 2.

    Compl exit y-th eore tic notion s, such as P, NP,log-space reducibility, etc. are defined briefly inthe Appendix.

    Exampl es

    The re la t ion RI ={( l ,O, O,O ) , ( O, l , l , O) , (O, l ,O , l ) ,( I , 0 , I , I ) } i s a ff in& s inc e~i ( u ,x ,y, z ) i s equ iva l-en t to (u~x: l )A(x~ymz:O}.

    The relation R 2 = { ( 0 , 0 , 0 ) , ( 0 , 0 , I ) , ( 0 , I , 0 ) ,(I ,I ,0 )} is bi jun ct ive and weakly negative, since~2(x ,y,z ) i s equ iva len t to ( - I xVy) A(4y v- l z ) .I~ is also, obviously, O-valid.

    The relation R 3={ (0 ,I ) , (I ,0 )} is defined bythe formula (xv y) ~(~ xv~ y) , or equivalent ly,x~ y= I. Hence this relat ion is bi junct ive andaf fi ne. I t i s not, however, weakly pos iti ve orweakly negat ive - - th is can be shown using Lemma3.1W.

    The relat ion R = {( 0, 0,0 ) , (I , I , I )} is definedby the formula (xmy) A(ymz), or eq uiv ale ntl y,( x v ~ y ) A ( y V I x ) A ( y v ~ z ) A ( Z ~ - I y ) . He nc e, i tis O-valid, l-valid, weakly positive, weakly nega-t ive, aff ine and bi junct ive.

    The re la t ion R~={(O,O, I ) , (O, I ,O) , (O, I , I ) ,(I ,O ,O ), (I ,O ,I ), (I ,T ,O )} is the complement of R 4.It does not have any of the six properties li st edfo r R -- th is can be proved using Lemmas 3.1A,3.1B, and 3.1W. Thus, t hi s example shows th at noneof these properties is preserved under complement.

    The re la t ion R ={(0 ,0 , I ) , (0 , I ,0 ) , ( I ,0 ,0 ) } i sthe re la ti on ex act ly one of three mentioned inthe In tr od uc ti on . I t can be shown, using Lemmas3.1A, 3.1B and 3.1W, tha t i t is not weakly po si ti ve ,not weakly negative, not affine and not bijunctive.

    By applyi ng Theorem 2.1 w it h S={R 5} and S={R6}re sp ec ti ve ly , i t can be deduced th at tee NOT-ALL vEQUAL and ONE-IN-THREE s a t i s f i a b i l i t y problems,def ined in Section I , age log- complete in NP. Weomit the proofs.

    Method of Proof

    The key question on which the proof of theDichotomy Theorem cent ers is : For a given S, whatrelat ions are definable by exis ten t i al l y quantif iedS-formulas? For example, i s S={R}, where R is therel ati on exactlY one of x, y, z, then the exist en-tially quantified S-formula (3Ul,U2,U3)(R(X,Ul,U3)AR(y,up,uR)A~(Ul,U2,Z)) defines the relation{(T ,I , I) , (T, O,O ),( O,I ,O) ,(O ,O, I)} , which in thenotation of Section 3 could be wri tte n [x ~y ~z =l ].

    Moreover, for this particular S, it turns out thatevery logic al rel at ion is definable by some exi s-ten t i al l y quantif ied S-formula. This fact readi lyimplies the NP-completeness of SAT(S).

    Another way to state this fact is as a closurepropert y: The smalle st set of rel at io ns which con-tain s S and is closed under certa in operations (con-junction and exi ste nti al quant ific atio n) is theset of all logical relat ions. From this point ofvie w, the general problem can be phrased as fo l -lows: What sets of l ogic al rel ati ons are closedunder these operations? I f we can obta in a reason-ably succinct classification of the sets of rela-tions that are closed in this way, then this mayserve as a basis for classifying the complexity ofSAT(S) for various S.

    We do in f act obtain a cl as si fi ca ti on theoremalong these li ne s. Section 3 is devoted to it sstatement and proof, and a refinement of it isgiven in Section 5. This theorem cl as si fi es thesets of logical relations that are closed undercomposition, substitution of constants for varia-bles, and ex is te nt ia l quant ifi cat ion . Although thecl as si fi ca ti on is not so thorough as to give a com-ple te enumeration of the sets having th is closu reproperty, it does permit the complexity of thecorresponding sa ti sf ia bi li t y problem to be deter-

    mined up to log-space equivalence in al l cases.The closure of the set S under these threeoperations is denoted Rep(S). I t is int e