Resolution Proofs of Generalized Pigeonhole Principles

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1 Resolutio Proofs of Geeralized Pigeohole Priciples Samuel R. Buss Departmet of Mathematics Uiversity of Califoria, Berkeley Győrgy Turá Departmet of Mathematics, Statistics, ad Computer Sciece Uiversity of Illiois, Chicago ad Automata Theory Research Group Hugaria Academy of Scieces Szeged, Hugary February 1988 Abstract We exted results of A. Hake to give a expoetial lower boud o the size of resolutio proofs for propositioal formulas ecodig a geeralized pigeohole priciple. These propositioal formulas express the fact that there is o oe-oe mappig from c objects to objects whe c > 1. As a corollary, resolutio proof systems do ot p-simulate costat formula depth Frege proof systems. 1. Itroductio S. Cook ad R. Reckhow [] itroduced propositioal formulas ecodig the pigeohole priciple. These propositioal formulas have polyomial size proofs i exteded resolutio proof systems (S. Cook, Reckhow []), i Frege proof systems (Buss [1]) ad i cuttig plae proof systems (W. Cook, Coullard, Turá [3]); however, A. Hake [4] showed they require expoetial size proofs Supported i part by NSF postdoctoral fellowship DMS

2 i a resolutio proof system. The purpose of this paper is to exted Hake s expoetial lower boud; i particular, we address the questio of lower bouds o the size of resolutio proofs of geeralized pigeohole priciples which state that for m >, if m pigeos sit i holes the some hole cotais more tha oe pigeo. For m > + 1 the geeralized pigeohole priciple is more true tha the usual pigeohole priciple (where m = + 1), ad hece might have shorter resolutio proofs. We show below that ay resolutio proof of the geeralized pigeohole priciple with m = c must be expoetial size i (for costat c > 1). This implies (usig results of Paris ad Wilkie [5] ad Paris, Wilkie ad Woods [6]) that resolutio does ot p-simulate costat formula depth Frege proof systems.. Resolutio ad the Pigeohole Priciple We begi by recallig the basic facts about resolutio (see Hake [4] for a more detailed expositio). A propositioal variable rages over the truth values True ad False. A literal is either a variable x or the egatio x of a variable x. A clause is a fiite set of literals; the meaig of a clause is the disjuctio of the variables i the clause. Hece a truth assigmet satisfies a clause if it assigs the value True to some variable i the clause or the value False to a variable whose egatio appears i the clause. The meaig of a set of clauses is the cojuctio of the clauses, so ay cojuctive ormal form formula ca be viewed as a set of clauses. The resolutio rule is a form of modus poes: if C 1 is a clause cotaiig x ad C cotais x the the clause (C 1 \ {x}) (C \ {x}) is iferred by resolvig o the variable x. Resolutio is a refutatio proof system. Give a formula φ i disjuctive ormal form, its egatio ca be expressed i cojuctive ormal form ad the as a set of clauses. A resolutio proof of φ is by defiitio a resolutio proof of the empty clause (a cotradictio) from the set of clauses expressig the egatio of φ. The completeess theorem for resolutio guaratees that every tautology i disjuctio ormal form has a resolutio proof; i.e., from ay set of clauses such that o truth assigmet ca simultaeously satisfy all of them there is a derivatio of the empty clause usig oly the resolutio rule. A resolutio proof ca be viewed as a sequece of clauses; each clause i the sequece is either a iitial clause (a assumptio) or is obtaied by resolutio from two earlier clauses. Alteratively a resolutio proof ca be viewed as a directed acyclic graph with a edge from oe clause to aother if the secod is obtaied by resolutio from the first together with some other clause. We shall use the followig fact: give a resolutio proof ad a truth assigmet α, there is a uique path C 1,C,...,C t through the proof (viewed as a directed acyclic graph) such that C 1 is a iitial clause ad C t is the

