On the Relative Strength of Pebbling and Resolution

On the Relati ve Strength of Pebblin g and Resolution ∗ Jakob Nordstr ¨ om † Computer Science and Artiﬁcial Intelligence Laboratory Massachusetts Institute of T echnology Cambridge, MA 02139, USA jakobn@mit .edu November 7, 2018 Abstract The last decade has seen a revi val o f interest in peb ble gam es in the context o f p roof com plexity . Pebbling has proven to be a useful tool for studying resolution- based proof systems when comparing the strength of dif feren t subsystems, sho wing bounds on proof space, and establishing size-space trade-offs. The ty pical a pproach h as been to encode the pebb le game play ed on a graph as a CNF f ormula an d then argue that p roofs of this form ula must in herit (various aspects of) the pe bbling proper ties of the underly ing grap h. Unfortu nately , the re ductions used here a re not tight. T o simulate reso lution proofs by pebblings, the full s treng th of nondetermin istic black-white pebbling i s needed, whereas resolution is only known to be able to simu late determin istic black p ebbling . T o o btain strong resu lts, on e ther efore needs to ﬁnd speciﬁc gr aph families wh ich e ither have essentially the same p roperties fo r black and black-white peb bling (no t at all tru e in gen eral) or whic h admit simu lations of black-wh ite pebblings in resolution. This pape r contributes to both these ap proach es. First, we design a restricted for m of black -white pebbling that can be simulated in resolu tion and show that there are gr aph families for wh ich such restricted pebb lings can b e asym ptotically b etter than black peb blings. This p roves that, p erhaps som e- what u nexpectedly , resolution can strictly beat b lack-on ly pebbling, and in p articular that th e space lo wer bound s on pebblin g formulas in [ Ben-Sasson and Nor dstr ¨ om 2008 ] are tight. Seco nd, we p resent a ver- satile p arametrized gr aph family with essentially the same proper ties for black and black-white peb bling, which gi ves sharp simultaneou s trade-offs for black and black-wh ite pebbling f or v arious parameter set- tings. Both of o ur contributions have been instrumen tal in obtain ing the time-space trade -off results for resolution- based proof systems in [Ben-Sasson and Nordstr ¨ om 2009] . 1 Introduc tion Pebbling is a tool for stud ying time-space relationsh ips by means of a game played on directed ac yclic graphs . This game models computation s where the execu tion is independ ent of the input and can be per- formed by s traight-l ine programs. Each such prog ram is encoded as a grap h, and a pebbl e on a v erte x in the graph indicate s that the correspo nding val ue is curren tly kept in memory . The goal is to pebble the output ver tex of the graph with minimal number of pebble s (amount of memory) and steps (amount of time). Pebble games were origin ally devis ed for studyi ng progra mming languages and compile r construction , b ut hav e later found a broad range of applicatio ns in computationa l complex ity theory . The pebble game ∗ This is the full-length version of the paper [Nor10b] to appea r at the 25th IEEE Confer ence on Computational Complexity . † Research supported by the Royal Swedish Academy of Sciences, the Ericsson Research Foundation, the Sweden-America Founda tion, the Found ation Ol le Engkvist Byggm ¨ astare, the Sv en and Dagmar Sal ´ en Founda tion, and the Foundation Blanceﬂor Boncompag ni-Ludovisi, n ´ ee Bildt. ON THE RELA TIVE S TRENGT H OF PEBBLING AND RESOLUTION model seems to hav e a ppeared for the ﬁrst time ( implicitly) in [PH70], whe re it was used to study ﬂo w charts and recursi ve schemata , and it was later employed to m odel regi ster allocation [Set75 ], and analyz e the relati ve power of time and space as T uring-mac hine resources [Coo74, HP V77]. More ove r , pebbling has been used to deri ve time-space trade-of fs for algori thmic concep ts such as linear recursi on [Cha73, SS83], fast Fourier transform [SS 77, T om78], matrix multiplication [T om78], and integer multiplicatio n [SS79]. An ex cellent surv ey of pebbling up to ca 1980 is [Pip80], and some more recent dev elopments are cov ered in the autho r’ s upcoming surve y [Nor10a]. The pebbling price of a directed acy clic graph G in the black pebble game captures the memory space, or number of registers, required to perfor m the determin istic computat ion describe d by G . W e will mainly be intereste d in the the m ore general blac k-white pebble game modelling nondeterminis tic compu tation, which was i ntroduce d in [CS76] and h as been studie d in [GT78, Kla8 5, L T80, L T82, Mey81, KS91, W il88] and other papers. Deﬁnition 1.1 (Peb ble game). Let G be a directed acycl ic graph (D A G ) with a unique sink ver tex z . T he blac k-white pebbl e game on G is the followin g one-playe r game. At any time t , we hav e a conﬁguratio n P t = ( B t , W t ) of black pebble s B t and w hite pebbles W t on the vertices of G , at most one pebbl e per ver tex. The rules of the game are as follo ws: 1. If all immediate predec essors of an empty verte x v ha ve pebb les on them, a black pebble may be placed on v . In parti cular , a black pebble can alwa ys be plac ed on a sourc e ver tex. 2. A black pebble may be remov ed from any ver tex at any time. 3. A white pebble may be placed on any empty vert ex at any time. 4. If all immed iate pre decessor s of a white -pebbled v ertex v ha ve pebbles on t hem, the w hite pebble on v may be remov ed. In particular , a white pebble can alw ays be remove d from a source verte x. A (complete) blac k-white pebbling of G , also calle d a pebbling str ate gy for G , is a sequence of pebble conﬁgura tions P = { P 0 , . . . , P τ } such that P 0 = ( ∅ , ∅ ) , P τ = ( { z } , ∅ ) , and for all t ∈ [ τ ] , P t follo w s from P t − 1 by one of the rules abov e. The time of a pebbling P = { P 0 , . . . , P τ } is simply time ( P ) = τ and the space is s pace ( P ) = max 0 ≤ t ≤ τ {| B t ∪ W t |} . T he bla ck- white pebb ling price (also kno wn as the pebb ling measur e or pebblin g number ) of G , denoted BW -P eb ( G ) , is the minimum space of any complete pebbling of G . A blac k pebbling is a pebbli ng usin g black pebbles only , i.e., having W t = ∅ for all t . The (blac k) pebbli ng price of G , denoted P eb ( G ) , is the m inimum spac e of an y complete black pebbling of G . In the last decade, there has been renewed intere st in pebbling in the contex t of proof comple xity . 1 A (non-e xhaus tiv e) list of pr oof compl exity papers using pebb ling in one way or an other is [AJPU 07, BEGJ00, BIPS10, Ben09, BIW04, BN08, BN 09a, BN 09b, BW01, E GM04, ET01, ET03, H U07, Nor09, NH08b, SBK04]. T he way pebbling results ha ve been used in proof complexi ty has mainly been by studying so- called pebblin g contrad ictions (also kno wn as pebblin g formulas or pebbling tautolog ies ). These are CNF formulas encodi ng the pebble game played on a DA G G by postula ting the sources to be true and the sink to be f alse, and specifyin g that truth propag ates throug h the graph acco rding to the pebbling rules. The idea to use such formulas seems to ha ve appeared for the ﬁrst time in [Koz7 7], and they were also studi ed in [RM99, BEGJ00] before being exp licitly deﬁned in [BW01]. Deﬁnition 1.2 (Peb bling contradiction). Suppose that G is a D A G w ith sources S and a unique sink z . Identi fy ev ery verte x v ∈ V ( G ) w ith a proposition al logic v ariable v . The pebblin g contrad iction over G , denote d Peb G , is the conj unction of the follo wing clauses: 1 W e remark that the pebb le game studied in this pa per should not be confused with the (very different) existential pebble games that hav e also been used in proof complexity , for instan ce, in the papers [Ats04, AD08, AKV 04, BG03, GT05]. 2 1 Introd uction z x y u v w (a) Pyramid graph Π 2 of heigh t 2. u ∧ v ∧ w ∧ ( u ∨ v ∨ x ) ∧ ( v ∨ w ∨ y ) ∧ ( x ∨ y ∨ z ) ∧ z (b) Peb bling contradiction Pe b Π 2 . ( u 1 ∨ u 2 ) ∧ ( v 2 ∨ w 1 ∨ y 1 ∨ y 2 ) ∧ ( v 1 ∨ v 2 ) ∧ ( v 2 ∨ w 2 ∨ y 1 ∨ y 2 ) ∧ ( w 1 ∨ w 2 ) ∧ ( x 1 ∨ y 1 ∨ z 1 ∨ z 2 ) ∧ ( u 1 ∨ v 1 ∨ x 1 ∨ x 2 ) ∧ ( x 1 ∨ y 2 ∨ z 1 ∨ z 2 ) ∧ ( u 1 ∨ v 2 ∨ x 1 ∨ x 2 ) ∧ ( x 2 ∨ y 1 ∨ z 1 ∨ z 2 ) ∧ ( u 2 ∨ v 1 ∨ x 1 ∨ x 2 ) ∧ ( x 2 ∨ y 2 ∨ z 1 ∨ z 2 ) ∧ ( u 2 ∨ v 2 ∨ x 1 ∨ x 2 ) ∧ z 1 ∧ ( v 1 ∨ w 1 ∨ y 1 ∨ y 2 ) ∧ z 2 ∧ ( v 1 ∨ w 2 ∨ y 1 ∨ y 2 ) (c) S ubstitut ion pebbling contradiction Peb Π 2 [ ∨ 2 ] with respect to binar y logical or . Figure 1: Example of pebb ling contradiction with substitution for t he pyramid g raph Π 2 . • for all s ∈ S , a unit clause s ( sour ce axioms ), • For all non-sourc es v with immediate predec essors pr e d ( v ) , the clause W u ∈ pr e d ( v ) u ∨ v ( pebb ling axioms ), • for the sink z , the unit clause z ( tar get or sink axiom ). For any nonconstan t Boolean functio n f d : { 0 , 1 } d 7→ { 0 , 1 } , the substi tution pebbli ng contradic tion with r espec t to f d is the CNF formul a P eb G [ f d ] obta ined by sub stituting f d ( x 1 , . . . , x d ) for e very va riable x and exp anding the result to con juncti ve normal form in the canonica l way . If the graph G has n vertices and maximal inde gree ℓ , P eb G [ f d ] is easily veriﬁed to be an unsatisﬁable formula over dn v ariables with less than 2 d ( ℓ +1) · n cla uses of size a t most d ( ℓ + 1) . An exa mple illus trating Deﬁnition 1.2 is gi ven in Figure 1. Giv en any black- only pebbling P of G , it is straightfo rward to simulate this pebbling in resolutio n to refute the corresp onding pebbl ing contrad iction Peb G [ f d ] in length O  time ( P )  and space O  space ( P )  . This was perhap s ﬁrst noted in [BIW04] for the simple Peb G formulas , b ut the simulation extend s readily to any formula Peb G [ f d ] , with the const ants hidden in the asymptotic notation depending on f d and the maximal indegree of G . In the other directi on, it was recently shown in [BN09b] (streng thening resul ts in [BN08]) that if f d has the right properties—fo r instance, if it is the exclus iv e or functi on or the threshold functi on e val uating to true if k out of d v ariables are true for 1 < k < d —then any resolution refutation π 3 ON THE RELA TIVE S TRENGT H OF PEBBLING AND RESOLUTION z x y u v w (a) { x i ∨ y j ∨ z 1 ∨ z 2 | i, j = 1 , 2 } . z x y u v w (b) { u i ∨ v j ∨ x 1 ∨ x 2 | i, j = 1 , 2 } . z x y u v w (c) { u i ∨ v j ∨ y k ∨ z 1 ∨ z 2 | i, j, k = 1 , 2 } . Figure 2: Black and white peb ble s and (intuitiv ely) corresponding sets of clauses. of Peb G [ f d ] can be transla ted into a black- white pebbli ng of G with time and spac e upper- bounded by the length and space of π , respecti ve ly (adjust ed for small multiplica tiv e constants dependi ng on the maximal inde gree of G ). There is an obvious gap in these reductions between pebbl ing and resolution. T o interp ret a resolution refutat ion of a pebbling contradictio n in terms of a pebbling of the under lying graph, the full power of black- white pebbling is needed to make the reduction work. If we want to translate pebblings of graphs into ref utations of the corr espondin g pebbling contradi ctions, ho wev er , we only kno w ho w to d o this for the weake r black pebb le game. T o see w hy resolu tion has a hard time simulating black-wh ite pebb lings, let us start by discuss ing a black- only pebbl ing P . W e can easily mimic such a pebbling in a resolution refutation of Peb G [ f d ] by deri ving that f d ( v 1 , . . . , v d ) is true whene ver the correspon ding ve rtex v in G is black-pebb led. W e end up deri ving that f d ( z 1 , . . . , z d ) is true for the sink z , at w hich point we can do wnload the sink axioms and deri ve a c ontradict ion. The intuition behind this t ranslatio n is that a black pebble o n v means that we know v , which in resol ution trans lates into truth of v . In the pebble game, ha ving a white pebble on v instead means that we need to assume v . By duality , we let this correspond to falsi ty of v in resolution . Focus ing on the pyr amid Π 2 and pebbling contradi ction Peb Π 2 [ ∨ 2 ] in Figure 1, our intuiti ve unders tanding then becomes that white pebble s on x and y and a black pebble on z should corresp ond to the set of clauses { x i ∨ y j ∨ z 1 ∨ z 2 | i, j = 1 , 2 } (1.1) which indeed encode that assuming x 1 ∨ x 2 and y 1 ∨ y 2 , we can deduce z 1 ∨ z 2 . See Figure 2(a) for an illustr ation of this. If we now p lace white pebble s on u and v , this allows us to remov e the white p ebble from x . Rephrasin g this in terms of resolu tion, w e can say that x follo ws if we assume u and v , which is encode d as the set of clause s { u i ∨ v j ∨ x 1 ∨ x 2 | i, j = 1 , 2 } (1.2) (see Figure 2(b)), and indee d, from the clause s in (1.1) and (1.2) we can deri ve in reso lution that z is black- pebble d and u , v and y are white pebbled, i.e., the set of clauses { u i ∨ v j ∨ y k ∨ z 1 ∨ z 2 | i, j, k = 1 , 2 } (1.3) (see F igure 2(c)). T his to y example indi cates one of the proble ms one runs into when one tries to simulate black- white pebbli ng in resolution : as the number of white pebbl es gro ws, there is an expon ential blow-u p in the number of clauses. The clause set in (1.3) is twice the size of those in (1.1 ) and (1.2), although it corres ponds to only one more white pebble. This suggest s that as a pebbling starts to make hea vy use of white pebbles, a resolutio n refutatio n will not be able to mimic such a pebbling in a length- and space- preser ving manner . 4 1 Introd uction This leads to the thought that perha ps black pebbling pro vides not only upper b ut also lo wer bounds on resolu tion refuta tions of pebbling c ontradict ions. T his would be co nsistent with what has b een kn own so far . For all p ebbling contradicti ons w ith prov en space lower b ounds, the unde rlying graphs hav e asympto tically the same b lack an d black-white peb bling pric e, and hence a ll kn own lo wer b ounds c an b e e xpres sed in terms of black peb bling. There ha ve been no ex amples of pebbli ng cont radiction s w here reso lution can do strictly better than black pebblin g and tightly match smaller bounds on space in terms of black- white pebblin g. 1.1 Our Results Our ﬁrst set of resu lts is that reso lution can in fact be strictly bette r than black- only pebblin g, both for time-spa ce trade- off s and with respect to space in absolute terms. W e prove this by designing a limited ver sion of black -white pebbling, where we explicitly restric t the amount of nondeterminis m, i.e., white pebble s, a pebbling strateg y can use. S uch restr icted pebblin g use “few white pebble s per black pebble” (in a sense that will be made formal below), and can therefore be simulated in a time- and space-pre serving manner by resoluti on, av oiding the expo nential blow-up just discussed. W e then sho w that for all known separa tion result s in the pebbl ing lite rature where black -white pebbling doe s asymptotically better than black- only pebbling , there are graphs exh ibiting these separat ions for which optimal black-white pebblings can be carried out in our limited version of the game. This means that resolu tion refuta tions of pebbling contra dictions ove r such D A Gs can do strictly asympto tically better than what is suggeste d by black-only pebbli ng, matching the lower b ounds in terms of (general) black-white pebbling. More precise ly , we obtai n such results for three familie s of grap hs. 2 The ﬁrst famil y are the bit r eversa l gra phs studie d by Lengauer and T arjan [L T82], for which black- white pebbli ng has quadratical ly bette r trade-o ffs than black pebbling . (W e refer to Section 3 for all formal notatio n and deﬁnitio ns used belo w .) Lemma 1.3 ([L T82]). T her e are DA Gs { G n } ∞ n =1 of size Θ ( n ) with blac k pebbling price P eb ( G n ) = 3 suc h that an y optimal b lac k pebblin g P n of G n e xhibits a trade-of f time ( P n ) = Θ  n 2 / space ( P n ) + n  b ut optimal blac k-white pebblings P n of G n ach ieve a trade -of f time ( P n ) = Θ  ( n/ space ( P n )) 2 + n  . Theor em 1.4. F ix any non-con stant Boolean function f and let Peb G n [ f ] be pebbling contrad ictions over the graphs in L emma 1.3. T hen for any monotonic ally nondecr easing function s ( n ) = O( √ n ) ther e are r esolu tion r efutati ons π n of Peb G n [ f ] in total space O( s ( n )) and length O  ( n/s ( n )) 2  , beating the lower bound Ω  n 2 /s ( n )  for blac k-onl y pebblings of G n . Focus ing next on abs olute bounds on spac e rather than t ime-space trade-of fs, the best kno wn separation between black and black -white pebbli ng for polynomia l-size graphs is the one sho wn by W ilber [W il88]. Lemma 1.5 ([W il88 ]). Ther e ar e D A Gs { G ( s ) } ∞ s =1 of size poly nomial in s with bla ck-wh ite pebblin g price BW -P eb ( G ( s )) = O( s ) and blac k pebb ling price P eb ( G ( s )) = Ω( s log s/ log log s ) . For p ebbling formul as over t hese graph s we do not kno w ho w to b eat the blac k pebbling space bou nd— we return to this some what intriguin g problem in Section 7—but using instead graphs in [KS91] exhi biting the same pebb ling proper ties, we can obtain the desired result. Theor em 1.6. F ix an y non-con stant Boolean funct ion f and let Peb G ( s ) [ f ] be p ebbling contr adiction s over the gra phs G ( s ) in [KS91] with pebbling pr operti es as in Lemm a 1.5. T hen ther e ar e res olution re futations π n of Peb G ( s ) [ f ] in total space O( s ) , beating the lower bound Ω( s log s/ log log s ) for b lac k-only pebbling . If we remov e all restrictio n on graph size, there is a quadratic separation of black and black-white pebbli ng establis hed by Kalyanasu ndaram and Schnitger [KS91]. 