Resolution and the Weak Pigeonhole Principle

Transcription

1 Resolutio ad the Weak Pigeohole Priciple Sam Buss 1 ad Toia Pitassi 2 1 Departmets of Mathematics ad Computer Sciece Uiversity of Califoria, Sa Diego, La Jolla, CA Departmet of Computer Sciece, Uiversity of Arizoa, Tucso, AZ Abstract. We give ew upper bouds for resolutio proofs of the weak pigeohole priciple. We also give lower bouds for tree-like resolutio proofs. We preset a ormal form for resolutio proofs of pigeohole priciples based o a ew mootoe resolutio rule. 1 Itroductio Tautologies expressig versios of the pigeohole priciple have bee importat test cases for obtaiig bouds o the legths of propositioal proofs ad for comparig the proof theoretic stregths of various propositioal proof systems. The semial paper of Cook ad Reckhow [5] showed that pigeohole priciples have polyomial legth exteded Frege systems; later, [3] showed that they also have polyomial legth Frege proofs. O the other had, the first superpolyomial lower boud o the legth of resolutio refutatios was Hake s proof [6], that resolutio proofs of the propositioal pigeohole priciple require expoetial legth. A sigificat stregtheig of Hake s lower boud was obtaied by Ajtai [1] who proved that costat-depth Frege proofs of the pigeohole priciple require superpolyomial size; this was later stregtheed by [2, 10, 7] to show that costat-depth Frege proofs of the pigeohole priciple require expoetial size. I some cases, fier distictios ca be made usig geeralized forms of the pigeohole priciple. Oe such priciple is the m ito geeralizatio which states that there is o oe-to-oe mappig of m objects ito objects, where m >. Tautologies, defied below, expressig this priciple are deoted PHP m. It is kow that the tautologies PHP 2 have quasipolyomial size costat Frege proofs [8, 9]. I additio, Hake s lower boud for resolutio proofs of the pigeohole priciple was geeralized by [4] who proved superpolyomial bouds o resolutio proofs of PHP m for m = o( 2 /log ). These prior upper ad lower bouds o the the size resolutio proofs leave ope the questio of the size of resolutio proofs of PHP m whe 2 /log m 2. It has bee a folklore cojecture that the shortest resolutio proofs of PHP m have the same legth as resolutio proofs of PHP +1 ; i other words, Supported i part by NSF grat DMS ad US-Czech Sciece ad Techology grat Research supported by NSF grat CCR

2 that whe m > + 1, optimal legth resolutio proofs ca be obtaied by igorig all but oe of the domai elemets. We show i this paper, however, that this cojecture is false for m = 2. The results of this paper are as follows. I sectio 3 we preset a ormal form for resolutio proofs of pigeohole priciple tautologies. Normal form resolutio proofs cotai oly positive occurreces of variables; the usual resolutio rule is replaced by a ew mootoe resolutio rule. The sizes of mootoe resolutio proofs are polyomially related to the sizes of resolutio proofs. As a corollary, we prove that resolutio proofs of the oto versio of the pigeohole priciple are ot sigificatly shorter tha resolutio proofs of the o-oto pigeohole priciple. I sectio 4, we give a polyomial upper boud o the size of resolutio proofs of PHP m for m = 2. This improves o the upper boud 2 2 for proofs of PHP +1 ; which shows that havig additioal domai elemets ca make the pigeohole priciple easier to prove. I sectio 5, we prove a expoetial lower boud o the size of tree-like resolutio proofs of PHP m. 2 Defiitios This paper deals exclusively with propositioal logic. A literal is either a propositioal variable or a egated propositioal variable. A clause is defied to be a set of literals ad is idetified with the disjuctio of its member literals. We assume that a clause ever cotais both a variable ad the egatio of that variable. We use capital letters, usually with subscripts, e.g., P i,j, to deote variables; lowercase letters such as x deote literals; ad clauses are deoted by letters A,B,C,... A resolutio iferece ifers A B from two clauses A x ad B x. A cojuctive ormal form (CNF) formula φ is idetified with the set of clauses which appear as cojucts of φ. A resolutio refutatio of φ cosists of a sequece C 1,...,C s of clauses, where each C i is either a cojuct of φ or is iferred from earlier clauses i the refutatio by a resolutio iferece. The size of the refutatio is equal to the umber, s, of clauses i the refutatio. A refutatio proof of a disjuctive ormal form (DNF) formula is defied to be a resolutio refutatio of the egatio of the formula. It is well-kow that resolutio is refutatioally soud ad complete, so a DNF formula has a resolutio proof if ad oly if it is a tautology. Resolutio refutatios are usually viewed as sequeces or directed acyclic graphs. However, they ca also be restricted to be tree-like with each clause i the refutatio beig used as a hypothesis of a iferece at most oce. Note that the same clause may appear multiple times i the tree-like proof; the size of a tree-like refutatio equals the umber of occurreces of clauses i the refutatio. Defiitio 1. Let m >. The tautology PHP m expresses the pigeohole priciple that there is o oe-to-oe mappig from a domai of m objects (called pigeos ) ito a rage of objects (called holes ). This is easily defied 2