3 empty clause ad each C i+1 is iferred by resolutio from C i ad oe other clause. This is proved by workig backwards startig at the root of the tree ad by otig that if α does ot satisfy a clause the α also does ot satisfy exactly oe of the two clauses from which it derived by resolutio. Sice we are workig i a resolutio proof system the geeralized pigeohole priciple PHP m eeds to be expressed as a usatisfiable propositioal formula i cojuctive ormal form. The variables of PHP m are x i,j with 1 i m, 1 j ; the variable x i,j is iteded to deote the coditio that pigeo i is sittig i hole j. The formula PHP m is defied to be m (x i1,j x i,j) i=1 j=1 x i,j j=1 1 i 1<i m where x i,j deotes the egatio of x i,j. The first part of PHP m expresses the coditio that every pigeo sits i oe or more holes; the secod part that o hole is occupied by more tha oe pigeo. It is easy to see that the geeralized pigeohole priciple for m pigeos ad holes is equivalet to PHP m beig usatisfiable. Note that the size of PHP m is O(m ). 3. A Lower Boud for Resolutio I this sectio we prove the mai result: Theorem 1 Every resolutio proof of the usatisfiability of PHP m at least ( ) m has legth Thus, i particular, PHP c requires expoetial legth resolutio proofs for ay costat c > 1. The lower boud is superpolyomial for m = o( /log ). We do ot kow whether PHP has polyomial legth proofs. (By the legth of resolutio proof we mea the umber of lies i the proof; however, this is polyomially related to the umber of symbols i the proof sice each clause i the proof will cotai at most oe istace of each variable.) The proof follows A. Hake s argumet. Although i his proof (ad i the subsequet work of Urquhart [7]) the existece of critical truth assigmets, which satisfy all but oe clause, seems to play a cetral role, it turs out that by suitably modifyig Hake s defiitios his ideas carry over to our case as well although here there are o critical truth assigmets. We shall picture the variables x i,j arraged i a m matrix with i (the pigeo) specifyig the colum ad j (the hole) the row. Each clause 3

4 i the resolutio proof is described by a m matrix partially filled with + s ad s, where a + (respectively, ) i a positio (i,j) meas that x i,j (respectively, x i,j ) occurs i the clause. A truth assigmet is pictured as a m matrix of 0 s ad 1 s which idicate assigig False or True (respectively) to the correspodig variable. Defiitio A truth assigmet α is maximal if it cotais exactly 1 s, all i differet rows ad colums. The m colums which cotai o 1 s (ad hece oly 0 s) are called the 0-colums of α. Note that a maximal truth assigmet assigs of the pigeos to distict holes ad leaves the other m pigeos uassiged. Now suppose we are give a arbitrary resolutio proof of the usatisfiability of PHP m. Recall that such a proof may be viewed either as a sequece of clauses edig with or as a directed acyclic graph with at the root. (The empty clause is ot satisfiable.) Each clause i the proof must either be a clause from PHP m or be deduced from prior clauses by resolutio. The iitial clauses from PHP m cosist either of oe colum filled with + s or of two s i oe row. Lemma For every maximal truth assigmet α there is a clause C i the resolutio proof such that (1) α makes C false, () C cotais at most + s i every 0-colum of α, (3) C cotais + s i exactly oe 0-colum of α. Proof I the resolutio proof there is a uique path of clauses C 1,...,C t such that α makes each C i false, C 1 is a iitial clause ad C t =. Because α is maximal C 1 must cosist of oe colum filled with + s; this will be a 0-colum of α. Let C be the last amog these clauses which cotais at least + s i some 0-colum of α. The C satisfies (1) by defiitio, ad it also satisfies () ad (3) as + s ca disappear from a clause oly oe at a time. If α is a maximal truth assigmet, let C α deote the first clause i the resolutio proof satisfyig the coditios of Lemma. Defie FS1 to be the set {S : S is a set of 4 variables, all i differet rows ad colums}. For S FS1, C S is the first clause i the proof sequece which is of the form C α for some maximal truth assigmet α which assigs 1 s to each variable i S. Ay such C S is called a complex clause. Lemma 3 Every complex clause has at least colums which cotai either a or at least + s. 4

5 Proof Let C S be a complex clause for S FS1 ad α be a maximal truth assigmet assigig 1 s to the variables i S such that C α = C S. Let COL ={l : colum l of C S cotais a }, COL + ={l : colum l of C S cotais at least + s ad o s ad is ot a 0-colum of α}, l 0 =the 0-colum of α which cotais exactly + s i C S, A={x i,j / S : α i,j = 1}. Sice α makes C S false, COL caot cotai ay 0-colum of α; thus COL, COL + ad {l 0 } are pairwise disjoit. By defiitio every 0-colum of α other tha l 0 cotais fewer tha + s i C S. As l 0 satisfies the coditios of the lemma, we have to show that COL + COL + 4. Claim 1: If COL + COL + < 4, the there exists a xi,j A such that (1) either x l0,j or x l0,j occurs i C S ad () i / COL COL +. Ideed, as (1) excludes elemets of A (i fact colum l0 cotais oly + s) ad () excludes < 4 elemets, the coditio A = 3 4 implies the existece of such a variable. To prove Lemma 3, suppose for the sake of a cotradictio that the coditios of Claim 1 hold ad let α be the maximal truth assigmet costructed from α by chagig the value of α i,j to 0 ad α l0,j to 1. Claim : (1) α assigs 1 s to all members of S ad makes C S false. () All 0-colums of α cotai less tha + s i C S. (1) follows by costructio. The 0-colums of α are the 0-colums of α, except l 0 beig replaced by i, but as i / COL +, it cotais less tha + s i C S, provig (). By the method of proof of Lemma, it is clear that C α is a clause precedig C S i the proof sequece which cotradicts the the defiitio of C S. { Proof of Theorem 1. Put g() = max C {S FS1 : C S = C} } ad h() = FS1. The as i [4], h()/g() is a lower boud to the legth of a resolutio proof, sice it is clearly a lower boud o the umber of distict complex clauses i the resolutio proof. Let k = 4. To compute h() ad g() suppose we have a particular complex clause C. By Lemma 3 we ca choose k + 1 colums which cotai a or at least + s. To cout the total umber of S FS1 we let the variable i deote the umber of variables i S i the chose k + 1 colums. The we have: h() = ( )( ) k + 1 m k 1! i k i ( k)! 5