2 All graphs discussed in this paper are explicitly constructible and hav e bounded vertex ind egree. Also, unless otherwise stated they ha ve a single, unique sink. W e do not repeat this in the formal statements here in order not to clutter the text un necessarily . 5 ON THE RELA TIVE S TRENGT H OF PEBBLING AND RESOLUTION Lemma 1.7 ([KS91]). Ther e ar e D A Gs { G ( s ) } ∞ s =1 of size exp (Θ( s log s )) such that BW -P eb ( G ( s )) ≤ 3 s + 1 b ut P e b ( G ( s )) ≥ s 2 . For peb bling formul as ov er these graphs , resol ution again matches the blac k-white pebb ling bound s. Theor em 1.8. F ix an y non-con stant Boolean funct ion f and let Peb G ( s ) [ f ] be p ebbling contr adiction s over the graphs G ( s ) in Lemma 1.7. Then ther e ar e r esolution r efutatio ns π n of Pe b G ( s ) [ f ] in total space O( s ) , beatin g the lower bound Ω  s 2  for blac k-onl y pebbling . In particu lar , T heorems 1.6 and 1.8 sho w that the lower bound on proof space for pebbling contradicti ons in terms of black-whi te pebbling price in [BN08] is tight (up to constan t factor s). T urning t o our second set o f results, we ﬁrst note that in sp ite of the theorems a bov e, fo r g eneral pebbling formulas we still do not kno w of any wa y of simulat ing black-white pebbling in resolu tion. Instead, we are limited to deri ving u pper bounds from bl ack-only pebblings whi le lower bounds ha ve to b e obtained in t erms of bl ack-white pebbling s. At ﬁrst sig ht, this migh t not loo k too bad since the space g ap betwee n the two c an be at most q uadratic , as shown by Mey er auf der Heide [Mey81]. Ho wev er , the t ranslatio n giv en in [Mey81] of a black-whi te pebblin g in space s to a black pebbling in space O  s 2  incurs an expo nential blow-up in pebbli ng time, destro ying all hope of obtaining nontri vial time-space trade-of f results for resolu tion in this way . H ence, to get meaningf ul trade- off s for pebbling formulas w e need graph families with strong dual trade-o ffs for black and black-white pebbling simultan eously . In this paper , we present such a family of graphs , bu ilding on and strengthe ning previ ous work by Carlso n and Sav age [CS80, CS82]. Theor em 1.9. Ther e is an e xplicitl y con structib le two-par ameter graph family Γ( c, r ) , for c, r ∈ N + , having uniqu e sink, verte x inde gr ee 2 , and size Θ  cr 3 + c 3 r 2  , and satisfyi ng the following pr operties: 1. Γ( c, r ) has black -white pebbling price BW -P eb (Γ( c, r )) = r + O (1) and blac k pebbling price P eb (Γ( c, r )) = 2 r + O (1) . 2. Ther e is a blac k-only pebb ling of Γ( c, r ) in time linea r in the gr aph size and in space O( c + r ) . 3. Suppose that P is a blac k-white pebbling of Γ( c, r ) with space ( P ) ≤ r + s for 0 < s ≤ c/ 8 . T hen time ( P ) ≥  c − 2 s 4 s +4  r · r ! . The graph family in Theorem 1.9 turns out to be surprising ly ve rsatile. For instance , we can use it to pro ve among othe r things the rather striking statement that for any arbitr arily slowly gr owing non-const ant functi on, there are exp licit graphs of such (arbitra rily small) pebblin g space complexity that ne verthe less exh ibit superpolyn omial time-spa ce trade -off s for black and black-white pebblin g simultaneou sly . Theor em 1.10. Let g ( n ) be any arbitr arily slowly gr owing 3 monoton e function ω (1) = g ( n ) = O  n 1 / 7  , and let ǫ > 0 be an arbitr arily small positive constant. Then ther e is a family of expli citly const ructible single -sink DA Gs { G n } ∞ n =1 of size Θ( n ) such that the followin g holds : 1. The graph G n has blac k-white pebbling price BW -P eb ( G ) = g ( n ) + O(1) and black peb bling price P eb ( G ) = 2 · g ( n ) + O(1) . 2. Ther e is a co mplete black pebblin g P of G n with time ( P ) = O( n ) and spa ce ( P ) = O  3 p n/g 2 ( n )  3. Any complete blac k-white pebbling of G n in space at most  n/g 2 ( n )  1 / 3 − ǫ r equir es pebbli ng time superp olynomial in n . More ex amples of inter esting trade- off s that can be obtain ed from the graphs in Theorem 1.9 are gi ven in Section 6. 3 Note t hat we also assume g ( n ) = O  n 1 / 7  , i.e., that g ( n ) does n ot g row to fast. This is just a simplifying technical assumption. If we all o w the minimal space to grow as fas t as n ǫ for some ǫ > 0 , then it is easy t o use our graph family with other pa rameter settings to obtain e ven stronger results. Hence, the interesting aspect here is that g ( n ) is allowed to gro w arbitrarily slowly . 6 2 Outline of Constructio ns and Proofs 1.2 Organization of This P aper In Section 2 we outline the main ideas behind our results, and Section 3 pro vides all the necessary pre- liminarie s for the formal proofs of these results giv en in the rest of the paper . Section 4 prov es our claims about the limited type of black-white pebbling s that can be simulated by resolution , and in Section 5 we sho w that there are such limited pebblings for some interestin g graph fa milies. In Section 6, we discuss the graphs e xhibit ing our ne w pebblin g trade-of f result s, and sho w ho w dif ferent parameter se ttings yie ld strong dual time-space tra de-of fs with upper bounds f or b lack p ebbling a nd mat ching lo wer b ounds f or b lack-white pebbli ng. W e conclude in Section 7 by di scussing some remaini ng open pro blems. 2 Outline of Constructions and Proofs W e wil l nee d to s et up a f air amo unt of technical machi nery befor e we c an gi ve the full, formal proofs of our results . In order not to obscure unnecessari ly what are in essenc e reasonably straight forward argumen ts, in this sectio n we try to gi ve an o vervie w of the main ideas, postponi ng the technica lities for later . 2.1 Limited Blac k-White P ebblings That Can Be Simulated by Re solution Let us start by discussin g the tools used to establish Theorems 1.4, 1.6, and 1.8 . The idea is to design a ver sion of th e black -white pebble game that is t ailor -made for res olution. This game is essentia lly just a for - malizatio n of the naiv e reso lution simulati on sketche d in Sectio n 1, b ut before stating the f ormal deﬁnit ions, let us try to pro vide some intuition w hy the rules of this ne w game look the way th ey do. First, if we want a game that can be mimicked by resoluti on, then place ments of isolated w hite vertice s do not make much sen se. What a resolu tion deri va tion can do is to do w nload axiom clauses , and intuiti vely this correspond s to placing a black pebble on a vertex together with white pebble s on its immediate pre- decess ors, if it has any . Therefore, we adopt such aggre gate moves as the only admissibl e way of placing ne w peb bles. For in stance, looking a t the g raph Π 2 and p ebbling con tradiction Peb Π 2 [ ∨ 2 ] in Figure 1 again, placin g a black pebble on z and w hite pebbl es on x and y correspo nds to do wnloading the axiom clauses in (1.1). Second, note that if we ha ve a black pebble on z with white pebbles on x and y correspond ing to the clause s in (1.1) an d a bl ack pebbl e on x w ith whit e pebbl es on u an d v correspondi ng to the c lauses in (1.2), we can deri ve the clauses in (1.3) correspond ing to z black-pebb led and u , v and y white-pebb led b ut no pebble on x . This s uggests th at a natural rul e for w hite pebble remov al i s t hat a w hite pebble c an be remo ved from a verte x if a black pebble is placed on that same verte x (and not on its immedia te prede cessors) . Third, if we th en just erase all clauses in (1.3), thi s correspond s to all pebble s disappea ring. On the face of it, this is very much unlike the rule for w hite pebble remov al in the standar d pebbl e game, where it is absolu tely crucial that a white peb ble can o nly be re move d when i ts pred ecessors are pe bbled. Howe ver , the importan t point here is that not only do the white pebbles disappea r—the black pebble that has been placed on z with the help of these w hite pebbles disapp ears as well. What this m eans is that we cannot treat black and white pebble s in isolation , b ut we hav e to keep track of for each black pebble which white pebble s it depen ds on, and m ake sure that the black pebble also is erased if any of the white pebbles supporting it is erased . The way we do this is to label each black pebble v with its supp orting white pebbles W , and deﬁne the pebble game in terms of mov es of such labelle d pebbl e subconﬁ gurati ons v h W i . Deﬁnition 2.1 (Pebble subconﬁguration). For v a ver tex and W a set of vertices, we say that v h W i is a pebble subcon ﬁgur ation with a black pebble on v supporte d by white pebbles on W . T he black pebble on v is said to be dependen t on the white pebbles in its support W . W e refer to v h∅i as an independe nt blac k pebble . 7 ON THE RELA TIVE S TRENGT H OF PEBBLING AND RESOLUTION Our next deﬁnition no w formaliz es the informal descrip tion of our new pebble game. W e remark that this d eﬁnition is quite similar t o the pebble game deﬁned in [Nor09], and th at we hav e borro wed freely fr om notati on and terminolog y there. Deﬁnition 2.2 (Labelled pebbling). For G any DA G with unique sink z , a (complete) labelled pebbling of G is a seque nce L = { L 0 , . . . , L τ } o f lab elled pebb le conﬁgura tions such that L 0 = ∅ , L τ = { z h∅i} , and for all t ∈ [ τ ] it holds that L t can be obtai ned from L t − 1 by one of the follo w ing rules: Introductio n L t = L t − 1 ∪ { v h pr e d ( v ) i} , where pr e d ( v ) is the set of immedia te prede cessors of v . Erasure L t = L t − 1 \ { v h V i} for v h V i ∈ L t − 1 . Merger L t = L t − 1 ∪  v h ( V ∪ W ) \ { w } i  for v h V i , w h W i ∈ L t − 1 with w ∈ V . W e den ote this s ubcon- ﬁguratio n merge ( v h V i , w h W i ) , and refer to it as a mer ger on w . Let Bl ( L t ) = S { v | v h W i ∈ L t } denote the set of all black-pebb led ver tices in L t and Wh ( L t ) = S { W | v h W i ∈ L t } the set of all white-pe bbled v ertices. Then the space of an labelled pebbling L = { L 0 , . . . , L τ } is max L ∈L {| Bl ( L ) ∪ Wh ( L ) |} and the time of L is time ( L ) = τ . Figures 2(a) and 2(b) are both ex amples of subconﬁgura tions resulti ng from introdu ction moves , and if we mer ge the two we get the subcon ﬁguration in Figure 2(c). The game in Deﬁnition 2.2 might look quite diffe rent from the standar d black-whit e pebble game, but it is not hard to sho w that labell ed pebblings are essentia lly just a restric ted form of black-white pebblings . (The proof of this is deferre d to Section 4.) Lemma 2.3. If G i s a singl e-sink D A G a nd L is a complete labelled pebb ling of G , then the r e is a compl ete blac k-white pebbling P L of G with time ( P L ) ≤ 4 3 time ( L ) and space ( P L ) ≤ space ( L ) . Ho wev er , the deﬁnitio n of space of labe lled pebbling s does not seem quite right from the poin t of vie w of resolut ion. Not only does the space measure fail to capture the exponen tial blo w-up in the number of white pebbles discussed above . W e also hav e the problem that if one white pebble is used to supp ort many dif ferent black pebbles, then in a resolu tion refutation simulatin g such a pebbli ng we hav e to pay multiple times fo r this si ngle white peb ble, once for e very blac k pebble su pported by it. T o get somet hing that can be simulate d by resolut ion, we therefore need to restrict the labelle d pebble game e ven furthe r . Deﬁnition 2 .4 (Bounded labelled pebblings). An ( m, S ) -boun ded labelled pebbl ing is a labelled pebblin g L = { L 0 , . . . , L τ } such that e very L t contai ns at m ost m pebble subco nﬁguration s v h W i and eve ry v h W i has white suppor t size | W | ≤ S . Observ e that bound edness automatica lly implies lo w space complexit y , since an ( m, S ) -bounded peb- bling L clea rly satisﬁes space ( L ) ≤ m ( S + 1) . And using the concep t of bound ed labelled pebblings, we can sho w that if there is such a pebbling of a graph G , then this pebbling can be used as a template for a resolu tion refuta tion of any pebb ling contra diction Peb G [ f ] . (W e again refer to Section 4 for the proof .) Lemma 2.5. Suppose that L is any complete ( m, S ) -bounde d pebbling of a graph G and that f is any nonco nstant B oolean function of arity d . Then ther e is a res olution r efutation π L of the formula Peb G [ f ] in simultaneous length L ( π L ) = time ( L ) · exp  O( dS )  and total space T otSp ( π L ) = m · exp  O( dS )  . In parti cular , ﬁxing f it holds that r esolut ion can simulate ( m, O(1)) -bounded pebblings in a time- and space- pr eservin g manner . The w hole problem thus boils do wn to the question w hether there are graphs with (a) asymptotical ly dif ferent prop erties for black and black -white pebbling for which (b) optimal black -white pebbling s can be 8 2 Outline of Constructio ns and Proofs s 1 s 2 γ 1 γ 2 γ 3 Figure 3: Base case for Carlson-Sa vage g raph with 3 spines and sinks. carried out in the bounded labelled pebbli ng framewor k. The answer to this questi on turns out to be yes, and the space upper boun ds for the pebbl ing con tradiction s in T heorems 1.4, 1.6, and 1.8 are all prov en by exhibiti ng bounded labelled pebblings for the correspo nding graphs. T he details concerning how these graphs are construct ed, as well as ho w they are pebbled , are somewhat intricate , howe ve r , and are theref ore presen ted separat ely in Section 5. 2.2 A Graph Family with Tight T r ade-offs f or Black and Blac k-White Pebb ling Let us next outline the proof of our graph pebblin g trade -off results in T heorem 1.9. W e remark that in what follows, we will di scuss a slightly dif ferent setting wher e gra phs may hav e multipl e sinks, no t just one, and where we only requir e that a pebbling visits ev ery sink once, touchin g it with a black or white pebble, instea d of lea ving a black pebble on the sink until the end of the pebbling. It is straightforw ard to translate results for such pebbl ings back to the setting in Theorem 1.9. (See Section 3 for the technical details.) Our graph family is b uilt on a construc tion by Carlson and Sa va ge [CS 80, CS82]. Carlson and S a vag e only prov e their trade-of f for black pebbling, ho wev er , and the extension of their results to black-white pebbli ng requires changin g the constructio n and doing a nontri vial amount of extra work (as is usually the case w hen one wants to lift a black pebbli ng result to black-white pebbling ). T he formal deﬁnition of the family of graphs, which we will refer to as Carlson-Sa vag e graphs , is prob ably easier to parse if the read er ﬁrst studie s the illustr ations in Figures 3 and 4. Deﬁnition 2.6 (Carlson-Sa vage g raphs). The two-paramet er gr aph f amily Γ( c, r ) , for c, r ∈ N + , is deﬁned by inducti on over r . The base case Γ( c, 1) is a D A G consist ing of two sources s 1 , s 2 and c sinks γ 1 , . . . , γ c with directed edges ( s i , γ j ) , for i = 1 , 2 and j = 1 , . . . , c , i.e., edges from both sourc es to all sinks. The graph Γ( c, r + 1) has c sinks and is bu ilt from the followin g compone nts: • c disjoint copi es Π (1) 2 r , . . . , Π ( c ) 2 r of a py ramid graph 4 of height 2 r with sinks z 1 , . . . , z c . • one copy of Γ( c, r ) , for which we denote the sinks by γ 1 , . . . , γ c . • c disjoint and identical spines , where each spine is composed of cr sec tions , a nd ev ery section contain s 2 c vertic es. W e let the vertic es in the i th sectio n of a spine be denot ed v [ i ] 1 , . . . , v [ i ] 2 c . The edges in Γ( c, r + 1) are as follo ws: • All “internal edges” in Π (1) 2 r , . . . , Π ( c ) 2 r and Γ( c, r ) are pres ent also in Γ( c, r + 1) . • For each spine, there are edges  v [ i ] j , v [ i ] j +1  for all j = 1 , . . . , 2 c − 1 w ithin each section i and edges  v [ i ] 2 c , v [ i + 1] 1  from the end of a secti on to the beginn ing of next for i = 1 , . . . , cr − 1 , i.e., for all sections b ut the ﬁnal one, where v [ cr ] 2 c is a sink. • For ea ch section i in each spine, there ar e edges  z j , v [ i ] j  from th e j th p yramid sink to the j th verte x in the section for j = 1 , . . . , c , as well as edges  γ j , v [ i ] c + j  from the j th sink in Γ( c, r ) to the ( c + j ) th verte x in the section for j = 1 , . . . , c . 9 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION z 1 γ 1 z 2 γ 2 z 3 γ 3 Π (1) 2 r Π (2) 2 r Π (3) 2 r Γ(3 , r ) Figure 4: Inductive deﬁnition of Carlson-Sav age gr aph Γ(3 , r + 1) with 3 spines and sinks. 