3 by a DNF formula, but sice it is more relevat for resolutio, we describe istead the set of clauses which are the cojucts of the CNF formula PHP m. The propositioal variables are P i,j, i m, j, with P i,j havig the ituitive meaig that pigeo i is mapped to hole j. The clauses of PHP m are: (1) P i,1 P i,2 P i,, for each i m; ad (2) P i,k P j,k, for each i,j m, k, i j. Note that the umber of clauses i PHP m is m + ( m 2) < m 2 < m 3. As metioed i the itroductio, Hake proved that resolutio proofs of require expoetial size. The first author ad Turá [4] showed that PHP +1 ) ay resolutio refutatio of PHP m requires size 2( 1 3 m 2. However, whe m 2 /log, this lower boud is oly polyomial, ad i fact there are o otrivial lower bouds kow i this case. To the best of the authors kowledge, prior to the preset paper, the best upper bouds kow for the sizes of resolutio proofs of PHP m was the boud 3 2 of Lemma 1 below. 3 A Normal Form Theorem I this sectio we defie a variatio of the resolutio proof system, called the mootoe resolutio system, which is tailored for proofs of pigeohole priciples. We prove that this system is complete for proofs of pigeohole priciple tautologies, ad that the sizes of mootoe resolutio proofs ad the sizes of resolutio proofs are polyomially related. The motivatio for itroducig the mootoe resolutio proof system is the hope that it will provide a better framework for obtaiig lower bouds o the sizes of resolutio refutatios of pigeohole priciples. We defie a mootoe resolutio proof for PHP m as follows. A mootoe clause is a clause which cotais oly positive variables. We let the m variables P i,j correspod to etries i a a -by-m array, with rows labeled by the holes, ad colums labeled by the m pigeos. Thus the variable P i,j correspods to the etry i the j-th row ad the i-th colum. A mootoe clause is visualized as a -by-m array, with + s i each etry correspodig to the occurreces of variables i the clause ad with array etries correspodig to variables ot occurrig i the clause left blak. For R {1,...,m} we let P R,j be the disjuctio of the variables P i,j for all i R. Let C 1 = A P R,j P S,j ad C 2 = B P R,j P T,j, where R, S ad T are disjoit ad where A ad B are both disjuctios of positive variables ot i row j. The the mootoe resolutio iferece rule allows us to derive C 3 = A B P R,j from C 1 ad C 2. I other words, we ca ifer the clause C 3 from C 1 ad C 2 by the mootoe resolutio rule with respect to hole j, provided C 3 cosists of the uio of all variables i C 1 C 2, mius all variables P i,j, which occur i exactly oe of C 1 ad C 2. Implicit i the mootoe resolutio rule is oe-to-oeess: if pigeo i is mapped to hole j, the o other pigeo i ca be mapped to hole j. 3