6 Similarly, to get the upper boud g() o the umber of S FS1 such that C S = C we let i be the umber of variables of S i oe of the k + 1 colums. I each of these k + 1 colums there are at most variables which ca be i such a S; this is because a + i C excludes the correspodig variable from S ad a i C implies that if S has a variable from that colum it must be the variable correspodig to the. Thus, ( )( ) k + 1 m k 1 i ( i)! g() i k i ( k)! So, ( )( ) k + 1 m k 1 h() g() i k i ( )( ) k + 1 m k 1 i ( i)! i k i! ( )( ) k + 1 m k 1 sice for i 4, i k i ( )( )( k + 1 m k 1 i k i 3 i ( i)!! ( ) i 3 The ratio of the (i 1)-st term over the i-th term i the summatio i the deomiator is i(m k + i 1) 3 (k i + 1)(k i + ) It is easily verified that this is less tha 1 for i 1 5 m, ad hece the terms i the deomiator are icreasig while i 1 5 m. Thus we ca give a weaker lower boud (with smaller umerator ad larger deomiator): ( )( ) k + 1 m k 1 ) i h() g() 1 i= 1 50 m i= 1 50 m ( ) m i k i ( )( k + 1 m k 1 i k i )( 3 ) i 6

7 which completes the proof of Theorem Resolutio versus Costat Formula Depth Frege Systems The otio of the depth of a formula is defied i terms of the alteratio of s ad s i the formula. A formula is of depth k iff it is i oe of the classes Σ k or Π k : Defiitio Σ k ad Π k are the smallest sets of propositioal formulas which satisfy the followig iductive defiitio: 1. A propositioal variable is i Σ 0 ad i Π 0,. If A ad B are i Σ k (respectively, i Π k ) the A is i Π k (resp., Σ k ), A is i Σ k+1 Π k+1, A B is i Σ k (resp., Σ k+1 ), A B is i Π k+1 (resp., Π k ). For istace, PHP m is i Π. A formula-depth k Frege proof system is a usual Frege proof system (see S. Cook, Reckhow []) with the additioal restrictio that every formula appearig i a proof be of depth k. Paris ad Wilkie [5] established the followig coectio betwee provability i Bouded Arithmetic ad provability i costat formula depth Frege proof systems. Let WPHP(f) be the setece x[x 0 ( y < x)(f(y) < x ) ( y)( z)(y z f(y) = f(z))]. Let I 0 (f)+ω 1 be the theory of arithmetic with iductio o bouded formulas with f a additioal fuctio symbol allowed i iductio formulas ad with a axiom assertig that x log x is a total fuctio; the a slight stregtheig of Theorem 6 of Paris-Wilkie [5] gives: Propositio 4 If I 0 (f) + Ω 1 WPHP(f) the there are costats k 1 ad k such that for all, PHP has Frege proofs of size O( (log 1) )k i which every formula is of depth k. Recetly, Paris, Wilkie ad Woods [6] established that I 0 (f) + Ω 1 does ideed prove WPHP(f). Combiig our Theorem 1 with these results gives: Theorem 5 There is a costat k such that resolutio does ot polyomially simulate formula depth k Frege proof systems. To the best of the authors kowledge, Theorem 5 is the oly kow separatio result applyig to costat formula depth Frege proof systems. 7

8 Refereces [1] S. R. Buss, Polyomial size proofs of the propositioal pigeohole priciple, Joural of Symbolic Logic, 5 (1987), pp [] S. A. Cook ad R. A. Reckhow, The relative efficiecy of propositioal proof systems, Joural of Symbolic Logic, 44 (1979), pp [3] W. Cook, C. R. Coullard, ad G. Turá, O the complexity of cuttig plae proofs, Discrete Applied Mathematics, 18 (1987), pp [4] A. Hake, The itractability of resolutio, Theoretical Computer Sciece, 39 (1985), pp [5] J. B. Paris ad A. J. Wilkie, Coutig problems i bouded arithmetic, i Methods i Mathematical Logic, Lecture Notes i Mathematics #1130, Spriger-Verlag, 1985, pp [6] 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 [7] A. Urquhart, Hard examples for resolutio, J. Assoc. Comput. Mach., 34 (1987), pp