10 2 Outline of Constructio ns and Proofs Let us focus on the trade -off lower bou nd in part 3 of Theorem 1.9, which is the hard part to prove , and let us start by trying to prov ide some intuitio n why this bound should hold. For simplicity , consid er ﬁrst black- only pebblings . Assume inducti vely that part 3 of T heorem 1.9 has been prov en for Γ( c, r − 1) and consid er Γ( c, r ) . Any pebbling strate gy for this D A G will ha ve to pebble throug h all sections in all spine s. Consider the ﬁrst section anywhere, let us say on spine j , that has been completely pebbled, i.e., there ha ve been pebb les placed on and remo ved fro m all vertic es in th e section. Let us say tha t this happens at time τ 1 . But this means that Γ( c, r − 1 ) and all pyra mids Π (1) 2( r − 1) , . . . , Π ( c ) 2( r − 1) must ha ve been complete ly pebbled during this part of the pebbl ing as well. Fix any pyramid and consider some point in time σ 1 < τ 1 when there ar e at least r + 1 pebbles on its v ertice s, w hich must happen because of kn own pe bbling lower bounds for pyramids [Coo74, Kla85]. At this point, the rest of the graph must contain ve ry fe w pebbl es (think of s here as being very small). In particular , there are ver y few pebbles on the subg raph Γ( c, r − 1) at time σ 1 , so for all practic al purpose s we can think of Γ( c, r − 1) as being essentially empty of pebbles. Consider now the next section in the spine j that is completed , say , at time τ 2 > τ 1 . Again, we can ar gue that some pyramid is complet ely pebbled in the time interv al [ τ 1 , τ 2 ] , and thus has r + 1 pebbl es on it at some time σ 2 > τ 1 > σ 1 . This m eans that Γ( c, r − 1) is essentiall y empty of pebbles at time σ 2 as well. But note that all sinks in the subgrap h Γ( c, r − 1) must hav e been pebbled in the time interv al [ σ 1 , σ 2 ] , and since we kno w that Γ( c, r − 1) is (almost) empty at times σ 1 and σ 2 , this allo ws us to apply the induct ion hypot hesis. Since P has to pebbl e through a lot of section s in dif ferent spines, we will be able to repeat the abo ve ar gument many times and apply the in duction hypothe sis on Γ( c, r − 1) e ach time. Adding up all t he lo wer bounds obtained in this way , the induction step goes through. This is the spirit of the proof of the black-onl y pebbling trade-o ff in [CS82]. When w e instead want to deal with black-whi te pebblin gs, things get much more complic ated. Black pebblin gs must by necessity pebble through a graph in a bottom-up fashion, and it is therefor e straightfor ward to measure “ho w far” a black pebb ling has progresse d. A black-white pebbli ng, ho wev er , can place and remove pebbles anywhe re in th e D A G at a ny time. Therefore, it is m ore dif ﬁ cult to co ntrol the progr ess of a black -white pebbling, and one has to use dif ferent ideas and work harder in the proof . W e establ ish part 3 of Theorem 1.9 by prov ing a slightl y strong er lemma, dealing with condition al pebbli ngs that start with some pebbles already present on the graph, and can also leav e some pebbles on the graph at the end of the pebbling. A crucial ingredient in the proof is that we assume belo w (without loss of general ity) that all pebblings are frugal , meanin g that no obvio usly redundant pebbl e placements are made, but that all pebbles placed on the graph are used to place other black pebbles on success ors or to remo ve white pebbles from successors . (Again, we refer to Section 3 for a more thorough discuss ion of these pebblin g techn icalities.) Lemma 2.7. Suppose that P = { P σ , . . . , P τ } is a conditio nal blac k-white pebbling on Γ( c, r ) such that 1. max  space ( P σ ) , space ( P τ )  < s for 0 < s ≤ c/ 8 − 1 . 2. P pebbles all sinks in Γ( c, r ) during the time interval [ σ, τ ] . 3. space ( P ) < r + s + 2 . Then it holds that time ( P ) = τ − σ ≥  c − 2 s 4 s +4  r · r ! . T o establish this resu lt we will need the follo wing four techn ical lemmas, the proo fs of w hich are post- poned to S ection 6. Lemmas 2.8 and 2.9 are easy , but Lemmas 2.10 and 2.11 are somewhat less immediate and pro vide the key to the proo f. 4 The formal deﬁnition will be giv en later in Deﬁnition 3.4, but as an ex ample the graph in Figure 1(a) is a pyramid of height 2 . 11 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION Lemma 2.8. Suppose v is a vert ex with a path Q to some sink such that all vertic es in Q have outde gr ee 1 . Then any fruga l blac k-white pebbling pebble s v exact ly once , and the path Q contains pebbles during one contig uous time interva l. Lemma 2 .9. Let H be a subgra ph of G such that the only edg es between V ( H ) an d V ( G ) \ V ( H ) eman ate fr om the uniq ue sink z h of H . Supp ose that P is a complet e pebbl ing of G such that H is complet ely empty of pebbles at some time τ ′ b ut at least one verte x of H has been pebbled during the time interva l [0 , τ ′ ] . Then P must have pebbled H completely during the interval [0 , τ ′ ] . Lemma 2.10. At all times during a p ebbling o f Γ( c, r ) as in Lemma 2.7, str ictly less than 4( s + 1) py ramids Π ( j ) 2 r contai n pebbles simultane ously . Lemma 2.11. A t all times during a pebbling of Γ( c, r ) as in Lemma 2.7, strictly less than 4( s + 1) spine sectio ns contain pebble s simultaneou sly . Pr oof of Lemma 2.7. Let P = { P σ , . . . , P τ } be a pebbling as in the statement of the lemma. W e sho w that time ( P ) ≥ T ( c, r, s ) =  c − 2 s 4 s +4  r · r ! by induction over r . For r = 1 , the assumptions in the lemma imply that more than c − 2 s sin ks are empty at times σ and τ . These sinks must be pebbled , which trivia lly require s strictly more than c − 2 s >  c − 2 s 4 s +4  = T ( c, 1 , s ) time steps. Assume that the lemma holds for Γ( c, r − 1) and cons ider any pebbl ing of Γ( c, r ) . Less than 2 s spines contai n pebbles at time σ or time τ . All th e other strictly more than c − 2 s spines ar e empty a t times σ and τ b ut must be completely pebbled during [ σ, τ ] since their sinks are pebbled during this time interv al. (This can be more formally ar gued by using L emma 3.12.) Consider the ﬁrst t ime σ ′ when an y spine gets a pe bble for the ﬁrst time. L et us d enote this spine by Q ′ . By Lemma 2.8 we kno w that Q ′ contai ns pebb les during a contigu ous time interv al until it is completely pebble d and emptied at, say , time τ ′ . During this whole interv al [ σ ′ , τ ′ ] less than 4 s + 4 sections contain pebble s at any one gi ven time by Lemma 2.11, so in particu lar less then 4 s + 4 spines contain pebbl es. Moreo ver , Lemma 2.8 says that e very spine containing pebb les will remain pebbled until completed . What this means is that if w e order the spines with respect to the time when they ﬁrst recei ve a pebble in group s of size 4 s + 4 , no sp ine in the second group can be pebbled until the at least one spine in the ﬁrst grou p has been complet ed. W e observ e that this di vides the spines that are empty at the begin ning and end of P into strictly more than c − 2 s 4 s +4 group s. Furthermor e, we claim that complete ly pebbling just one empty spine requires at least r · T ( c, r − 1 , s ) time steps. Giv en this claim we ar e done, since it follo ws that the total pebbli ng time must then be lo wer-b ounded by c − 2 s 4 s +4 r · T ( c, r − 1 , s ) = T ( c, r , s ) . This is so since at least one spine from each group is pebbl ed in a time interv al totally disjoint from the time interv als for all spines in the next group . It remains to estab lish the claim. T o this end, ﬁx any spin e Q ∗ empty at times σ ∗ and τ ∗ b ut completely pebble d in [ σ ∗ , τ ∗ ] . Consider the ﬁrst time τ 1 ∈ [ σ ∗ , τ ∗ ] when any sectio n in Q ∗ , let us denote it by R 1 , has been complete ly pebb led (i.e., all vertices has been touched by pebbles b ut a re no w empty a gain). During the time interv al [ σ ∗ , τ 1 ] all pyramid sinks z 1 , . . . , z c must be pebbl ed (since they are immediate predecess ors). Since less than 2 · (4 s + 4) < c pyra mids contain pebbles at times σ ∗ or τ 1 (Lemma 2.10), at least one pyr amid is pebb led completely (Lemma 2.9), which requires r + 1 pebbl es. Moreo ver , there is at least one pebble on the section R 1 during this whole interva l. Hence, there must exist a point in time σ 1 ∈ [ σ ∗ , τ 1 ] when there are stric tly less than ( r + 2) + s − ( r + 1 ) − 1 = s pebbles on the subgraph Γ( c, r − 1) . Also, at this time σ 1 less than 4 s + 4 sections contain pebbles (L emma 2.11), and in particular this means that there ar e pebbl es on less tha n 4 s + 3 othe r section of o ur spin e Q ∗ . T his p uts an up per bound on the number of sections of Q ∗ that can hav e been touched by pebbles this fa r , since eve ry section is completely pebb led during a contiguo us time interv al before being emptie d again, and we chose to focus on the ﬁrst section R 1 in Q ∗ that was ﬁnished . 12 3 Preliminaries Look no w at the ﬁrst section R 2 in Q ∗ other than the less than 4 s + 4 section s containin g pebbl es at time σ 1 that is completely pebbled , and let the time w hen R 2 is ﬁ nished be denoted τ 2 (clearl y , τ 2 > τ 1 ). During [ σ 1 , τ 2 ] all sinks of Γ( c, r − 1) must hav e been pebbled, and at time τ 2 − 1 less than 4 s + 3 other sectio n in Q ∗ contai n pebbl es. Finally , consider the ﬁrst ne w section R 3 in our spine Q ∗ to be completely pebbled among those not yet touched at time τ 2 − 1 . Suppose that R 3 is ﬁnished at time τ 3 . Then during [ τ 2 , τ 3 ] some pyr amid is complete ly pebbled, and thus there is some time σ 3 ∈ ( τ 2 , τ 3 ) when there are at least r + 1 pebb les on this pyr amid and at least on e pebble on the spin e Q ∗ , lea ving less than s pebbles for Γ( c , r − 1 ) . But this means that we can apply t he indu ction hypothes is on t he inter val [ σ 1 , σ 3 ] an d deduce that σ 3 − σ 1 ≥ T ( c, r − 1 , s ) . Note also that at time σ 3 less than 8 s + 8 < c sect ions in Q ∗ ha ve been ﬁnishe d. Continui ng in this way , for e very group of 8 s + 8 < c ﬁnished sections in the spine Q ∗ we get one pebbli ng of Γ ( c, r − 1) in spac e less than r + s + 1 and with less than s pebble s in the start and end conﬁgura tions, which all ows us t o apply the indu ction hypo thesis a total numbe r of at least cr 8 s +8 > r times. (Just to ar gue that we get the consta nts right, note that 8 s + 8 < c implies that after the ﬁnal pebbl ing of the sink s of Γ( c, r − 1) has been done, there is still some empty section left in Q ∗ . When this ﬁ nal section is taken care of, w e will again get at least r + 1 pebbles on some pyramid while at least one pebble resides on Q ∗ , so we get the space on Γ( c, r − 1) do wn belo w s as is need ed for the induc tion hypot hesis.) This prov es our claim that pebbli ng one spine tak es time at least r · T ( c, r − 1 , s ) . Lemma 2.7 now follo w s. 3 Preliminaries In this section , we collec t all the basic deﬁnition s and facts we need about resol ution and pebblin g. 3.1 The Resolution Pr oof System A literal is either a propositi onal logic var iable or its negatio n, denote d x and x , respecti vely . A clause C = a 1 ∨ · · · ∨ a k is a set of literals. A clause contain ing at most k literals is called a k -clause . A CN F formula F = C 1 ∧ · · · ∧ C m is a set of cla uses. A k -CNF formu la is a CNF formula cons isting of k -clauses. W e say that F implies C , denot ed F  C , if any truth v alue assignmen t satisfyin g F must also satisfy C . When we want to st udy lengt h and s pace si multaneous ly , the follo wing deﬁnition of the r esolutio n proof system is very con venient . Deﬁnition 3.1 (Resolution ([ABRW02 ])). A seque nce of clause conﬁgur ation s (sets of clauses) π = { C 0 , . . . , C τ } is a r esolution r efutation of a CNF formul a F if C 0 = ∅ , C τ contai ns the contradi ctory empty clause 0 without any literals, and for all t ∈ [ τ ] , C t is obtained from C t − 1 by one of the follo wing rules: Axiom Do wnload C t = C t − 1 ∪ { C } for some C ∈ F (an axiom cla use). Erasure C t = C t − 1 \ { C } for some C ∈ C t − 1 . Infere nce C t = C t − 1 ∪ { D } for some D inferred from C 1 , C 2 ∈ C t − 1 by the r esolut ion rule , i.e., D = C 1 ∪ C 2 \ { x, x } for some variab le x such that x ∈ C 1 and x ∈ C 2 . Deﬁnition 3.2 (Length and space). The length L ( π ) of a resolution deri v ation π is the total number of axiom do wnloads and inferenc es made in π , i.e., the total numbe r of claus es coun ted with repetit ions. The c lause spac e Sp ( C ) o f a c lause conﬁgu ration C is | C | , i.e., the number of clauses in C , and th e total space T otSp ( C ) is P C ∈ C | C | , i.e., the total number of literals in C counted with repetiti ons. T he clause 13 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION space (total space) of a deriv ation π is the m aximal clause space (total space) of any clause conﬁgurat ion C ∈ π . T aking the m inimum over all refutations of a formula F , we deﬁne L ( F ⊢ 0) = min π : F ⊢ 0 { L ( π ) } , Sp ( F ⊢ 0) = min π : F ⊢ 0 { Sp ( π ) } , and T otSp ( F ⊢ 0) = min π : F ⊢ 0 { T otSp ( π ) } as the le ngth, clause space, and total space of refut ing F in resolu tion, respec tiv ely . It is sometimes techni cally con venient to add a w eak ening rule to Deﬁnition 3.1, allowin g a resoluti on deri v ation to deri ve a weak er clau se C ′ % C from C . W e can allow or disallo w this rule as we see ﬁt, since any such weaken ing steps can always be eliminated without increa sing the length or space of a refutat ion. In particular , the follo wing upper bounds on resoluti on length and space are cleane r to state if we assume that weake ning can be used . Pro position 3.3. Suppose C is a set of clau ses and C is a clause, both over a set of variables of size n . Then C  C if and o nly if the r e e xists a res olution derivat ion of C fr om C . F urthermo r e, if C can be derived fr om C then it can be derive d in length at most 2 n +1 − 1 and total space at most n ( n + 2) simult aneously . The proof of this propo sition is standard and can be found in, for instance , [BN09b]. 3.2 Graph T ermi nology and Notation W e write G to denote a graph with vertic es V ( G ) and edges E ( G ) . All graphs in this paper are directed unless otherwis e stated, and ( u, v ) denotes a directe d edge from u to v . W e let suc c ( v ) denote the immediate successo rs and pr e d ( v ) denote the imm ediate predec essors of a ver tex v in G . W e say that vertic es of G with inde gree 0 are sour ces and that vertices with outd egree 0 are sinks . (In the liter ature, sources are als o referr ed to as in puts and si nks as tar gets or outputs ). In the no tation just introduc ed, a source verte x s in G is a ve rtex with pr e d ( s ) = ∅ , and for a sink z we ha ve suc c ( z ) = ∅ . W e will w rite S ( G ) to deno te the sourc e vertices of G and Z ( G ) to denote the si nk verti ces. For bre vity , we will sometimes refer to a D A G with a unique sink as a single-si nk DA G . Some mor e no tational con ventions are th at the parameter ℓ denotes the maximal ind egree of a D A G, a nd that when not stated otherwise, n will denote the size, i.e., the number of vertices , of a D A G (or , if more con veni ent, the size to within a small cons tant factor ). W e write Q : v w to denote a path Q starting at the vert ex v and ending at the verte x w . The pyr amid gr aphs already mentioned se veral times in this paper are formally deﬁned as follo w s. Deﬁnition 3.4 (Pyramid graph). The pyramid gr aph Π h of heigh t h is a layered D AG with h + 1 lev els, where there is one vertex on the highest lev el (the sink z ), two vertices on the nex t le vel et cetera down to h + 1 vertice s at the lowest lev el 0 . The i th ve rtex at lev el L has incoming edges from the i th and ( i + 1 ) st ver tices at lev el L − 1 . 3.3 P ebbling T echnicalities The ﬂav our of the pebble game presented in Deﬁnition 1.1 is the vers ion that we are intereste d in for our applic ations in proof comple xity , b ut for the purposes of stating and prov ing our results we need a sligh tly more general deﬁnition . Deﬁnition 3.5 (General pebbling deﬁn ition). Suppose that G is a D A G with sources S and sinks Z (one or many). A black -white pebbl ing from ( B 0 , W 0 ) to ( B τ , W τ ) in G is a sequence of pebble conﬁgu rations P = { P 0 , . . . , P τ } such that P 0 = ( B 0 , W 0 ) , P τ = ( B τ , W τ ) , and for all t ∈ [ τ ] , P t follo w s from P t − 1 by one of t he r ules i n D eﬁnition 1.1. The space o f a pebble con ﬁguration P = ( B , W ) is space ( P ) = | B ∪ W | and the space of the pebblin g P is space ( P ) = max t ∈ [ τ ] { space ( P t ) } . 