4 A mootoe resolutio proof is a sequece of mootoe clauses, where the fial clause is the empty clause; ad where every clause is either a iitial clause of the form j=1 P i,j, or follows from two previous clauses by the mootoe resolutio rule. Strictly speakig, mootoe resolutio is ot a proof system, sice it is ot complete for arbitrary sets of clauses; however, it follows from the ext theorem that mootoe resolutio is sufficiet to prove pigeohole priciple tautologies. Oly such tautologies are cosidered i this paper. Two proof systems are said to be polyomially equivalet for a class Φ of formulas if ad oly if there is a polyomial q(x) such that if φ Φ has a proof of size s i oe of the systems, the it has a proof of size q(s) i the other system. Theorem 1. The resolutio proof system ad the mootoe resolutio proof system are polyomially equivalet for the pigeohole tautologies PHP m. Proof. Let us first show that if we have a mootoe refutatio, the we also have a resolutio refutatio of the usual kid.. For this it suffices to simulate a mootoe resolutio iferece by oly polyomially may ordiary resolutio ifereces. Suppose that C 3 is obtaied from C 1 ad C 2 by the mootoe resolutio rule, where C 1 = A P R,j P S,j ad C 2 = B P R,j P T,j ad C 3 = A B P R,j, ad where R, S ad T are disjoit ad A ad B are sets of variables ot ivolvig hole j. We shall show how to obtai C 3 from C 1, C 2 ad the iitial clauses with oly polyomially may resolutio steps. First, geerate the clauses C1 t = A P R,j P t,j, for all t T. Each clause C1 t is obtaied by S may resolutio ifereces from C 1 ad the iitial clauses ( P t,j P s,j ) for all s S. The from the clauses C1, t where t T, ad from C 2, geerate C 3 = A B P R,j i T additioal ifereces. Sice S, T m, the above costructio shows that a mootoe resolutio iferece ca be simulated with m 2 usual resolutio ifereces. I the other directio we wat to show that if P is a resolutio refutatio of PHP m, the there exists a mootoe resolutio refutatio P of PHP m of size polyomial i the size of P. As a first step, we will trasform every clause i P ito a totally mootoe clause as follows: if C = A B is a clause i P, where A is the disjuctio of positive variables, ad B is the disjuctio of egative variables, the C m = A B m, where B m is obtaied by replacig every egative literal P i,k i B by the (disjuctio of the) set of literals {P l,k l i}. Note that the iitial clauses of the form k=1 P j,k will remai uchaged, ad the iitial clauses of the form ( P i,k P j,k ) will become m l=1 P l,k. Note that i the latter case, the clause is ot a valid iitial clause for a mootoe resolutio refutatio. Now suppose that C 3 is iferred from the clauses C 1 ad C 2 i the origial resolutio refutatio. We wat to show how to derive C3 m from C1 m ad C2 m. Suppose that C 1 cotais P i,k ad C 2 cotais P i,k, where P i,k is the variable resolved upo to obtai C 3. We must show how to derive a subclause of C3 m from C1 m ad C2 m. (It suffices to derive a subclause of C3 m, sice it is obvious that 4

5 the subsumptio priciple applies to mootoe resolutio.) There are two cases to cosider. Firstly, suppose C 2 is a iitial clause of the form ( P i,k P j,k ). I this case, it is easy to check that Cm 1 is a subclause of Cm, 3 so this resolutio refutatio does ot eed to be simulated by ay mootoe resolutio steps. More geerally, if the array represetatio of C2 m has a + i the positio correspodig to P i,k, the it has + s i every positio i row k ad hece C1 m is a subclause of C3 m ad o additioal mootoe resolutio iferece is eeded. Secodly, suppose C2 m does ot have + i the positio for P i,k. Let C3 be the clause obtaied from C1 m ad C2 m by usig the mootoe resolutio iferece with respect to row k. We shall show that C3 is a subclause of C3 m. I this case, we ca write C1 m = A P R,k P i,k where i / R, ad ca write C2 m = B P R,k P T,k where T is the complemet of R {i}. Thus Cm is the clause A B P R,k. Each member P j,k of P R,k is preset i C1 m because it is already i C 1 or because P j,k is i C 1 for some j j. The same literal also appears i C 3 ad therefore P j,k is also i C3 m. This shows that C3 is a subclause of C3 m. Therefore, we have show that a resolutio iferece ca be simulated by (at most) a sigle mootoe resolutio iferece. The oto versio of the pigeohole priciple is obtaied by takig the clauses P 1,k P 2,k P,k as additioal iitial clauses. However, these clauses are just the mootoe traslatio of the iitial clauses P i,k P j,k of the usual pigeohole priciple. Examiatio of the above proof shows that we have proved that ay (ordiary) resolutio refutatio of the oto pigeohole priciple of size ifereces, ca be traslated ito a mootoe resolutio of size. From this, the followig theorem is a immediate corollary of Theorem 1. Theorem 2. The shortest resolutio proofs of PHP m have size polyomially bouded by the size of resolutio proofs of the oto pigeohole priciple with m pigeos ad holes. 4 A Upper Boud Theorem 3. There is a d > 0 such that whe m = 2, the PHP m has a resolutio proof with m d steps. Thus, for m 2, PHP m has a resolutio proof of size polyomially bouded by the umber of variables. Sice Hake [6] proved a size lower boud of 2 ǫ for proofs of PHP +1, where ǫ is a costat, Theorem 3 implies that the size of resolutio proofs of PHP +1 must be superpolyomially loger tha the shortest resolutio proof of PHP m where m = 2. By Theorem 1, it will suffice to prove Theorem 3 for mootoe resolutio proofs istead of ordiary resolutio proofs; ideed, sice there are m pigeos, the legth of the shortest ordiary resolutio refutatio is o more tha m 2 5