14 3 Preliminaries W e say that a pebbling P = { P 0 , . . . , P τ } is condi tional if P 0 6 = ( ∅ , ∅ ) and uncondition al otherwise . A complete black-white pebbling visiting Z is a pebbl ing such that P 0 = P τ = ( ∅ , ∅ ) and such that for e very z ∈ Z , there exists a time t z ∈ [ τ ] when z ∈ B t z ∪ W t z . The minimum space of such a visiting pebbli ng is denote d BW -P eb ∅ ( G ) , and for the black pebble game we ha ve the measure P eb ∅ ( G ) . A per sistent pebb ling of G is a pebbling P such that P τ = ( Z, ∅ ) . The minimum space of an y complet e persis tent black- white or black-on ly pebbling of G is denoted BW- P e b ( G ) and P eb ( G ) , respecti vely . W e think of the moves in a pebbling as occurrin g at integra l time interv als t = 1 , 2 , . . . and talk about the pebbli ng m ov e “at time t ” (which is the move resulting in conﬁgura tion P t ) or the moves “during the time interv al [ t 1 , t 2 ] . ” A vi siting pebbli ng touches all s inks b ut lea ves the graph empty at time τ , wher eas a p ersistent pebbli ng lea ves black pebbles on all sinks at the end of the pebbling. If G has m sinks, then it clearl y holds that BW -P eb ( G ) ≤ BW -P eb ∅ ( G ) + m and P e b ( G ) ≤ P eb ∅ ( G ) + m . Also, if G has a unique sink, it is easy to see that P eb ( G ) = P eb ∅ ( G ) . The only pebblings w e are really interested in are complete pebblings of G . Howe ver , when we prov e lo wer bounds on pebbling price it will sometimes be con ven ient to be able to reason in terms of partia l pebbli ng mov e sequenc es, i.e., condition al pebbli ngs. One can think of conditiona l pebblin gs as pebbl ings that receiv e the start conﬁgura tion ( B 1 , W 1 ) “as a gift”, and are also allo wed to leav e ( B 2 , W 2 ) w ithout “clean ing up” when they ﬁnish. It is clear that we can assume that ( B 1 , W 1 ) = ( B 1 , ∅ ) and ( B 2 , W 2 ) = ( ∅ , W 2 ) since we can freely place white pebble s on G and freely remov e black pebbl es. The way the gift can help us is that we get black pebbles at the begin ning for free, and are allo wed to lea ve white pebbles without ha ving to do the hard work of remov ing them. The reason w e need visiting pebbling s and not just persistent ones is that the graphs of intere st will be const ructed in terms of smaller subgraph compone nts with useful pebbl ing prope rties, and that for such subgra phs we ha ve the follo w ing easy observ ation (the proof of which is omitted). Observ ation 3.6. Suppos e that G is a D AG and that P is any complete pebbli ng of G . Let U ⊆ V ( G ) be any subset of vertices of G and let H = H ( G, U ) denote the induced subgr aph with vertices V ( H ) = U and edges E ( H ) =  ( u, v ) ∈ E ( G )   u, v ∈ U  . Then the pebbling P r estricted to the vertices in U is a complete visitin g pebbl ing stra te gy for H . Some proofs are facilitate d by observing that visiting pebblin gs hav e a certain “duality ” property . The nex t propo sition is an imm ediate consequen ce of the anti-symmetri c nature of the pebbling rules in Deﬁni- tion 1.1 (just observ e that the rules for placing and removing a black pebble are the duals of the rules for remov ing and placing a white pebble, respecti vely ). Pro position 3.7 ([CS76]). If P is a black -white pebbling fr om ( B 1 , W 1 ) to ( B 2 , W 2 ) , then we can get a dual pebbl ing P fr om ( W 2 , B 2 ) to ( W 1 , B 1 ) in exact ly the same time and spac e by re ver sing the seque nce of move s and switching the colo urs of the pebble s. In partic ular , if P is a comple te visiting pebbling of G , then so is P . For the applicat ions in proof comple xity , we often want results stated for D A G s with one unique sink, b ut most pebbli ng resu lts are more natural to state and prov e for D A Gs with multiple sinks. This small techni cality is easily take n care of as follo ws. Deﬁnition 3.8 (Single-sink versio n). Let G be a D A G with sinks Z ( G ) = { z 1 , . . . , z m } for m > 1 . The single -sink version b G of G consists of all verti ces and edges in G plus the extra vertices z ∗ 1 , . . . , z ∗ m − 1 and the edges ( z 1 , z ∗ 1 ) , ( z 2 , z ∗ 1 ) , ( z ∗ 1 , z ∗ 2 ) , ( z 3 , z ∗ 2 ) , ( z ∗ 2 , z ∗ 3 ) , ( z 4 , z ∗ 3 ) , et cetera up to ( z ∗ m − 2 , z ∗ m − 1 ) , ( z m , z ∗ m − 1 ) . That is, b G consis ts of G with a binary tree of minimal size added on top of the sink s. See Figure 5 for a small examp le. T he follo wing observ ation is immediate. 15 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION z 1 z 2 z 3 z 4 z 5 z 6 z ∗ 1 z ∗ 2 z ∗ 3 z ∗ 4 z ∗ 5 G Figure 5: Schematic illustration of single-sink version b G of grap h G . Observ ation 3.9. Let G be a D AG w ith sinks Z ( G ) = { z 1 , . . . , z m } for m > 1 . Then for any ﬂavou r of pebbli ng (visiting or pers istent) it holds that BW- P e b  b G  ≤ BW -P eb ( G ) + 1 and P eb  b G  ≤ P eb ( G ) + 1 . Mor eover , if ther e is a pebbling strate gy P (visiting or per sistent) fo r G tha t can pebble th e s inks in ar bitrar y or der , then ther e is a pebblin g stra te gy b P of the same type (black or blac k-white, visiting or persi stent) for b G with time  b P  ≤ time ( P ) + 2 m and space ( b P ) ≤ space ( P ) + 1 . The nex t proposition is con venient when compos ing pebblin gs of smaller subgrap hs into a pebbling of a lar ger graph. Pro position 3.10. Suppos e that G is a D A G with unique s ink z . Then for any complete bla ck or blac k-white pebbli ng P of G ther e is a complete pebbling P ′ with the same colour s such that time ( P ′ ) = time ( P ) , space ( P ′ ) = space ( P ) , and ther e is a time t during P ′ when z has a pebble bu t the pebblin g space is strictl y less than space ( P ) . Pr oof. For blac k pebbling s this statemen t is obvious . Once we place a black pebble on the sink z , we can remov e all other pebbles from the D A G. Suppose for a black-whit e pebblin g P that the pebbling space reaches the maximum s precise ly when a pebble is placed on z at time t . Then the move at time t + 1 must be a pebble remov al. If a pebble is remov ed from a vertex other than z , we are done. Otherwise, ﬁx some verte x w ∈ pr e d ( z ) hav ing z as its only succes sor . Suppose that w contains a white pebble during some interv al [ σ, τ ] ⊇ [ t, t + 1] (and if not, run the dual pebbli ng in Propositio n 3.7 instead). T o obtain P ′ , we change P as follo ws. The pebble placemen t on w at time σ is omitted. At time t , a white pe bble is placed on z . In betwee n times t and t + 1 , w is white pebb led, and then the white pebb le on z is removed at time t + 1 . It is immediate from the deﬁnition of the black pebble game that black pebbling s always proceed in a bottom-u p fashio n in the follo wing sense . Observ ation 3.1 1. Sup pose that Q : u v is a pat h in G and th at P = { P σ , P σ +1 , . . . , P τ } i s a black -only pebbli ng such that the whole path Q is completely fr ee of pebble s at time σ but a pebble is placed on the endpo int v at time τ . Then the starting point u must have been pebbled during the time interval ( σ, τ ) . A simple b ut importa nt lemma, lying at the heart of essentiall y all b lack-white p ebbling lo wer bou nds, is the follo w ing generaliza tion of Observ ation 3.11 to black-white pebbling: In order to pebble the endpoint v of a some path, one needs to pebb le all vert ices on this path at some poi nt prior to or afte r pebbling v . 16 3 Preliminaries Lemma 3.12 ([GT78]). Suppos e that Q : u v is a path in G and that P = { P σ , P σ +1 , . . . , P τ } is a blac k-white pebblin g suc h that the w hole path Q is completely fr ee of pebbles at times σ and τ but the endpo int v is pebb led at some poin t during ( σ, τ ) . T hen the sta rting point u is pebbled dur ing ( σ , τ ) as well. Pr oof. By indu ction over t he length of t he path Q . T he ba se case u = v is tri vial. For the in duction step, let w be the immediate succes sor of u on Q . By the inducti on hypoth esis, w is pebbled and unpebb led again some time duri ng ( σ, τ ) . Then u must be cov ered by a pebble either when the pebb le on w is placed there (if this pebble is black) or when it is remov ed (if it is white). The lemma follo w s. When proving lower bounds on pebblin gs, it often helps to assume that the pebblings under consideratio n do not pe rform any o bviou sly redund ant moves. The follo wing deﬁnition, which forma lizes this notion, is a genera lization of [GL T80] from black -only to black -white pebbli ng. Deﬁnition 3.13 (Fruga l pebbling). Let P be a complete peb bling of a D AG G . T o e very pebb le placement on a verte x v at time σ we associate the pebb ling interval [ σ, τ ) , where τ = τ ( σ, v ) is the ﬁrst time after σ when the pebb le is remov ed from v again (or τ = ∞ , say , if this ne ver happens). If a sink z i ∈ Z ( G ) is pebb led for the ﬁrst time at time σ , then the pebb le on z i is essen tial durin g the pebbli ng interv al [ σ , τ ) . A pe bble on a non-sink v ertex v is ess ential during [ σ , τ ) if eit her an essential black pebble is place d on an immediate succes sor of v during ( σ, τ ) or an essenti al white pebbl e is remo ved from an immediate succes sor of v during ( σ, τ ) . Any other pebb le placement s on any v ertices are non-es sential. The pebbl ing strate gy P is fruga l if all pebble s in P are essentia l at all times. W ithout loss of gene rality, we can ass ume that all pebbli ngs are frugal. Lemma 3.14. F or any complete pebbling P (blac k or blac k-white, visiting or persiste nt) ther e is a frugal pebbli ng P ′ of the same type suc h that time ( P ′ ) ≤ time ( P ) and space ( P ′ ) ≤ space ( P ) . Pr oof sketch . Just delete any non- essential pebble s and ver ify that what remai ns is a le gal pebbling . One min or tech nical snag is that we will need t o assume not only th at comple te pebblin gs are fruga l, but that this also holds for conditional p ebblings (Deﬁnition 3.5 ). T his is no real problem, ho wev er , sinc e we can alw ays assume that the conditional pebblings we are dealing with are subpebbl ings of larg er , uncon ditional pebbli ngs. In fact , an alternat iv e way of deﬁning frugal pebblings (uncondit ional or conditiona l) is to say that a pebbl e placemen t on a non-sink verte x v is essentia l if the pebble stays until either a black pebble is placed on an immediate successor of v or a white pebble is remove d from an immediate successor of v . If a pebbling contains non-es sential mov es, then it is easy to see that such mov es can be eliminat ed to get a shorter pebbling that is still legal. This new pebbling might contain other non-essen tial moves, bu t after applyi ng the proc edure a ﬁni te number of times we obtain a pebbling with only essent ial mov es. Adding the requir ement that each sink should only be pebbled once, we reco ver Deﬁnition 3.13. W e conclude this section by recalling the follo w ing results for pebblings of pyramid graphs. Theor em 3.15 ([Coo74 , Kla85 ]). The black pebbling price of the pyramid Π h of height h is P eb (Π h ) = h + 2 , and ther e is a lin ear -time pebblin g achie ving this bound. The black- white pebbl ing price of Π h is BW -P eb (Π h ) = h/ 2 + O(1) . F or pyramids of odd height the e xact bound BW- P e b (Π 2 h +1 ) = h + 3 h olds, and for even heigh t we have BW -P eb ∅ (Π 2 h ) = h + 2 . W e remark that the exact bound s for black-whi te pebblin g abo ve are not stated or prov en by Klawe in [Kla85], b ut can be read of f from the exposit ion of Klawe’ s proof in (the full-length ve rsion [NH08a] of) [NH08b]. 17 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION 4 Labelled Blac k-White P ebbli ngs and Resolution Sim ulations Let us now prov e the claims made in Section 2.1 about the labelled black-whit e pebble game in Deﬁni- tion 2.2 , namely that this game is just a limited versi on of standa rd black-whi te pebblin g (Lemma 2.3) and that resolutio n refutations of pebbling contrad ictions can simulate labelled pebbli ngs if all labelled pebble subco nﬁgurations ha ve bound ed size (Lemma 2.5). 4.1 Pr oof of Lemm a 2.3 Recall that we want to prov e that if L is a complete labelled pebbli ng of a single -sink D A G G , then we can transfo rm L into a complete standa rd black-whi te pebblin g P L of G with time ( P L ) ≤ 4 3 time ( L ) and space ( P L ) ≤ space ( L ) . The proof of this fact is not hard, and much of the needed material can be ext racted from similar argumen ts in [Nor09]. Since what is actually prov en in [Nor09] is something diffe rent and slight ly weaker , ho wev er , we prov ide a full, explici t proof of Lemma 2.3 belo w . The ﬁrst modi ﬁcation of the peb ble game w hen goin g from Deﬁnition 1.1 to Deﬁnitio n 2.2 is that in t he conte xt of resol ution, a more natural rule for w hite pebble remov al appears to be that a w hite pebble can be remov ed fr om a verte x when a black pebble is placed on that same v erte x. It s eems intu itiv ely fair ly ob vious that this rule chang e shoul d not really af fect the pebble game, and indeed it does not. Lemma 4.1. Let us say that a superp ositione d blac k-white pebbling of G is a peb bling as in Deﬁn ition 1.1, e xcept that a verte x m ay have both a blac k and a w hite peb ble on itself , and that rule (4) is c hange d to: 4’. A white peb ble on v can be r emove d only if ther e is a blac k pebbl e on v . Then for any complete superpositio ned pebblin g S of G ther e is a standa r d complete blac k-white pebbli ng P w ith time ( P ) ≤ time ( S ) and space ( P ) ≤ space ( S ) . Pr oof. Suppose that we are gi ven a superp ositione d pebbling S = { S 0 , . . . , S τ } of G . W e construct a standa rd black-white pebblin g P = { P 0 , . . . , P τ } such that for P t = ( B t , W t ) and S t = ( B ′ t , W ′ t ) it holds that B t ⊇ B ′ t , B t ∪ W t = B ′ t ∪ W ′ t and (as require d by Deﬁnition 1.1) B t ∩ W t = ∅ . In parti cular , this means that space ( P ) = spa ce ( S ) , and that if S is a complete pebbling , then so is P . The construc tion is by forwar d induction ov er S . W e set P 0 = S 0 = ( ∅ , ∅ ) and then make the indu cti ve step by a case analys is ov er the pebblin g move s. 1. If S places a black pebble on v at time t + 1 , the vertices in pr e d ( v ) must be pebble d in S t and thus by indu ction also in P t . If v ∈ W t , we remov e the w hite pebble from v in P . T hen we place a black pebble on v . 2. If S removes a black pebble from v at time t + 1 , by induc tion v is black-pe bbled in P t . W e remo ve the black pebble from v in P , unless v ∈ W ′ t in which case we lea ve the black pebbl e on v . 3. If S places a white pebble on v at time t + 1 , w e place a w hite pebble there in P if v 6∈ B t and otherwis e do nothing . 4. When a white pebble is remov ed from v in S it holds that v ∈ B ′ t . Thus, by indu ction v ∈ B t , so the white pebble has already been remov ed from v in P , or was ne ver place d there. It clearl y holds that time ( P ) ≤ time ( S ) , since P makes at most as many pebb ling mov es as S . The second step in the proof of Lemma 2.3 is to sho w that if we take a complete labe lled pebbl ing L = { L 0 , . . . , L τ } of a D A G G and look at the vertices  Bl ( L t ) , Wh ( L t )  cov ered by black and white 18 4 Labelled Black-White Pebbling s and Resolutio n Simulations pebble s for all t ∈ [ τ ] , we can extra ct a legal complete (superpo sitioned ) black-white pebbling of G in essent ially the same time and space. W e prov e this formally in the next two lemmas. The ﬁrst lemma says that w ithout loss of generality w e can assume that all labelled pebblings are non- r edund ant in the sens e that if a subconﬁgu ration v h V i is deri ved at time t , then this subcon ﬁguration is not just thro w n away b ut is used at some time t ′ > t furth er on in the pebbl ing before being erased. Lemma 4.2. L et L = { L 0 , . . . , L τ } be any complete labelled pebbling of a DA G G . Then we can con- struct a comple te labelled pebbling L ′ = { L ′ 0 , . . . , L ′ τ ′ } of G w ith time ( L ′ ) ≤ time ( L ) and space ( L ′ ) ≤ space ( L ) that has the following pr operty: If v h V i is erase d at time t in L ′ , i.e., v h V i ∈ L ′ t \ L ′ t +1 , then this subco nﬁgur ation has been used in a mer ger or re ver sal m ove immediat ely befo r e being erased, and the subco nﬁgur ation r esulti ng fr om this move is pr esent in L ′ t +1 . Pr oof. This is easy if formally some what tedious, so let us ﬁrst try to visualize the proof. For any labelled pebbli ng L , we can construct a D AG G L encod ing the pebblin g as follows. For e very subconﬁgur ation v h V i appear ing a t time t 1 and staying in the graph until time t 2 when it is erased, we create a verte x ( v h V i , [ t 1 , t 2 ]) . For e ach mer ger u h U i = merge ( v h V i , w h W i ) , we dra w edges from v h V i and w h W i to u h U i . The sou rces in G L are vertice s ( v h pr e d ( v ) i , [ t 1 , t 2 ]) , and by assumptio n there is a sink ( z h∅i , [ t 1 , τ ]) . Note that w ithout loss o f ge nerality we can a ssume tha t we nev er deri ve a subco nﬁguration that i s alre ady prese nt in t he g raph, so all ver tices in G L ha ve inde gree 0 or 2 correspond ing to intro ductions and mer gers, respecti vely . Consider the subgrap h of G L consis ting of all vertic es from which the sink verte x ( z h∅i , [ t 1 , τ ]) is reacha ble. W e construc t L ′ to be the subpebblin g correspond ing exa ctly to the mov es in this subgraph, exc ept that we reor der moves i f neede d so that erasure s are al ways perf ormed as soon as po ssible. S ince the mov es in L ′ are a sub set of the mov es in L , clearly time ( L ′ ) ≤ time ( L ) . Formally , this amounts to the followin g. W e construct the modiﬁed pebb ling L ′ by backward induction ov er L = { L 0 , . . . , L τ } . Let L ′ τ = L τ = { z h∅i} . Our induction hypothesi s is that L ′ t ∗ ⊆ L t ∗ for t ∗ > t . T he backw ard induction step from t + 1 to t is a case analysis ove r the mov es L t L t +1 in L . For simplic ity , we allo w using fractiona l time steps in the interv al [ t, t + 1] in the inducti ve constru ctions belo w . Introductio n L t +1 = L t ∪ { v h pr e d ( v ) i} : Set L ′ t = L ′ t +1 \ { v h pr e d ( v ) i} . Note that w e m ight ha ve L ′ t = L ′ t +1 if v h pr e d ( v ) i 6∈ L ′ t +1 . In any cas e, the indu ction hypo thesis holds for L ′ t . Merger L t +1 = L t ∪ { v h ( V ∪ W ) \ { w }i} : If v h ( V ∪ W ) \ { w }i 6∈ L ′ t +1 , set L ′ t = L ′ t +1 . The induct ion hypothesis triv ially remains true. Otherwise, if the merg ed subconﬁgur ation is present in L ′ t +1 set L ′ t =  L ′ t +1 ∪ { v h V i , w h W i}  \ { v h ( V ∪ W ) \ { w }i} . W e can go from L ′ t to L ′ t +1 in at most three steps via intermediate L-conﬁgurations L ′ t +1 / 3 = L ′ t ∪ { v h ( V ∪ W ) \ { w }i} and L ′ t +2 / 3 = L ′ t +1 ∪ { w h W i} by ﬁrst merg ing v h V i an d w h W i , then poss ibly erasin g v h V i and ﬁnally possib ly erasing w h W i . Erasure L t +1 = L t \ { v h V i} : All erasure mov es in L ′ are tak en care of in conne ction with mer gers, so set L ′ t = L ′ t +1 . W e claim that all move s in L ′ constr ucted in this way are legal. For if u h U i ∈ L ′ t , then u h U i ∈ L t and we kno w that this subconﬁgura tion m ust hav e been deriv ed at some point in time t ∗ ≤ t in L . Thus the backw ard constru ction of L ′ will yield a correct deri vat ion of u h U i . Also note that by construc tion, w hen a subco nﬁguration in L ′ is erase d it has just been used in some mer ger move. Finally , by constructi on L ′ t ⊆ L t , and for the intermedia te fraction al time step L -conﬁgurati ons L ′ t + a/b in the mer ger moves in L ′ we ha ve L ′ t + a/b ⊆ L t +1 . It follo w s that space ( L ′ ) ≤ space ( L ) . For labelle d pebblings as in Lemma 4.2, if we ignore all relatio ns between black and white pebbles in the subcon ﬁgurations and consid er  Bl ( L t ) , Wh ( L t )  for t ∈ [ τ ] , this is a le gal superposi tioned pebbling . 19 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION Lemma 4.3. Suppo se that L is a complete labelled pebb ling of a D A G G . Then ther e is a complet e super- positi oned pebbling S of G suc h that time ( S ) ≤ 4 3 time ( L ) and space ( S ) ≤ spa ce ( L ) . Pr oof. By Lemma 4.2, w ithout loss of generality we can assume that each v h V i is eras ed from L precisely after it has been used in a merger , and that v h V i is erased before w h W i when both subcon ﬁgurations are eliminate d after a move v h ( V ∪ W ) \ { w }i = merge ( v h V i , w h W i ) , so that the white pebble on w is remov ed before the black pebbl e on w . It is clear that we are done if we can construc t a superpo sitioned pebbling S with moves m atchin g the mov es in L exac tly . L et S 0 = ( ∅ , ∅ ) and construct S t +1 induct iv ely by lookin g at the mov es in L t L t +1 . Introductio n L t +1 = L t ∪ { v h pr e d ( v ) i} : P lace white pebbles on pr e d ( v ) and then a black pebble on v in S . Merger L t +1 = L t ∪ { v h ( V ∪ W ) \ { w }i} for v h V i , w h W i ∈ L t : No p ebbling mov es i n S , b ut note that if v h V i is now remo ved, the chang e in pebble s on G in L is exact ly the same as after an applicat ion of rule (4’) on w . Erasure L t +1 = L t \ { v h V i} : T his is the only nontri vial case. In genera l, an erasure move in an labelled pebbli ng can remov e an a rbitrary number of w hite pebbles without a ny black pebbles being e ven close to these white pebble s, and there is no way we can match such a mov e in a superpo sitioned pebblin g. But since w e can assume that L is an labelled pebblin g as descr ibed in L emma 4.2, we kno w that v h V i has just been used in a merge r . Consequ ently , the only pebb le that disappe ars when going from  Bl ( L t ) , Wh ( L t )  to  Bl ( L t +1 ) , Wh ( L t +1 )  is either the black pebble on v , which is alw ays a legal pebble remov al, or some white pebble on w ∈ V which has just been eliminate d in the merger move by a black pebble, and this is a lega l pebble remov al according to rule (4’). W e see that S generate d in this way is a lega l superp ositione d pebbling, if we m odify each introd uction step into | pr e d ( v ) | + 1 pebb le placement moves. Clearly , space ( S ) ≤ space ( L ) . T o see that time ( S ) ≤ 4 3 time ( L ) , cons ider any v erte x v . The way S is cons tructed from L , e ver y time v is pe bbled it is both black- pebble d and white-pebble d, after which the pebbles are remov ed. T his takes 4 move s in S . In L , a single introd uction mov e can place pebbles on many vertices. Howe ver , to remov e the pebbles from v requ ires 3 mov es, namely 1 merge r followed by 2 erasures . T his gi ves the time bo und, and the lemma follo ws. No w Lemma 2.3 follo ws from combining Lemmas 4.1 and 4.3. 4.2 Pr oof of Lemm a 2.5 The assumption in L emma 2.5 is that we are giv en a complete ( m, S ) -bounded labelled pebb ling L = { L 0 , . . . , L τ } of a D A G G . W e want to prov e that for any non constant Boolean function f of arity d , there is a resoluti on refutation π L of Peb G [ f ] in length L ( π L ) = time ( L ) · exp  O( dS )  and total space T otSp ( π L ) = m · e xp  O( dS )  . Let us ﬁrst adopt the notation that for a verte x v , we let v [ f ] denote the set of clauses obtained when substi tuting f ( v 1 , . . . , v d ) for v and exp anding to conjuncti ve normal form, and similarly for v [ f ] . W e ex tend this notation to clauses by deﬁning ( C ∨ D )[ f ] = { C ′ ∨ D ′ | C ′ ∈ C [ f ] , D ′ ∈ D [ f ] } . Note that if a clause C contains K literals , then C [ f ] has at most 2 dK clause s contai ning at most dK literals each. The proof is by induction o ver the pebblin g L . W e maintain the in v ariant that if L t is the set of subcon- ﬁguratio ns at time t , then then π will contai n exactly the clauses C t =  W w ∈ W w ∨ v  [ f ]   v h W i ∈ L t  . Since L is an ( m, S ) -bounded pebblin g, this means that C t will contain at most m 2 d (1+ S ) clause s, each clause of size at most d (1 + S ) . T o simplify the notation in the proof, we w ill implici tly use fract ional time steps in π , making sure that it nev er takes more than exp  O( dS )  time steps to get from C t − 1 to C t . Consider the pebbli ng mov e made in L at time t : 20 5 Separations of Black Pebbling and Bounded Labelled Pebbling 1. If L introd uces v h pr e d ( v ) i , w e downlo ad all the axiom cl auses i n  W w ∈ pr e d ( v ) w ∨ v  [ f ] . By a ssump- tion we ha ve | pr e d ( v ) | ≤ S , so the numbe r of axiom claus es are at most 2 d (1+ S ) . 2. Suppose L merge s v h V i , w h W i ∈ L t − 1 with w ∈ V into v h ( V ∪ W ) \ { w } i . By the inducti ve hypot hesis, we hav e the clause s  W u ∈ V u ∨ v  [ f ] and  W x ∈ W x ∨ w  [ f ] in memory . T ogether , these clause s clearly imply  W u ∈ ( V ∪ W ) \{ w } u ∨ v  [ f ] . Let D be any clau se in the set  W u ∈ ( V ∪ W ) \{ w } u ∨ v  [ f ] . By P roposi tion 3.3, we can deri ve D from the clauses corres ponding to v h V i and w h W i in length exp  O( dS )  and additional total space O  ( dS ) 2  . Doing this in turn for all the 2 d (1+ S ) clause s D ∈  W u ∈ ( V ∪ W ) \{ w } u ∨ v  [ f ] estab lishes the induct ion step. 3. If L er ases a su bconﬁgurat ion v h V i , we just erase all clauses in  W w ∈ pr e d ( v ) w ∨ v  [ f ] from memory . At the end of the pebbling L , w e hav e C τ = { z [ f d ] } for z the sink of G . W e conclud e the refutati on by do wnloadin g all the sink axioms in z [ f d ] and deriv ing the empty clause 0 in length exp (O( d )) and total space O  d 2  . This prov es the lemma. 5 Separations of Blac k P ebbling and Bounded Labelled P ebbling The second component in our proof that resoluti on refut ations of pebbling contradicti ons can be strictl y more ef ﬁcient than black pebblings of the correspond ing graphs is to sho w that there are graph famili es which separate bla ck pebb ling and bounded blac k-white labe lled pebbl ing. In this sectio n, we b rieﬂy r evi ew the graph families exhibit ing the separation s between black and black-wh ite pebbli ng in L emmas 1.3, 1.5, and 1.7, and then pro ve that the black- white pebbling s for these graphs can be carried out in the bounded labelle d pebbling frame work. From this T heorems 1.4, 1.6, and 1.8 immediately follo w by appea ling to Lemma 2.5. W e ﬁ rst attend to Lemma 1.3 and Theorem 1.4 in Secti on 5 .1, a nd th en t ake care o f Lemmas 1.5 and 1.7 and Theorems 1.6 and 1.8 in Section 5.2. 5.1 Bounded P ebblings for Time-Space T rade-offs The trade- offs in Lemma 1.3 are obtained for graphs buil t from permuta tions in the foll owing way . Deﬁnition 5.1 (Pe rmutation graph ([L T82])). Let π denote some permutati on of { 0 , 1 , . . . , n − 1 } . T he permutat ion graph ∆( n, π ) ov er n elements with re spect to π is de ﬁned as fol lows. ∆ ( n , π ) has 2 n v ertices di vided into a lower r ow with vertice s u 0 , u 1 , . . . , u n − 1 and an upper r ow with vert ices w 0 , w 1 , . . . , w n − 1 . For all i = 0 , 1 , . . . , n − 2 , there are dire cted ed ges ( u i , u i +1 ) and ( w i , w i +1 ) , and for all i = 0 , 1 , . . . , n − 1 , there are edges  u i , w π ( i )  from the lo wer ro w to the upper row . Thus, the only source in ∆( n, π ) is u 0 and the only sink is w n − 1 . All vertices in the lower row except the leftmost one ha ve inde gree 1 and all vertices in the upper ro w except the leftmost one ha ve inde gree 2 . Any D AG of f an-in 2 must ha ve pe bbling pr ice at least 3 . It i s no t too hard t o s ee that the g raphs ∆( n, π ) ha ve pebbl ings in this minimal space: keep ing one pebble on v ertex w i of the upper ro w , mov e two pe bbles consec uti vely on the lower row until u π − 1 ( i +1) is rea ched, and then pebbl e w i +1 . Generalizing this p ebbling strate gy leads to the followin g upper bound on the time-sp ace trade-o ff for any permutat ion graph. 5 5 All results revie wed below are from [L T82, S ection 2]. Our statements of the results dif fer slightly in the constants, t hough, since there are some (minor) t echnical dif ferences in the deﬁnitions in [L T82] as compared to the present paper . Proofs of the lemmas and theorems as stated here can be found in [Nor10a]. 21 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 Figure 6: Bit rev ersal g raph ∆(8 , rev) on 8 elements. Lemma 5.2 ([L T82]). Let ∆( n, π ) be the permutatio n graph ove r n elements for any permut ation π . Then the blac k pebbl ing price of ∆( n, π ) is P eb (∆( n, π )) = 3 , and for any space s , 3 ≤ s ≤ n , ther e is a blac k pebbli ng stra te gy P for ∆( n, π ) with space ( P ) ≤ s and time ( P ) ≤ 2 n 2 s − 2 + 2 n . T o pro ve lower bounds for permutatio n graphs, Lengauer and T arjan focus on permutatio ns deﬁned in terms of re vers ing the bina ry repre sentation of the inte gers { 0 , 1 , . . . , n − 1 } when n is an e ven power o f 2 . Deﬁnition 5.3 (Bit re versa l graph ([L T82])). The m -bit r ever sal of i , 0 ≤ i ≤ 2 m − 1 , is the integer rev m ( i ) obtained by writing the m -bit binary represen tation of i in rev erse order . The bit r ever sal grap h ∆(2 m , rev m ) is the permutation graph ov er n = 2 m with respe ct to rev m . W e will denote the bit rev ersal graph by ∆( n, rev ) for simplic ity , implicitl y assuming that n = 2 m . An exa mple of a bit re ve rsal graph can be found in Figure 6. For bit re versal graphs, the trade-of f in Lemma 5.2 for black pebbling is asymptotic ally tight. Theor em 5.4 ([L T82]). Suppose that P is any complet e blac k pebbl ing of the bit r eve rsal graph ∆( n, rev ) ove r n = 2 m elements such tha t spa ce ( P ) = s for s ≥ 3 . Then time ( P ) ≥ n 2 8 s . Note, in p articular , that if we want to bla ck-pebbl e ∆ ( n , rev ) in line ar time, then linear spac e is needed . The proof of Theorem 5.4 relies on the fac t that a black pebbling must alway s proceed through a graph in topolo gical order . For a black- white pebbling this is no longer true, since pebb les may be placed any where at any time. Adjusting the ar gument used in the proof of Theorem 5.4 accordin gly , one instead gets the follo w ing, weaker lo wer bound. Theor em 5.5 ([L T82]). Let P be any complete blac k-white pebbling of ∆( n, rev ) with space ( P ) = s for s ≥ 3 . T hen time ( P ) ≥ n 2 18 s 2 + 2 n . When ﬁrst looking at the proof of Theorem 5.5, it might seem that the bound should not really ha ve to be weaker than in Theorem 5.4 but that this could plausibly be just a consequenc e of the analysis being harder to carry out in the black- white pebb ling case. Somewhat s urprising ly , howe ver , Lengaue r and T arjan pro ve that Theorem 5.5 is in fa ct tight. That is, one can do (much) bette r using white pebbles in addition to the black ones. In particular , there is a linear -time black-white pebbl ing strategy for ∆ ( n, rev ) using only order of √ n pebbl es. Moreo ver , it is possible to transfo rm the pebbling strate gy in [L T82] into a bounded labelle d pebblin g. W e co nclude our discussio n of permut ation graphs by statin g and p rovin g this as a formal theore m. Theor em 5 .6. Let ∆( n, rev) be the b it r ever sal graph o ver n = 2 m elements . Then for a ny space par ameter s ≥ 3 ther e is a complete (2 + 2 s/ 3 , 2) -bou nded labelled pebbl ing L of ∆( n, rev) with space ( L ) ≤ s and time ( L ) ≤ 288 n 2 s 2 + 22 n . 22 5 Separations of Black Pebbling and Bounded Labelled Pebbling Theorem 5.6 is an easy corol lary of the next lemma. W e establi sh the lemma ﬁrst and then exp lain ho w it implies the theore m. W e also remark that our proof follo w s [L T82] fairly closely . Thus, our contrib ution consis ts in adaptin g the arg ument to the bo unded labelled pebb ling frame work. Lemma 5.7. F or all s , 3 ≤ s ≤ 3 √ n , ther e is a complete (2 + 2 s/ 3 , 2) -bounde d labell ed pebbling L of ∆( n, rev ) with space ( L ) ≤ s and time ( L ) ≤ 288 n 2 s 2 + 6 n . Pr oof of Lemma 5.7. Write m = log n and let r be the non-ne gati ve inte ger such that 3 · 2 r ≤ s < 3 · 2 r +1 . Div ide the upper ro w of ∆ ( n, rev ) into 2 r interv als I j =  w j · 2 m − r + k   k = 0 , 1 , . . . , 2 m − r − 1  (5.1) of size 2 m − r for j = 0 , . . . , 2 r − 1 and then subdi vide each inter val into 2 m − 2 r chu nks by deﬁnin g C i j =  w j · 2 m − r + i · 2 r + k   k = 0 , 1 , . . . , 2 r − 1  (5.2) for i = 0 , . . . , 2 m − 2 r − 1 . (Note th at 2 m − 2 r ≥ 1 sin ce s ≤ 3 √ n by assumption.) F igure 7 exempli ﬁes these deﬁnitio ns on the 32 -eleme nt bit rev ersal D A G with 2 2 interv als and 2 chunk s per interv al. The pebbling st rategy wil l p roceed in 2 m − 2 r phase s correspo nding to t he 2 m − 2 r chunk s in each inter val , and in 2 r sta ges within each phase correspon ding to the dif ferent interv als. All the phases in the pebbling are completely analogo us exc ept for some minor tweaks in the ﬁrst and ﬁnal phases, which we refer to as the 0 th and (2 m − 2 r − 1) st phases, respec tiv ely . T o help the reader parse the nota tion, we note that in what follo w s superscri pts i will corres pond to phases/c hunks and subscripts j to stage s/interv als. W e reserv e 2 r indepe ndent black pebble s for the lower row , 2 r depen dent black pebbles for the “current chunks” in the upper row , and 2 r − 1 support ing white pebbl es for theses depende nt black pebble s. These white pebbles will be placed on t he rightmost vertices in I 0 , I 1 , . . . , I 2 r − 2 . By t he way we chose r , this l eav es one auxiliary pebble to help with adv ancing the other pebbles. W e start the 0 th stage in the 0 th phase by doing what is in essence a complete black- only pebbling of the lo wer ro w , leav ing 2 r indepe ndent black pebb les on U 0 0 = { u rev m ( k ) h∅i | k = 0 , 1 , . . . , 2 r − 1 } . (5.3) More formally , this is done as follo ws. Introduce the subco nﬁgurations u 0 h∅i and u 1 h u 0 i , and then merge them to get u 1 h∅i . Next, introduc e u 2 h u 1 i and mer ge w ith u 1 h∅i to get u 2 h∅i . W e continue in this way along the lo wer row , erasing all subconﬁgurat ions u i h u i − 1 i as we go, as well as all subconﬁgurati ons u i h∅i not found in U 0 0 . Once we hav e the indepen dent black pebbles in U 0 0 , we use them to “sweep” a black pebble past the 0 th chunk of I 0 in the up per ro w , leavi ng it on th e rightmost verte x w 2 r − 1 . In formal notation, we introduce w 0 h u 0 i , mer ge with u 0 h∅i to get w 0 h∅i , and then erase w 0 h u 0 i . Next, we introduc e w 1  w 0 , u rev m (1)  and mer ge ﬁrst w ith w 0 h∅i and then with u rev m (1) h∅i , resultin g in w 1 h∅i . The depend ent black pebbles on w 1 are then erased. N ext, we intr oduce w 2  w 1 , u rev m (2)  and merge w 1 h∅i and u rev m (2) h∅i to get w 2 h∅i , after w hich the dependen t black pebbles on w 2 are erased. Moving right in this fash ion, we ﬁnally deri ve w 2 r − 1 h∅i , noting that all the indepe ndent black pebbles u rev m ( i ) h∅i that we need for this are pres ent in U 0 0 . This concl udes the 0 th stage of our labelle d pebbling . In the ne xt stage, we mov e all in depende nt black pebbles in U 0 0 on the lo wer row e xactly one step to th e right to t he v ertices u k for k = 1 , r ev m (1) + 1 , rev m (2) + 1 , . . . , rev m (2 r − 1) + 1 . For k = 1 , this is done by introducin