6 times the legth of a mootoe resolutio refutatio. First, we eed the followig lemma: Lemma 1. PHP +1 has a mootoe resolutio refutatio of size O(2 ). Note that the lemma ad the proof of Theorem 1 imply that PHP +1 a ordiary resolutio proof of size O( 3 2 ). Proof. Let P S,T deote the disjuctio of the variables P i,j, where i S, j T. Also, P i,t deotes P {i},t. Let [i,j] deote the set {i,i + 1,...,j}. The iitial clauses of the mootoe resolutio refutatio are P i,[1,] for all i [1, + 1]. The mootoe refutatio first derives the clauses P S,[2,], for (2) all sets S (2) [1, + 1] of size 2. Each is obtaied by oe mootoe resolutio step from the iitial clauses. Next, we geerate the clauses P S (3),[3,] for all S (3) [1, + 1] of size 3. Each is obtaied by two mootoe resolutio steps from clauses derived i the previous stage. Cotiuig i this fashio, we evetually derive P S (), for all S () [1, + 1] of size. Fially, we derive the empty clause from this last set of clauses. The total umber of mootoe resolutio ifereces derived is bouded by (( ) + ( ) ( )) = O(2 ). has Proof. We will ow prove Theorem 3 by iductio o. Let a = b log for a fixed b > 1. The base case, = 2, is trivial for d sufficietly large. The iductio step is argued as follows: The mootoe resolutio refutatio we costruct has two stages. The first stage splits the m pigeos ito disjoit blocks of a + 1 pigeos., so as to remove a 1 s (rage elemets) from the colums. That is to say, for each block S of a + 1 colums, we derive the clause C S,T where T is [1, a]. The size aalysis for this part is equal to the umber of blocks times the complexity of provig For each block, we ru a resolutio refutatio of PHP a+1 a PHP a+1 a ; i.e., (m/(a + 1))O((a + 1)2 a ) = O(2 (b+1) ). I the secod stage, we use the iductio hypothesis applied to a holes; we do this by keepig the disjoit blocks of a + 1 colums (pigeos) grouped together; i essece, we have divided the umber of colums by a+1. Therefore, we are provig a istace of PHP m/(a+1) a : the iductio hypothesis tells us that this ca be proved with the umber of mootoe resolutio ifereces bouded by 2 d ( a) log( a) < 2 d( a(1+log )/(2 )) = 2 d 0.5db(1+log ) = o(2 d ) 6