g u 1 h u 0 i , mergin g with u 0 h∅i to get u 1 h∅i , and then erasin g u 1 h u 0 i and u 0 h∅i . T he general case is of course completely analogo us. Using the fact that 1 = rev m (rev r (1) · 2 m − r ) , we see that we no w ha ve indep endent black pebble s on U 0 1 =  u rev m (rev r (1) · 2 m − r + k ) h∅i   k = 0 , 1 , . . . , 2 r − 1  , (5.4) 23 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION I 0 I 1 I 2 I 3 00000 00000 00001 00001 00010 00010 00011 00011 00100 00100 00101 00101 00110 00110 00111 00111 01000 01000 01001 01001 01010 01010 01011 01011 01100 01100 01101 01101 01110 01110 01111 01111 10000 10000 10001 10001 10010 10010 10011 10011 10100 10100 10101 10101 10110 10110 10111 10111 11000 11000 11001 11001 11010 11010 11011 11011 11100 11100 11101 11101 11110 11110 11111 11111 Figure 7: Inter vals I j fo r r = 2 in ∆(32 , rev ) and 0 th chunks in I 0 and I rev r (1) = I 2 with in verse images. which by (5.2) is the se t of all predecesso rs in the lo wer r ow of t he 0 th ch unk C 0 rev r (1) of the i nterv al I rev r (1) . This cruci al fac t is illust rated in Figure 7. Intuiti vely , w hat we want to do no w is to place a white pebble on the rightmost verte x of the in- terv al I rev r (1) − 1 and use this white pebble plus the lo wer -row black pebble s on U 0 1 to sweep a black pebble all the way to the rightmost ver tex in the 0 th chunk of I rev r (1) . T o acco mplish this, ﬁrst intro- duce w rev r (1) · 2 m − r  w rev r (1) · 2 m − r − 1 , u rev m (rev r (1) · 2 m − r )  and mer ge this sub conﬁguratio n with the inde pen- dent black pebble u rev m (rev r (1) · 2 m − r ) h∅i , which is present in U 0 1 , to deri ve w rev r (1) · 2 m − r  w rev r (1) · 2 m − r − 1  . Then introduc e w rev r (1) · 2 m − r +1  w rev r (1) · 2 m − r , u rev m (rev r (1) · 2 m − r +1)  and merge to get t he s ubconﬁgura tion w rev r (1) · 2 m − r +1  w rev r (1) · 2 m − r − 1  . Continuing in this way , e rasing dependent black pebbles in the upper ro w as soon as the y are no longer nee ded, we adv ance a black pe bble al ong a ll the v ertices o f the 0 th ch unk of the interv al I rev r (1) , ﬁ nally arri ving at the pebble subcon ﬁguration w rev r (1) · 2 m − r +2 r − 1  w rev r (1) · 2 m − r − 1  . T his conclu des stage 1 of phase 0 . The res t of the sta ges of phase 0 are completely analogo us. In the j th stage, we can mo ve the lower -row pebble s from U 0 j − 1 to U 0 j where this notatio n is general ized to mean U 0 j =  u rev m (rev r ( j ) · 2 m − r + k ) h∅i   k = 0 , 1 , . . . , 2 r − 1  (5.5) for all j ≤ 2 r − 1 , and then place black pebbles on the rightmost verte x in e ver y chunk C 0 rev r ( j ) with the help of a white pebble on the righ tmost ver tex in I rev r ( j ) − 1 , i.e., , deriv e pebble sub conﬁguratio ns w rev r ( j ) · 2 m − r +2 r − 1  w rev r ( j ) · 2 m − r − 1  . At the end of the ﬁnal stage of phase 0 , we thus hav e black pebble s on the rightmo st vertic es of all 0 th chunks and white pebbles on the righ tmost vert ices of I 0 , I 1 , . . . , I 2 r − 2 . Later phases will mov e the black pebbles to the right, chunk by chunk, while leav ing the w hite pebble s in place. W e observe that during phase 0 , w e m ade at most n intro duction moves and n merger mov es on the lo wer ro w to get t he pe bbles into “starting po sition” U 0 0 , and the n ex actly 2 r introd uctions and m er gers mor e on the lo wer row in ea ch of the other 2 r − 1 stages. Induct iv ely , suppose at the begin ning of phase i that there are depe ndent black pebb les on the rightmost ver tices in all ( i − 1) st chunks, i.e., subcon ﬁgurations w rev r ( j ) · 2 m − r + i · 2 r − 1  w rev r ( j ) · 2 m − r − 1  for all j > 0 and w i · 2 r − 1  ∅  for j = 0 . Let us e xtend the lower -ro w pe bble co nﬁguration n otation ab ove to full generality and deﬁne U i j =  u rev m (rev r ( j ) · 2 m − r + i · 2 r + k ) h∅i   k = 0 , 1 , . . . , 2 r − 1  =  v h∅i   v ∈ rev − 1 m  C i rev r ( j )  , (5.6) 24 5 Separations of Black Pebbling and Bounded Labelled Pebbling where the second equality is easily veriﬁed from (5.2) . In stage 0 of phase i , we rearrange the lo wer-ro w black pebbles to obtain the c onﬁguratio n in U i 0 . Since there a re already 2 r indepe ndent bla ck pe bbles present some where on the lower row , th is can be achiev ed with at most n − 2 r introd uctions and merge rs (esse ntially by moving the black pebbles to the closest new position to the right—we refer to [L T82] for the details). This allo ws us to adv ance the independe nt black pebble in I 0 on the upper row from the rightmos t verte x in chunk i − 1 to the rightmos t vertex in chunk i . Movin g the indepen dent black pebbles in U i 0 one step to the righ t in each follo wing stage to U i 1 , U i 2 , et ceter a, we can sweep depende nt black pebbles across the i th chunk s of the other interv als I j in the order I rev r (1) , I rev r (2) , . . . , I rev r (2 r − 1) = I 2 r − 1 . All in all, we make at most ( n − 2 r ) + (2 r − 1) · 2 r introd uctions and merger mov es on the vert ices in the lower row during phase i for i ≥ 1 . In the ﬁ nal (2 m − 2 r − 1) st phase, we note that there are support ing white pebb les on the rightmos t ver tex of the chunk in ev ery interv al except I 2 r − 1 (where the rightmost vert ex is the sink). Therefore, in e very stage exce pt the ﬁnal one, when we make an introduct ion move on a rightmost ve rtex, we mer ge the introdu ced subconﬁgu ration with the subconﬁgur ations on its two prede cessors of this vert ex to remov e the white pebble. In the very ﬁnal stage, w e obtain an independ ent black pebble on w n − 1 . Removing all other pebbles from the DA G, whi ch a re al l ind ependent bla ck pebb les, we hav e ob tained a complete la belled pebbli ng of ∆( n, rev ) . The s pace of t his peb bling is 3 · 2 r ≤ s by constru ction. All subc onﬁguration s v h W i hav e white support size | W | ≤ 2 , a nd there are always a t mos t 2 · 2 r ≤ 2 s/ 3 “static” s ubconﬁgura tions plus 2 auxiliary one s. As to the time bound, it is easy to verify that we make an introdu ction for each upper ro w vertex exa ctly once, and 2 m er gers are needed to eliminate the white pebb les in the support of the intro duced subco nﬁguration . The number of introdu ctions and merge rs in the lo wer row is at most 2 n + (2 r − 1) · 2 r +1 during phase 0 and at most 2( n − 2 r ) + (2 r − 1) · 2 r +1 for each of the other 2 m − 2 r − 1 phases , and summing up we get a total of at most 2 m − 2 r  (2 n − 2 r +1 ) + (2 r − 1) · 2 r +1  + 2 r +1 + 3 n < 2 m − 2 r  2 n + 2 2 r +1  + 3 n < n ( s/ 6) 2  2 n + 2( s/ 3) 2  + 3 n ≤ 144 n 2 s 2 + 3 n (5.7) introd uction and merge r m ov es in total, where we used that 2 m − 2 r ≥ 1 , 2 r ≤ s/ 3 < 2 r +1 , and s ≤ 3 √ n . Multiply ing by 2 to take the remov al m ov es into accou nt gi ves the time boun d stated in the lemma. Pr oof of Theor em 5.6. For s ≤ 3 √ n this was pro ven in Lemma 5.7. T o get the statement for s > 3 √ n , use the same pebbl ing strateg y as in the proo f of Lemma 5.7 but choo se r so that √ n/ 2 < 2 r ≤ √ n . T hen the number of chunk s 2 m − 2 r is at most 2 , and the time boun d deri ved from (5.7) reduces to 22 n . T o obtain the grap hs G n of size Θ( n ) in Lemma 1.3, we set m = ⌈ log 2 n ⌉ and let G n = ∆(2 m , rev m ) . As noted at the beginn ing of this section , Theorem 1.4 no w follo ws if we combine Lemma 2.5 with Lemma 5.7. 5.2 Bounded P ebblings for Absolute Separations of P ebbling Space T o o btain resu lts for re solution match ing the p ebbling sepa rations of Lemma 1.5 by [W il88 ] an d Lemma 1.7 by [KS91 ], it is sufﬁcie nt to consider a m ore general graph family studied in the latter paper . T o describe ho w this graph family is cons tructed we ﬁrst need an auxilia ry deﬁnition. Deﬁnition 5.8 ( m -line and ( n, m ) -spiral mesh). An m -line is a D AG with verte x set v 1 , v 2 , . . . , v m and edge set { ( v i , v i +1 ) | i = 1 , 2 , . . . , m − 1 } . 25 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION An ( n, m ) -spir al mesh is a D A G on v ertices { v i,j | i ∈ [ n ] , j ∈ [ m ] } with ed ges ( v i,j , v i,j +1 ) fo r i ∈ [ n ] and j ∈ [ m − 1] , ( v i,j , v i +1 ,j ) for i ∈ [ n − 1] and j ∈ [ m ] , and ( v i,m , v i +1 , 1 ) for i ∈ [ n − 1] . T he i th column of the ( n, m ) -spiral mesh consists of the vertices v i,j for j ∈ [ m ] . W e n ow present the three-pa rameter graph f amily Λ( p, q , k ) in [KS91]. The co nstructio n is by induction ov er q . Deﬁnition 5.9 ( Λ( p, 0 , k ) -graph). The graph Λ( p, 0 , k ) is a (1 , p ) -mesh, that is, a p -line, the ﬁrst r ow f 1 , f 2 , . . . , f p and last r ow l 1 , l 2 , . . . , l p of which are both deﬁned to be the verti ces of the p -lin e. For q > 0 , the grap h Λ( p, q , k ) consi sts of a number of identic al bu ilding bloc ks N ( p, q , k ) , which all contain a copy each of Λ( p, q − 1 , k ) . In the recursi ve deﬁnition s below , we will be some what slopp y with the indices in order not to clutter the notation . In particular , if we wanted to be formally correct, all subgra phs and verti ces belo w should be labelled by their “lev el of recurs ion” q within the constr uction, as well as by a number indicati ng which of the identic al copies on recursio n lev el q the vertex resid es in, bu t we belie ve that adding these e xtra indices would lead to more confus ion than clarity . The N ( p, q , k ) -block graph constru ction, deﬁned next, is illustrated in Figure 8. W e remark that this graph has been slight ly modiﬁed as compared to [KS91]. 6 Deﬁnition 5.10 ( N ( p, q , k ) -block [KS91]). Suppose that Λ( p, q − 1 , k ) has bee n deﬁned. The bloc k graph N ( p, q , k ) , where k ≤ p , consists of the follo wing compon ents: • a copy of Λ( p, q − 1 , k ) with ﬁrst ro w f 1 , f 2 , . . . , f m and last ro w l 1 , l 2 , . . . , l m , • a  ( p + 1) 2 , p  -spiral mesh B on ve rtices b i,j , i ∈  ( p + 1) 2  , j ∈ [ p ] , • a  ( p + 1) 3 , p  -spiral mesh A on vert ices a i,j , i ∈  ( p + 1) 3  , j ∈ [ p ] , • k copies R 1 , . . . , R k of a ( p + 1) -line , with the i th copy ha ving vertic es r i,j for j ∈ [ p + 1] . For ease of no tation, in what follows we will write n b = ( p + 1) 2 and n a = ( p + 1) 3 for th e number of rows in B and A . The subgraph components are connected by edges as follo ws (where we use the notation  u ; v  for the edge from u to v for clarity): •  b n b ,j ; f j  for j ∈ [ p ] , •  b n b ,j ; r i,p +2 − j  for i ∈ [ k ] and j ∈ [ p ] , •  l j ; a 1 ,j  for j ∈ [ p ] , •  l ⌊ ip/k ⌋ ; r i, 1  for i ∈ [ k ] , and •  r i,p +1 ; a 1 ,j  for all i ∈ [ k ] and all j such that ( i − 1) p/k < j ≤ ip/k . The i th column of N ( p, q , k ) cons ists of the i th colu mns of B , Λ( p, q − 1 , k ) , and A . W e glue the N ( p, q , k ) -blocks together to form the graph Λ( p, q , k ) as foll ows. 6 Again, proofs of the results as stated here can be found in [Nor10a]. 26 5 Separations of Black Pebbling and Bounded Labelled Pebbling B Λ( p, q − 1 , k ) A R 1 R 2 R k b 1 , 1 b 2 , 1 b n b , 1 a 1 , 1 a 2 , 1 a n a , 1 f 1 l 1 b 1 , 2 b 2 , 2 b n b , 2 a 1 , 2 a 2 , 2 a n a , 2 f 2 l 2 b 1 , 3 b 2 , 3 b n b , 3 a 1 , 3 a 2 , 3 a n a , 3 f 3 l 3 b 1 , 4 b 2 , 4 b n b , 4 a 1 , 4 a 2 , 4 a n a , 4 f 4 l 4 b 1 ,p b 2 ,p b n b ,p a 1 ,p a 2 ,p a n a ,p f p l p b 1 ,p − 1 b 2 ,p − 1 b n b ,p − 1 a 1 ,p − 1 a 2 ,p − 1 a n a ,p − 1 f p − 1 l p − 1 r 1 , 1 r 1 , 2 r 1 , 3 r 1 ,p +1 r 1 ,p r 1 ,p − 1 r 1 ,p − 2 r 2 , 1 r 2 , 2 r 2 , 3 r 2 ,p +1 r 2 ,p r 2 ,p − 1 r 2 ,p − 2 r k, 1 r k, 2 r k, 3 r k,p +1 r k,p r k,p − 1 r k,p − 2 Figure 8: Building bl ock N ( p, q , k ) in graph separating b lack and b lac k-white pebbl ing ( here k = p/ 2 ). 27 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION Deﬁnition 5.11 ( Λ( p, q , k ) -graph [KS91]). For q ≤ p an d k ≤ p , the graph Λ( p, q , k ) consis ts of ⌈ p/k ⌉ + 1 copies of the block graph N ( p, q , k ) , w hich we denote N (1) ( p, q , k ) , N (2) ( p, q , k ) , . . . , N ( ⌈ p/k ⌉ +1) ( p, q , k ) . The edg es between the blo cks are  a ( i ) n a ,j ; b ( i +1) 1 ,j  for i = 1 , . . . , ⌈ p/k ⌉ and j = 1 , . . . , p , i.e., the last v ertex in ev ery column in the i th N -block is conn ected to the ﬁrst vert ex in the same colu mn in the ( i + 1) st N -block . W e deﬁne the ﬁrst row f 1 , f 2 , . . . , f m of Λ( p, q , k ) to consist of the ﬁrst ro w b (1) 1 , 1 , b (1) 1 , 2 , . . . , b (1) 1 ,p , of the ﬁrst N -block and the last ro w l 1 , l 2 , . . . , l m , to consist of the last row a ( ⌈ p/k ⌉ +1) n a , 1 , a ( ⌈ p/k ⌉ +1) n a , 2 , . . . , a ( ⌈ p/k ⌉ +1) n a ,p of the last N -block. The i th column of Λ( p, q , k ) is deﬁned to be the union of the i th columns of all the N -block s. Let us no w ﬁrst state the properti es that we need from th e Λ( p, q , k ) -graphs , then show how Lemmas 1.5 and 1.7 and Theorems 1.6 and 1.8 follo w from these prop erties, and ﬁnally gi ve the proof that there are ef ﬁcient bounde d labelled pebblings of the graphs. Pro position 5.12 ([KS91]). The graphs Λ( p, q , k ) , have size O  p oly( p )( p/k ) q  , maximal verte x i nde gr ee 3 , and a uniqu e sink. Theor em 5.13 ([KS91]). A ny complet e blac k pebbling of Λ( p, q , k ) r equir es at least pq pebb les. Theor em 5.14. E very gra ph Λ( p, q , k ) has a complete ( p + kq + 2 , 3) -boun ded labelled pebbling. If w e set k = p log log p/ log p and q = log p/ log log p in Deﬁnition 5.11, it follo ws from Propo- sition 5.12 and Theorem 5.13 that we obtain graphs of size polynomia l in p with black pebblin g price Ω( p log p/ log log p ) , as claimed in Lemma 1.5. Since these graph s ha ve (O( p ) , O(1)) -bounde d labell ed pebbli ngs by Theorem 5.14, we can appeal to Lemma 2.5 to deduc e that resol ution refutat ions of pebbli ng contra dictions ov er these graphs can m atch the black-white peb bling spac e bounds, which prov es Theo- rem 1.6. If we instead choose k = 1 and q = p in Deﬁnition 5.11, we get graphs of size exp(Θ( p log p )) that ha ve black pebb ling price Ω  p 2  b ut admit (O( p ) , O(1)) -bounded labell ed pebblings . This gi ves us Lemma 1.7 and Theorem 1.8. Hence, all that remains is to establish Theorem 5.14, and we conc lude this section by doing so. Again, we point out that the pebb ling strate gy presented below follo ws the one in [KS91 ] closely , and that our contri but ion is thus not in design ing a completely ne w pebbling strate gy , b ut in taking an existing strateg y and turnin g it into a bounde d labelled pebblin g. Before presentin g the formal proof, let us sk etch the main idea. Observe that if there w ere no R -graphs in Λ( p, q , k ) but only the vertice s in the p columns, then it woul d be straight forward to do a complete bottom-u p black-o nly pebbling with just p + 1 pebbl es. Howe ver , this strateg y is impossible to implement in the blac k pebbl e game. V ery brie ﬂy , the reason for this is that an y black pebbli ng has to pebble the grap h in topolo gical orde r , but si nce the pred ecessors of the ver tices in the R -graph s hav e their order re versed—with the source of R havi ng its predecesso r in Λ( p, q − 1 , k ) , whereas the successor vertic es hav e predecessors in the precedi ng subgraph B —this constantly throws the black pebbling off-b alance. Using the power of white pebbles, ho wev er , w e can av oid this problem and place black pebbles on the sinks of all graphs R i , i ∈ [ k ] , at all lev els of recurs ion in the graph constru ction, and then do the black bottom-up pebbling of the ver tices in the co lumn-part of the grap h. The formal details follo w . Pr oof of Theor em 5.14. The labelled pebb ling strateg y is cons tructed by induction ov er q . The base case is tri vial since Λ ( p, 0 , k ) is just a p -lin e. For the the s ake of o ur in duction hypothesis , let us do some e xtra work and note that we can in fact e ven ﬁ ll the whole p -line w ith independent b lack pebbles and s till s tay within our space bounds. That i s, i f l 1 , . . . , l p are the v ertice s of Λ( p, 0 , k ) , we can int roduce l 1 h∅i and l 2 h l 1 i and mer ge them to get l 2 h∅i , after which l 2 h l 1 i is erased, then introduce l 3 h l 2 i and merge with l 2 h∅i to obtain l 3 h∅i , 28 5 Separations of Black Pebbling and Bounded Labelled Pebbling after which l 3 h l 2 i is erased, et cetera, until we hav e the whole ro w { l j h∅i | j ∈ [ p ] } of independ ent black pebble s. Induct iv ely , suppose that we hav e const ructed for Λ( p, q − 1 , k ) a pebbling L startin g with indepen dent black pebb les { f j h∅i | j ∈ [ p ] } on the ﬁrst row , endin g with independe nt black pebbles { l j h∅i | j ∈ [ p ] } on the last row , and ne ver using more than p + k ( q − 1) + 2 subc onﬁguratio ns v h W i at any time, all with bound ed white supp ort size | W | ≤ 3 . It is s ufﬁcie nt to c onstruct from L a labell ed pebbling L ′ for th e block graph N ( p, q , k ) m ovi ng independ ent black pebbles from the ﬁrst ro w of B to the last ro w of A using no more than p + k q + 2 subco nﬁgurations w ith bounded sup port size. S uch a pebbling is then easily e xtend ed to pebblin g for all of Λ( p, q , k ) by pebbling the bloc ks one by one in a bottom-up fashio n. (This is so since we can easily shift independ ent black pebble s from the last ro w of an N -block to the ﬁrst ro w of the next N -block using the same kind of labelle d pebb ling mov es that will be discus sed more in detail belo w .) Thus, suppose that we hav e independen t black pebb les { b 1 ,j h∅i | j ∈ [ p ] } on all v ertices in the ﬁrst ro w of B . W e move these pebbles up one row as follo ws. First introduce b 2 , 1  b 1 , 1 , b 1 ,p  and merge with b 1 , 1 h∅i and b 1 ,p h∅i to get b 2 , 1 h∅i , erasing b 1 , 1 h∅i and the dependen t black pebbles on b 2 , 1 . Next, introduce b 2 , 2  b 2 , 1 , b 1 , 2  and merge with b 1 , 2 h∅i and the newly deri ved subconﬁgu ration b 2 , 1 h∅i to get b 2 , 2 h∅i , after which the dependent black pebbles on b 2 , 2 are erased, as well as b 1 , 2 h∅i . Continu ing in this way , erasing pebble s ubconﬁgura tions as s oon as they are no long er needed and using only 2 auxiliary sub conﬁguratio ns, we can shift the whole ro w , and we keep on shifting the pebbles ro w by ro w , from left to right for each ro w , until the last ro w of B has all ver tices cove red by indep endent black pebb les { b n b ,j h∅i | j ∈ [ p ] } . Next, we want to place black pebbles on the sinks of all the R i -subgr aphs. Fix some i and consider R i . Introd uce r i, 2  r i, 1 , b n b ,p  and merg e with b n b ,p h∅i to obtain r i, 2  r i, 1  , erasin g r i, 2  r i, 1 , b n b ,p  . Con- tinue by introduc ing r i, 3  r i, 2 , b n b ,p − 1  and mergin g it with b n b ,p − 1 h∅i to obtain r i, 3  r i, 2  , and then mer ge this subconﬁgur ation with r i, 2  r i, 1  to deriv e r i, 3  r i, 1  , where the subconﬁgur ations r i, 3  r i, 2 , b n b ,p − 1  , r i, 3  r i, 2  , and r i, 2  r i, 1  are erased as soon as the y are no longer needed. W orking our way up R i in this fashion, we ﬁnally deri ve r i,p +1  r i, 1  . Note that w e use here that w e hav e all the indepe ndent black pebble s b n b ,j h∅i , j ∈ [ p ] , a va ilable. W e repeat thes e pebbling moves for all the R i -graph s to obtain { r i,p +1  r i, 1  | i ∈ [ p + 1] } . For this part of the pebbling we again use 2 auxiliary subconﬁgu rations, and we end up with a total of k subconﬁgu rations on all the subgrap hs R i , i ∈ [ k ] . No w , sh ift the independen t black pebbles { b n b ,j h∅i | j ∈ [ p ] } fr om the last ro w of B to { f j h∅i | j ∈ [ p ] } on the ﬁrst row of Λ( p, q − 1 , k ) (by the same kind of mov es that ha ve been describ ed in detail abo ve), and then appeal to the induc tion hypo thesis to obtain a pebbling moving these black pebbles further upward to { l j h∅i | j ∈ [ p ] } on the last row of Λ( p, q − 1 , k ) . By the induct ion hypothesis , such a pebbling uses at most p + k ( q − 1) + 2 pebble subconﬁgurat ions. W e note that adding the k pebble subcon ﬁgurations on the R i -subgr aphs , the total number of subconﬁgura tions exactl y meets the upper bound we are aiming for in the induc tiv e step. T o ﬁnish the pebblin g of N ( p, q , k ) , we ﬁ rst want to eliminate all the white pebbles on r i, 1 , i ∈ [ k ] , which is pos sible since there are (indepe ndent) black p ebbles on the p redecess ors of th ese v ertices in the l ast ro w of Λ( p, q − 1 , k ) . T hus, for all i ∈ [ k ] in turn, introduce r i, 1  l ⌊ ip/k ⌋  and merge r i,p +1  r i, 1  with the introd uced subcon ﬁguration as well as with l ⌊ ip/k ⌋ h∅i to deri ve r i,p +1 h∅i , where we erase r i, 1  l ⌊ ip/k ⌋  and r i,p +1  r i, 1  and any intermediate subcon ﬁgurations as soon as they are no longer needed. Next, we shift the black pebbles { l j h∅i | j ∈ [ p ] } from the last row of Λ( p, q − 1 , k ) to { a 1 ,j h∅i | j ∈ [ p ] } on the ﬁrst row of A . T his is done in the same way as pre vious “shif ting” mov es, and we use that in addition to the pebb les on the last row of Λ( p, q − 1 , k ) we also hav e independ ent black pebbles on the sinks of all R i -subgr aphs . In this part of the pebbling w e will need subconﬁgurati ons w ith white support size 3 , since that is the inde gree of the verti ces in the ﬁrst row of A . When we are done shifting, we erase the pebbles r i,p +1 h∅i from the sinks of the R i -subgr aphs . Finally , we move all the black pebbles in A row by row upward, using 2 auxilia ry subcon ﬁgurations, until the last ro w of A has all v ertices cov ered by independe nt black pebbles. This concl udes the induc tiv e step, and the theorem follo ws. 29 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION 6 Carlson-Sa v age Graphs and Strong Dual T r ade-offs In this sec tion, we p resent a full proof of Theorem 1.9 and s ho w ho w the Carlson- Sav age gra phs can be us ed to ob tain graphs with s trong dual pebbli ng trade-of fs where the u pper bounds are in terms of blac k pebbling and the lo wer bounds are in terms of black-white pebbling . W e ﬁrst list the statements that we want to prove in order to establish T heorem 1.9 in L emmas 6.1, 6.2, and 6.3 below . Note that the lemmas are stated for the graph family Γ ( c, r ) in Deﬁnition 2.6. It is straigh tforward to transl ate the lemmas to what is needed for Theorem 1.9 by using the single-sink versio n of Γ( c, r ) in Deﬁnition 3.8 and appea ling to Observ ation 3.9. Then , we sho w ho w these lemmas yield pebbli ng time-spac e trade-o ffs . Finally , we pr ovide the formal proofs of the lemmas. Let us start by recall ing the size and pebbli ng price bounds. Lemma 6.1. The graph s Γ ( c, r ) ar e of size | V (Γ( c , r )) | = Θ  cr 3 + c 3 r 2  , and have blac k-white pebbling price BW -P eb ∅  Γ( c, r )  = r + 2 and blac k pebb ling price P eb ∅  Γ( c, r )  = 2 r + 1 . Note that Lemma 6.1 says that the minimum pebbling space requi red grows linearly w ith the recursio n depth r b ut is indepen dent of the number of spines c of the D A G. Next, we need the fact that there is a linear -time completely black pebbli ng of Γ( c, r ) in space linear in c + r . This is in fac t a strict impro vement (thoug h easily obtain ed) of the corres ponding result in [CS82]. Lemma 6.2. The graph s Γ ( c, r ) have persist ent blac k pebbling strate gies in simultaneo us space O( c + r ) and time linea r in the size of the grap hs. Our main result for t he Carlson-Sa vag e graph s is the f ollo wing trad e-of f fo r b lack-white pebbling, which pro vides us w ith a v ariety of pebblin g trade-of f results if we choose the parameters c and r appropria tely . Lemma 6.3. Suppose that P is a complete visiting blac k-white pebbling of Γ( c, r ) with space ( P ) < BW -P eb ∅  Γ( c, r )  + s = ( r + 2) + s for 0 < s ≤ c/ 8 − 1 . Then the time re quir ed to perfor m P is lower -bounded by time ( P ) ≥  c − 2 s 4 s + 4  r · r ! . Observ e that Lemma 6.3 is j ust a specia l case of Lemma 2.7, obtain ed by setting P σ = P τ = ( ∅ , ∅ ) , and we already ga ve a proof of Lemm a 2.7 in Section 2.2, assuming some auxiliar y technical lemmas. Hence, for Lemma 2.7 all we need to do is to establ ish the lemmas stated without proof in Section 2.2. Before showing any lemmas, howe ver , let us now see ho w we can prov e Theorem 1.10 by appealing to Lemmas 6.1, 6.2, and 6.3. Theor em 1.10 (res tated). L et g ( n ) be any arbitr arily slowly gr owing monotone function ω (1) = g ( n ) = O  n 1 / 7  , and let ǫ > 0 be an arbit rarily small positive con stant. T hen ther e is a family of exp licitly constr uctible single -sink D AGs { G n } ∞ n =1 of size Θ( n ) with consta nt verte x inde gr ee such that : 1. The graph G n has blac k-white pebbling price BW -P eb ( G ) = g ( n ) + O(1) and black peb bling price P eb ( G ) = 2 · g ( n ) + O(1) . 2. Ther e is a co mplete black pebblin g P of G n with time ( P ) = O( n ) and spa ce ( P ) = O  3 p n/g 2 ( n )  3. Any complete black- white pebblin g P of G n in space at most  n/g 2 ( n )  1 / 3 − ǫ r equir es pebbling time superp olynomial in n . 30 6 Carlson- Sav age Graphs and Strong Dual T rade-o ffs Pr oof. Consider th e graphs Γ( c, r ) in Deﬁnitio n 2.6. W e wan t to c hoose the parameters c an d r in a suitable way so that get a famil y of graphs in size n = Θ  cr 3 + c 3 r 2  (using the bound on the size of Γ( c, r ) from Lemma 6.1). If we choose r = r ( n ) = g ( n ) for g ( n ) = O  n 1 / 7  , this forces c = c ( n ) = Θ  3 p n/g 2 ( n )  . Consider the graph family { H n } ∞ n =1 deﬁned by H n = Γ( c ( n ) , r ( n )) as abov e and let G n = c H n be the single -sink versi on of H n . This is a famil y of sing le-sink D AGs o f size Θ( n ) . By Lemma 6.1 combin ed with Observ ation 3.9, it holds that P eb ( G n ) = g ( n ) + O(1) . A lso, the black pebbli ng of H n in Lemma 6.2 yields a linear -time pebbling of G n in space O  3 p n/g 2 ( n )  . No w set the paramete r s in Lemma 6.3 to s = c 1 − ǫ ′ for ǫ ′ = 3 ǫ . T hen for lar ge enough n we hav e s ≤ c/ 8 − 1 and Lemma 6.3 can be applied. W e get that if the pebbling space is less than  n/g 2 ( n )  1 / 3 − ǫ , then the required time for t he black-white pebblin g gro ws as  Ω  c ǫ ′  r =  Ω  n/g 2 ( n )  ǫg ( n ) which is su perpolyn omial in n for any g ( n ) = ω (1) . The theorem follo ws. W e also note that using dif ferent parameter setting s, we can obtain graphs w ith very r ob ust trade-of fs in the sense that the lower bound in the trade -of f applies ov er a very wide space range, namely all the way from log n up to ≈ 3 √ n . Theor em 6.4. T her e is a family of exp licitly constr uctible single-sink DA Gs { G n } ∞ n =1 of size Θ ( n ) with consta nt verte x inde gr ee such that : 1. P e b ( G n ) = O(log n ) . 2. Ther e is a complete black pebbling P of G n with time ( P ) = O( n ) and space ( P ) = O  3 q n/ log 2 n  . 3. Ther e is a constant K > 0 such that any complete blac k-white pebbling P of G n in space at most K 3 q n/ log 2 n must take time n Ω(log log n ) . Pr oof. Consider th e gr aphs Γ( c, r ) in Deﬁnition 2.6 with pa rameters cho sen so that c = 2 r . T hen the size of Γ( c, r ) is Θ  r 2 2 3 r  by Lemma 6.1. Let r ( n ) = max { r : r 2 2 3 r ≤ n } and deﬁne the graph family { G n } ∞ n =1 to be the single- sink versio n of Γ(2 r , r ) for r = r ( n ) . T ranslati ng from G n back to Γ( c, r ) we ha ve paramet ers r = Θ (log n ) and c = Θ  ( n/ log 2 n ) 1 / 3  , so Lemma 6.1 yields that P e b ( G n ) = O(log n ) . Hence, the linear -time persi stent black pebbling of G n in Lemma 6.2 has space O  ( n/ log 2 n ) 1 / 3  . Setting s = c/ 8 − 1 in Lemma 6 .3 shows that there is a const ant K such that if th e space o f a black-white pebbli ng P drops belo w K · ( n / log 2 n ) 1 / 3 ≤ ( r + 2) + s , then we must ha ve time ( P ) ≥ O(1) r · r ! = n Ω(log log n ) (6.1) (where we used that r = Θ(log n ) for the ﬁnal equality ). T he theo rem follo w s. As a ﬁ nal application of Theorem 1.9, we show that it can be used to construct D AGs with not only superp olynomial but ev en expo nential trade-of fs. A simple counti ng argu ment can be used to show that we can ne ver expect to ge t expon ential trade-of fs from D A Gs with poly logarithmic pebbling price. Howe ver , if we move to g raphs wit h p ebbling p rice Ω( n ǫ ) for so me con stant ǫ > 0 , such graphs c ould p otentially e xhibit exp onential t rade-of fs. W e obtai n such a family of graphs by a gain adjusting the p arameters i n D eﬁnition 2.6 approp riately . Theor em 6.5. T her e is a family of exp licitly constr uctible single-sink DA Gs { G n } ∞ n =1 of size Θ ( n ) with consta nt verte x inde gr ee such that : 1. P e b ( G n ) = O  8 √ n  . 31 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION 2. Ther e is a complete blac k pebb ling P of G n with time ( P ) = O( n ) and space ( P ) = O  4 √ n  . 3. Ther e is a constant K > 0 su ch th at any comple te blac k-white pebblin g of G n in sp ace at most K 4 √ n must take time  8 √ n  ! . Pr oof. Use the single- sink versio n of Γ( c, r ) as abov e w ith parameters c = 4 √ n and r = 8 √ n . W e remark that there is nothing magic in our particu lar choice of parameters c and r in Theorem 6.5. Other parameter s could be plugged in instead and yield slightly diff erent results . Note also that again we ha ve a certain rob ustness in the trade-of f results in that it holds for space from 8 √ n to 4 √ n , at which point it drops sharply to allo w a linear -time pebb ling. W e no w turn to the proofs of Lemmas 6.1, 6.2 , and 6.3. In the proofs we will need a fe w useful auxili ary lemmas, the ﬁrst of which giv es us information about ho w the spin es in the Carlson-Sa v age D A Gs are being pebble d. W e will use this informat ion repeate dly in what follo w s. Lemma 6.6 (Rephrasing of Lemma 2.8). Suppose that G is a DA G and that v is a verte x in G with a path Q to some sink z i ∈ Z ( G ) suc h that all vertices in Q \ { z i } have outde gr ee 1 . T hen any fruga l blac k- white pebbling str ate gy pebbles v e xactly on ce, an d the p ath Q contains p ebbles durin g one contiguous time interv al. Pr oof. By induction from the sink backw ards. The induction base is imm ediate . For the induc tiv e step, suppo se v has immediate successo r w and that w is pebble d exactl y once. If w is black-p ebbled at time σ , then v has been pebbled before this and the ﬁrst pebble placed on v stays until time σ . No secon d placement of a pebble on v after time σ can be essentia l since v has no othe r immediate successor than w . If w is white-pebble d and t he p ebble is remove d at time σ , then the ﬁrs t peb ble placed on v stays until time σ and no second placement of a pebble on v after time σ can be essential. Thus each vertex on the path is pebbled exactly once, and the time interv als when a verte x v and its succes sor w ha ve pebbles on them ov erlap. The lemma follo w s. The second auxiliary lemma speaks about subg raphs H of a D A G G whose only connectio n to the rest of the graph G \ H are via the sink of H . Note that the pyr amids in Γ( c, r ) satisfy this cond ition. Lemma 6.7 (Reph rasing of Lemma 2.9). Let G be a D A G and H a subgra ph in G such that H has a uniqu e sink z h and the only edges between V ( H ) and V ( G ) \ V ( H ) emanate fr om z h . Suppose that P is any fru gal complete pebbli ng of G h aving the pr operty tha t H is complete ly empty of pebble s at some given time τ ′ b ut at least one verte x of H has been pebbled during the time interval [0 , τ ′ ] . Then P pebbles H complete ly during the interval [0 , τ ′ ] . Pr oof. Suppose that v ∈ V ( H ) is pebbled at time σ ′ < τ ′ . Note that all paths starting in v must hit z h soone r or later , since z h is the uniqu e sink of H and there is no other way out of H into the rest of G . Consider the longes t path from v to z h . If th is pa th h as le ngth 1 , clearly z h must be p ebbled b efore ti me τ ′ since otherwise the p ebble plac ement on v is no n-essent ial. The same statement follo ws for an y v by ind uction ov er the path length . But since H is empty at times 0 and τ ′ and z h is pebbled during (0 , τ ′ ) , H is completely pebbled during this time interv al. Let us no w establish that the size and pebblin g price of the Carlson-Sa v age D AGs are as cl aimed. Pr oof of Lemma 6.1. The base case graph Γ( c, 1) in Deﬁnition 2.6 has size c + 2 . A pyramid of height h has ( h + 1)( h + 2) / 2 vertices, so the c p yramids of height 2 ( r − 1) in Γ( c, r ) con trib ute cr (2 r − 1) ve rtices. 