7 (The first iequality is obtaied by lettig f(x) = xlog x ad usig the fact that f( a) < f() af () sice f is cocave dow.) Addig the size bouds from the two stages of the mootoe resolutio refutatio gives the desired upper boud of 2 d, provided d is sufficietly large. It is still left to verify that the use of the iductive hypothesis was valid, i.e., that m/(a + 1) > 2 ( a) log( a). The lefthad side is equal to 2 /(b log + 1). By the calculatio above, the righthad side is less tha or equal to 2 / b/2, sice d > 1. Thus the desired iequality holds sice b > 1. 5 A Lower Boud Theorem 4. For ay m >, ay tree-like resolutio refutatio of PHP m requires 2 steps. Proof. We ll prove the stroger statemet that ay tree-like mootoe resolutio refutatio P of PHP m has at least 2 ifereces. The proof is by iductio o. For = 1, the statemet is easy to verify. Now suppose > 1. Let the last iferece of P ifer the empty clause from two clauses C 1 ad C 2 by a mootoe resolutio iferece. We have C 1 = P S,k ad C 2 = P T,k for disjoit oempty subsets S ad T of [1,]. Let p i,k S ad p j,k T. Let P 1 ad P 2 be the subproofs of P which derive C 1 ad C 2 respectively. We form a ew refutatio from P 1 by restrictig p j,k to be true: this ivolves (1) removig from P 1 every clause which cotais p j,k ad (2) erasig from the clauses of P 1 every occurrece of the variables p j,k with j j. The result is (easily modified to be) a valid resolutio proof of PHP 1 m 1. By the iductio hypothesis, this proof ad hece P 1 must have at least 2 1 ifereces. Similar reasoig shows that P 2 must have at least 2 1 ifereces. Therefore, P has at least ifereces. 6 Further Research Subsequetly to the preset paper, Razborov, Widgerso ad Yao [11] have ivestigated relatioships betwee restricted resolutio refutatios of the pigeohole priciple ad restricted read-oce brachig programs. They idetified several restricted versios of resolutio, icludig a rectagular resolutio calculus, ad they geeralized the upper boud of Theorem 3 to the rectagular calculus ad proved a early matchig lower boud o the size of rectagular refutatios for the weak pigeohole priciple. For the (urestricted) resolutio calculus, the problem of provig expoetial lower bouds for the weak pigeohole priciple, PHP m, where the umber of pigeos, m, is polyomially large (e.g., m = 2 ) remais ope. 7

8 Refereces 1. M. Ajtai, The complexity of the pigeohole priciple, i Proceedigs of the 29-th Aual IEEE Symposium o Foudatios of Computer Sciece, 1988, pp P. Beame, R. Impagliazzo, J. Krajíček, T. Pitassi, P. Pudlák, ad A. Woods, Expoetial lower bouds for the pigeohole priciple, i Proceedigs of the 24th Aual ACM Symposium o Theory of Computig, 1992, pp S. R. Buss, Polyomial size proofs of the propositioal pigeohole priciple, Joural of Symbolic Logic, 52 (1987), pp S. R. Buss ad Győrgy Turá, Resolutio proofs of geeralized pigeohole priciples, Theoretical Computer Sciece, 62 (1988), pp S. A. Cook ad R. A. Reckhow, The relative efficiecy of propositioal proof systems, Joural of Symbolic Logic, 44 (1979), pp A. Hake, The itractability of resolutio, Theoretical Computer Sciece, 39 (1985), pp J. Krajíček, P. Pudlák, ad A. Woods, Expoetial lower boud to the size of bouded depth Frege proofs of the pigeohole priciple, Radom Structures ad Algorithms, 7 (1995), pp J. B. Paris ad A. J. Wilkie, 0 sets ad iductio, i Ope Days i Model Theory ad Set Theory, W. Guzicki, W. Marek, A. Pelc, ad C. Rauszer, eds., 1981, pp J. B. Paris, A. J. Wilkie, ad A. R. Woods, Provability of the pigeohole priciple ad the existece of ifiitely may primes, Joural of Symbolic Logic, 53 (1988), pp T. Pitassi, P. Beame, ad R. Impagliazzo, Expoetial lower bouds for the pigeohole priciple, Computatioal Complexity, 3 (1993), pp A. Razborov, A. Widgerso ad A. Yao, Read-oce brachig programs, rectagular proofs of the pigeo-hole priciple ad the trasversal calculus, i Proceedigs of the 29th Aual ACM Symposium o Theory of Computig, 1997, pp