32 6 Carlson- Sav age Graphs and Strong Dual T rade-o ffs The c spines with cr sections of 2 c ver tices each contrib ute a total of 2 c 3 r vert ices. And then there are the ver tices in Γ( c, r − 1) . Summing up, the total number of vertices in Γ( c, r ) is ( c + 2) + r X i =2  ci (2 i − 1) + 2 c 3 i  = Θ  cr 3 + c 3 r 2  (6.2) as is stated in the lemma. Clearly , B W -P eb ∅ (Γ( c, 1)) = P eb ∅ (Γ( c, 1)) = 3 , since pebbling a ve rtex with fan-in 2 requires 3 pebble s and Γ( c, 1) can be completely pebbled in this way by placin g pebbles on the two sources and then pebbli ng and unpebb ling the sinks one by one. Suppose inducti vely that BW -P eb ∅ (Γ( c, r )) = r + 2 and consi der Γ( c, r + 1) . It is straightf orward to see t hat BW -P e b ∅ (Γ( c, r + 1)) ≤ r + 3 . Every p yramid Π ( j ) 2 r can b e complet ely pebbled with r + 2 p ebbles (Theorem 3.15). W e can pebble each spine bottom-up in the follo wing way , sectio n by section. S uppose by indu ction that we ha ve a black pebble on the last vert ex v [ i − 1] 2 c in the ( i − 1) s t sectio n. Ke eping the pebble on v [ i − 1] 2 c , perfo rm a complete visiting pebb ling of Π (1) 2 r . At some point during this pebbling we must hav e a pebbl e on the p yramid s ink z 1 and at mo st r o ther pe bbles o n the pyra mid (by Proposition 3 .10). At th is time, plac e a black peb ble on v [ i ] 1 and r emov e the peb ble from v [ i − 1] 2 c . Complete the p ebbling of Π (1) 2 r , lea ving the p yramid empty . Performing complete visit ing pebblings of Π (2) 2 r , . . . , Π ( c ) 2 r in an ana logous fash ion all ows us to mo ve the black p ebble along v [ i ] 2 , . . . , v [ i ] c , nev er e xceedi ng total pebbling space r + 3 . In th e same w ay , for ev ery v isiting pebbl ing P of Γ( c, r ) there must exist t imes σ i for a ll i = 1 , . . . , c , when space ( P σ i ) < space ( P ) and the sink γ i contai ns a pebble. Performing a m inimum-spa ce pebbling of Γ( c, r ) , possibly c times if necess ary , this allo ws us to adva nce the black pebble along v [ i ] c +1 , . . . , v [ i ] 2 c , ne ver exc eeding total pe bbling s pace r + 3 . This sho ws that Γ( c, r + 1) can be completely pebbled wit h r + 3 pebble s. A simple syntac tic adap tation of this ar guments for black pebbl ing (appealing to Theorem 3.15 for the black pebb ling price of pyr amids) also yield s P eb ∅ (Γ( c, r )) ≤ 2 r + 3 . T o prov e tha t there are matching lo wer bo unds for the pebbling c onstructe d abov e, it is s ufﬁcie nt to show that s ome p yramid Π ( j ) 2 r must be comple tely pebb led while th ere is at lea st on e pebb le on Γ( c , r + 1) outs ide of Π ( j ) 2 r . T o see why , note that if we can pro ve this, th en simply by using the the fact that BW -P e b ∅ (Π 2 r ) = r + 2 and BW -P eb ∅ (Π 2 r ) = 2 r + 2 and adding one for the pebble outside of Π ( j ) 2 r we hav e the matching lo wer bound s that we need. W e present the ar gument for black-white pebbli ng, which is the harder case. The black- only pebblin g case is handled completely analog ously . Suppose in order to get a contradicti on that there is a visiting pebbl ing strategy P for Γ( c, r + 1) in space r + 2 . B y Obser vat ion 3.6 , P performs a complete visiti ng pebb ling of e very p yramid Π ( j ) 2 r . Consider the ﬁrst time τ 1 when some pyra mid has been completely pebbled and let this pyramid be Π ( j 1 ) 2 r . Then at some time σ 1 < τ 1 there are r + 2 pebbles on Π ( j 1 ) 2 r and the rest of the graph Γ( c, r + 1) must be empty of pebble s at this point. W e claim that this implies that no ver tex in Γ ( c, r + 1) outside of the pyramid Π ( j 1 ) 2 r has been pebbled before time σ 1 . Let us pro ve this crucial fact by a case anal ysis. 1. No vert ex v in any other pyramid Π ( j ′ ) 2 r can ha ve been pebbled before time σ 1 . For if so, Lemm a 6.7 says that Π ( j ′ ) 2 r has been complete ly pebbled before time σ 1 , contradi cting our choice of Π ( j 1 ) 2 r as the ﬁrst such pyra mid. 2. No verte x on a spine has been pebbled before time σ 1 . This is so since Lemma 6.6 tells us that if some v ertex on a spine has be en pebbled , then the whole spine must ha ve been pebb led in vie w of th e fact that it is empty at time σ 1 . But then Lemma 3.12 implies that all pyr amid sinks must ha ve been pebble d. This case has already been exclude d. 33 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION 3. Finally , no vert ex v in Γ( c, r ) can ha ve been pebbled before time σ 1 . Otherwise the frugali ty of P implies (by pattern matching on the argu ments in the proofs of L emmas 3.12 and 6.6) that some succes sor of v must ha ve been peb bled as well, and some succ essor of that suc cessor et cetera, all the way up to where Γ( c, r ) connect s with the spines. But we ha ve ruled out the possib ility that a spine ver tex has been pebbled. This establish es the claim, and we are no w almost done. T o clinch the argu ment, we need a couple of ﬁnal observ ations. Note ﬁrst that by frugality , at some time in the interv al ( σ 1 , τ 1 ) some verte x in some spine must hav e been pebbled. This is so since the pyramid sink z j 1 has been pebbled before time τ 1 , all of Π ( j 1 ) 2 r is empty at time τ 1 , and all spines are empty at time σ 1 < τ 1 . But then Lemma 6.6 tells us that there will remain a pebb le on this spine until all of the spine has been complete ly pebbled . Consider n ow the se cond p yramid Π ( j 2 ) 2 r complete ly pebbled by P , sa y , at time τ 2 . At some point in time σ 2 < τ 2 we ha ve r + 2 pebbl es on Π ( j 2 ) 2 r , and moreov er σ 2 > τ 1 since Π ( j 2 ) 2 r is empty at time τ 1 . But now it must hold that either there is a pebble on a spine at this point, or , if all spines are completely empty , that some spine has been complet ely pebbled. If some spine has been completely pebbled , howe ver , this in turn implies (a ppealing to Lemma 3.12 aga in) that t here must b e some pebble some where in some other p yramid Π ( j ′ ) 2 r at time σ 2 . T hus t he p ebbling spac e exc eeds r + 2 and we ha ve ob tained ou r contra diction. The le mma follo w s. Studying the pebblin g strategie s in the proof of Lemma 6.1, it is not hard to see that they are very inef ﬁcient. The subgr aphs in Γ( c, r ) w ill be pebbled ov er and over ag ain, and for eve ry step in t he recursio n the time required multiplie s. W e next sho w that if we are a bit more generous w ith the pebblin g space, then we can get do wn to linear time. Pr oof of Lemma 6.2. W e want to pro ve that Γ( c, r ) has a persi stent black pebbl ing strate gy P that pebbles e very v erte x in Γ( c, r ) exac tly once and uses spac e O ( c + r ) . Suppose that there is suc h a pebblin g strate gy P r for Γ( c , r ) . W e des cribe ho w to construct a p ebbling P r +1 for Γ( c , r + 1) inducti vel y . Note th at the base case for Γ( c, 1) is tri vial. The constructio n of P r +1 is very straig htforwa rd. First use P r to make a persist ent pebbl ing of Γ( c, r ) in space O( c + r ) . At the end of P r , we hav e c pebbl es on the sinks γ 1 , . . . , γ c . Keeping these pebbles in place, pebble the pyra mids Π (1) 2 r , . . . , Π ( c ) 2 r persis tently one by one in space O( r ) with a strategy pebbli ng each v ertex exactly o nce (for i nstance, by p ebbling the pyramid bot tom-up lev el by le vel). W e lea ve pebbles on all pyramid sink s z 1 , . . . , z c . T his stage of the pebbling only requires space O( c + r ) and at the end we ha ve 2 c black pebble s on all pyramid sinks z 1 , . . . , z c and all sinks γ 1 , . . . , γ c of Γ( c, r ) . Ke eping all these pebble s in place, we can pebble all c spines in parallel in linear time with c + 1 extra peb bles. It remains to ﬁll in the gaps in the proof of Lemma 2.7 and its special case Lemma 6.3. Recall that the proof of Lemma 2.7 presen ted in Section 2.2 hinged on the claims that not too many pyramids can be pebble d simultane ously in a space -efﬁci ent pebbling , and that this is true for the spines as well. Assuming these two claims, we could show that that as any spine was pebbled, the pebblin g had to alternate back and forth between time interv als when there are a lot of pebble s on some pyr amid and time interv als when all sinks in Γ( c, r ) are pe bbled. This all owed us to apply the induction hypo thesis multiple ti mes an d obtai n the requir ed lo wer bound. Hence, al l tha t remains to compl ete the proof of Lemma 2.7 is to establi sh the tw o tec hnical lemmas that upper -bound how many pyra mids and spine sections can contain pebbles simultaneo usly at any one giv en time in a pebbling subjected to space constr aints as in Lemma 2.7. The claims in the two lemmas are very similar in spirit, as are the proofs , so we state the lemmas togethe r and then present the proofs in sequen ce. 34 6 Carlson- Sav age Graphs and Strong Dual T rade-o ffs Lemma 6.8 (Rephrasing of Lemma 2.10). Suppos e that P = { P σ , . . . , P τ } is a conditi onal blac k-white pebbli ng on Γ( c, r ) an d that s is a c onstant satis fying the cond itions in Lemma 2 .7. Then at a ll times dur ing the pebbli ng P strictly less than 4( s + 1) pyramid s Π ( j ) 2 r contai n pebbl es simultaneo usly . Lemma 6.9 (Rephrasing of Lemma 2.11). Suppos e that P = { P σ , . . . , P τ } is a conditi onal blac k-white pebbli ng on Γ( c, r ) an d that s is a c onstant satis fying the cond itions in Lemma 2 .7. Then at a ll times dur ing the pebbli ng P strictly less than 4( s + 1) spine sections contain pebbles simultaneo usly . Note that Lemma 6.9 provide s a total bound on the number of pebble d sections in all c spines . There might be spines contai ning se veral sections being pebb led simultaneo usly (in fact, this is exactly w hat one would e xpect a black-white pebbling to do to optimiz e the time giv en the space constraint s), b ut what Lemma 6.9 says t hat if we ﬁx an arb itrary time t ∈ [ σ, τ ] , add up th e number of sectio ns containing pebbles at time t in each spine, and sum ov er all spines, we ne ver exc eed 4( s + 1) sections in tota l. Pr oof of Lemma 6.8. Suppose that on the contrar y , there is some time t ∗ ∈ ( σ, τ ) when at least 4 s + 4 pyr amids Π ( j ) in Γ( c, r ) contain pebble s. O f these pyra mids, at least 2 s + 4 are empty both at time σ and at time τ since space ( P σ ) < s and space ( P τ ) < s . B y Lemma 6.7, these pyramid s, which we denote Π (1) , . . . , Π (2 s +4) , are completely pebbled during [ σ, τ ] . Moreov er , we can conclude that for ev ery Π ( j ) , j = 1 , . . . , 2 s + 4 , there is an interv al [ σ j , τ j ] ⊆ [ σ, τ ] such that t ∗ ∈ ( σ j , τ j ) and Π ( j ) is empty at times σ j and τ j b ut contains pebbles througho ut the interv al ( σ j , τ j ) during which it is completely pebbled. For eac h Π ( j ) there must exist some time t ∗ j ∈ ( σ i , τ i ) when there are at least r + 1 = B W -P eb ∅  Π ( j )  pebble s. F ix such a time t ∗ j for e very pyramid Π ( j ) and assume that all t ∗ j , j = 1 , . . . , 2 s + 4 , are sorted in increa sing order . W e hav e two possibl e cases: 1. At least half of all t ∗ j occur before (or at) time t ∗ , i.e., the y satisfy t ∗ j ≤ t ∗ . If so, look at the larges t t ∗ j ≤ t ∗ . At thi s time there are at l east r + 1 pebbles on Π ( j ) and at least 2 s +4 2 − 1 = s + 1 pebbl es on other pyramid s, which means that space  P t ∗ j  ≥ ( r + 2) + s . In other words, P e xceeds the space restric tions in Lemma 2.7. Contradic tion. 2. At leas t half of all t ∗ j occur af ter time t ∗ , i.e., they sati sfy t ∗ j > t ∗ . If we co nsider the s mallest t ∗ j lar ger than t ∗ we can agai n conclude that spa ce  P t ∗ j  ≥ ( r + 1) + ( s + 1) , which is again a contrad iction. Hence, if P is a pebbling that complies with the restrict ions in Lemma 2.7, it can nev er be the case that 4 s + 4 py ramids Π ( j ) in Γ( c, r ) cont ain pebbles simultan eously . Pr oof of Lemma 6.9. Suppose that at some time t ∗ ∈ ( σ, τ ) at least 4 s + 4 sections conta in pebbles . At least 2 s + 4 of these sections are empty at times σ and τ . Let us denote these sectio ns R 1 , . . . , R 2 s +4 . Appealin g to L emma 6.6, we conclude that there are interv als [ σ j , τ j ] ⊆ [ σ, τ ] for j = 1 , . . . , 2 s + 4 , such that t ∗ ∈ ( σ j , τ j ) and R j is empty at times σ j and τ j b ut contains pebbles througho ut the interv al ( σ j , τ j ) during which it is completel y pebbl ed. By Lemma 6.8, we kno w that less than 4 s + 4 pyramid s contain pebb les at time σ j and similarly at time τ j . S ince all c pyramids in Γ( c, r ) must hav e their sinks peb bled during ( σ j , τ j ) b ut it holds that 2 · (4 s + 4) < c by the assu mptions in Lemma 2.7, we conclude from Lemma 6.7 that for ev ery section R j we c an ﬁnd some pyramid Π ( j ) that is completely pebble d during the int erv al ( σ j , τ j ) . This, in tur n, implies that there is some time t ∗ j ∈ ( σ j , τ j ) when the pyramid Π ( j ) contai ns at least BW -P eb ∅  Π ( j )  = r + 1 pebble s. (W e note that many t ∗ j can be equal and e ven refer to the same py ramid, b ut this is not a proble m.) As in the proof of Lemma 6.8, we no w sort the t ∗ j , j = 1 , . . . , 2 s + 4 , in increasing order and consider the two poss ible cases. If at least half of all t ∗ j satisfy t ∗ j ≤ t ∗ , we look at the larg est t ∗ j ≤ t ∗ . At this time there ar e at le ast r + 1 pebble s on Π ( j ) and at least 2 s +4 2 = s + 2 pebble s on dif ferent se ctions, which means 35 ON THE RELA T IVE STRENGTH OF PEBBLING AND RESOLUTION that space  P t ∗ j  ≥ r + s + 3 exc eeds the space restriction s. If, on the other hand, at least half of all t ∗ j satisfy t ∗ j > t ∗ , then for the smallest t ∗ j lar ger than t ∗ we can again conclud e that spa ce  P t ∗ j  ≥ r + s + 3 , which is a contr adiction. The lemma follo ws. As we discuss ed at the start of this section, T heorem 1.9 now follows by applying Observ ation 3.9 on the single- sink versio n of Γ( c, r ) . As a ﬁ nal note, w e remark that not only do our proofs get much more in volv ed when going from the black- only pebbling trade-o ff in [CS82] to our black -white pebbling trad e-of f, b ut the added complic ations also lead to our bound for black -white pebb ling being slig htly worse than the one in [CS82] for black pebbli ng. More speciﬁcally , Carlson and Sa vag e are able to prov e their results for D A Gs ha ving only Θ( r ) sectio ns per spi ne, whereas we need Θ( cr ) sectio ns in Γ( c, r ) . This blo ws up the numb er of vertice s, w hich in turn weak ens the trade-of fs measured in terms of graph size. It would be intere sting to ﬁnd out whether our proof could in fac t be made to work for graphs with only O ( r ) sections per spine. If so, this would immediatel y improve the trade-of fs for the graphs in Theore ms 1.10, 6.4, and 6.5, as well as the resolution trade-o ffs deri ved from these graphs in [BN09b]. 7 Conc luding Remarks It is known that the black -white pebbl ing price is always a lower bound on the resolution space of refuting pebbli ng contra dictions Peb G [ f ] with respect to the “right” functio ns f , as prov en in [BN08]. Also, for all graphs studied in this conte xt so far there ha ve been sho wn to exis t refutatio ns of the correspo nding peb- bling contr adiction s in spac e upper- bounded by the black-white pebbling price—tri vially for graphs where the black and black-white pebbling prices coincide , and more interestin gly for the graphs in the curren t paper where the black-white pebbling price is asymptoticall y smaller than the black pebbling price. T his natura lly raises the question whethe r it holds in genera l that the refutation space of pebbling contrad ictions is asympto tically equal to the black- white pebblin g price of the underlyi ng graphs. Open Q uestion 1. Is in true for any D AG G with bounded verte x inde gr ee and any (ﬁxed) Boolean function f that the pebbli ng contr adiction P eb G [ f ] can be r efuted in total space O( BW -P e b ( G )) ? More speciﬁcally , one could ask—as a natural ﬁ rst line of attack if one wants to in ves tigate whether the answer to the abov e quest ion could be yes—if it holds that bounded labelled pebbling s are in fact as po werful as general black-white pebblings. In a sense, this is asking whether only a very limited form of nonde terminism is suf ﬁcient to realize the full potential of black-whit e pebbling . Open Question 2. Does it hold that any complete blac k-white pebbling P of a single-sink D AG G with bound ed verte x inde gr ee can be simulated by a (O( space ( P )) , O(1)) -bound ed pebbling L ? Note that a positi ve answer to this second question would immediatel y imply a positi ve answer to the ﬁrst quest ion as well by Lemma 2.5. W e hav e no strong intuition either way reg arding Open Question 1, b ut as to Open Question 2 it would perhap s be somewha t surprising if bounde d labelled pebblings turned out to be as strong as general black- white pebblings . Inter estingly , althou gh the optimal black-whi te pebblings of the graphs in Lemma 1.7 can be simulated by bounded pebbli ngs, the same appro ach does not work for the origina l graphs separating black- white from black-onl y pebbling in [W il88]. Indeed, these latter graphs might be a candidat e graph family for answeri ng Open Ques tion 2 in the ne gati ve, i.e., sho wing that sta ndard blac k-white pebblin gs can be asymptoti cally stronger than bounded labell ed pebblin gs. Finally , we are intrigued by the question of whether the propert ies of the formulas Peb G [ f ] shown to hold in [BN08, BN09b] for “the right kind” of functio ns f in fact exten d to the simpler formulas Peb G [ ∨ ] deﬁned in terms of non-e xclusi ve or . 36 Referenc es Open Question 3. Is it true for any DA G G tha t any res olution re futation π of Peb G [ ∨ ] can be tr anslated into a black -white pebbling o f G with time and space upp er- bounded asymptot ically by th e len gth and space of π ? Earlier results in [Nor09, NH08b] can be interp reted as indicating that this should be the case, b ut the results there only apply to limited classes of graphs and only capture space lo wer bounds, not time-space trade-o ffs . And the papers [BN08, B N09b] do not shed any light on this question , as the technique s used there inher ently cannot work for formu las deﬁned in terms of non- exclus iv e or . If the answer to O pen Que stion 3 i s yes —which we w ould caut iously exp ect it to be—then this could be useful for settling the comple xity of decision problems for resol ution proof space , i.e., the problem gi ven a CNF formula F and a space boun d s to dete rmine w hether F has a resolution refutation in space at most s . Reducing from pebbling space by way of formulas Peb G [ ∨ ] would av oid the blo w-up of the gap between upper and lo wer bounds on pebblin g space that cause problems when using, for instance , exclusi ve or . Ac kno wledgements I would like to thank David Carlson, N icholas Pippenger , and John Sav age for helpfu l corresp ondence re- gardin g their papers on pebbling , and Eli Ben-Sasson and Johan H ˚ astad for stimulatin g discussion s about pebbli ng and proof complexit y . 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