Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
Susanna F. de Rezende, Or Meir, Jakob Nordström, Robert Robere
aa r X i v : . [ c s . CC ] J a n Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling ∗ Susanna F. de RezendeMathematical Institute of theCzech Academy of Sciences Or MeirUniversity of HaifaJakob Nordstr¨omUniversity of Copenhagen andKTH Royal Institute of Technology Robert RobereDIMACS andInstitute for Advanced StudyJanuary 9, 2020
Abstract
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatzrefutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t andspace s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use thiscorrespondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to thebest of our knowledge are the first such results for this proof system. In this work, we obtain strong trade-offs in proof complexity by making a connection to pebble gamesplayed on graphs. In this introductory section we start with a brief overview of these two areas and thenexplain how our results follow from connecting the two.
Proof complexity is the study of efficiently verifiable certificates for mathematical statements. Moreconcretely, statements of interest claim to provide correct answers to questions like: • Given a CNF formula, does it have a satisfying assignment or not? • Given a set of polynomials over some finite field, do they have a common root?There is a clear asymmetry here in that it seems obvious what an easily verifiable certificate for positiveanswers to the above questions should be, while it is not so easy to see what a concise certificate for anegative answer could look like. The focus of proof complexity is therefore on the latter scenario.In this paper we study the algebraic proof system system
Nullstellensatz introduced by Beame etal. [BIK + Nullstellensatz refutation of a set of polynomials P = { p i | i ∈ [ m ] } with coefficientsin a field F is an expression m X i =1 r i · p i + n X j =1 s j · ( x j − x j ) = 1 (1.1)(where r i , s j are also polynomials), showing that lies in the polynomial ideal in the ring F [ x , . . . , x n ] generated by P ∪ (cid:8) x j − x j (cid:12)(cid:12) j ∈ [ n ] (cid:9) . By (a slight extension of) Hilbert’s Nullstellensatz, such a refutationexists if and only if there is no common { , } -valued root for the set of polynomials P . ∗ A preliminary version of this work appeared in CCC 2019.
ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING
Nullstellensatz can also be viewed as a proof system for certifying the unsatisfiability of CNF formulas.If we translate a clause like, e.g., C = x ∨ y ∨ z to the polynomial p ( C ) = (1 − x )(1 − y ) z = z − yz − xz + xyz , then an assignment to the variables in a CNF formula F = V mi =1 C i (where wethink of as true and as false) is satisfying precisely if all the polynomials { p ( C i ) | i ∈ [ m ] } vanish.The size of a Nullstellensatz refutation (1.1) is the total number of monomials in all the polynomials r i · p i and s j · ( x j − x j ) expanded out as linear combinations of monomials. Another, more well-studied,complexity measure for Nullstellensatz is degree , which is defined as max { deg( r i · p i ) , deg( s j · ( x j − x j )) } .In order to prove a lower bound d on the Nullstellensatz degree of refuting a set of polynomials P ,one can construct a d -design , which is a map D from degree- d polynomials in F [ x , . . . , x n ] to F suchthat1. D is linear, i.e., D ( αp + βq ) = αD ( p ) + βD ( q ) for α, β ∈ F ;2. D (1) = 1 ;3. D ( rp ) = 0 for all p ∈ P and r ∈ F [ x , . . . , x n ] such that deg( rp ) ≤ d ;4. D ( x s ) = D ( xs ) for all s ∈ F [ x , . . . , x n ] such that deg( s ) ≤ d − .Designs provide a characterization of Nullstellensatz degree in that there is a d -design for P if and only ifthere is no Nullstellensatz refutation of P in degree d [Bus98]. Another possible approach to prove degreelower bounds is by computationally efficient versions of Craig’s interpolation theorem. It was shownin [PS98] that constant-degree Nullstellensatz refutations yield polynomial-size monotone span programs,and that this is also tight: every span program is a unique interpolant for some set of polynomials refutableby Nullstellensatz. This connection has not been used to obtain Nullstellensatz degree lower bounds,however, due to the difficulty of proving span program lower bounds.Lower bounds on Nullstellensatz degree have been proven for sets of polynomials encoding com-binatorial principles such as the pigeonhole principle [BCE + + polynomial calcu-lus [CEI96, ABRW02], where the proof that lies in the ideal generated by P ∪ (cid:8) x j − x j (cid:12)(cid:12) j ∈ [ n ] (cid:9) can be constructed dynamically by a step-by-step derivation. However, the Nullstellensatz proof systemhas been the focus of renewed interest in a recent line of works [RPRC16, PR17, PR18, dRMN + resolution proof system [Bla37], which operates directly on the clauses of a CNFformula and repeatedly derives resolvent clauses C ∨ D from clauses of the form C ∨ x and D ∨ x untilcontradiction, in the form of the empty clause without any literals, is reached. For resolution, size ismeasured by counting the number of clauses, and width , measured as the number of literals in a largestclause in a refutation, plays an analogous role to degree for Nullstellensatz and polynomial calculus.By way of background, it is not hard to show that for all three proof systems upper bounds on de-gree/width imply upper bounds on size, in the sense that if a CNF formula over n variables can be refutedin degree/width d , then such a refutation can be carried out in size n O( d ) . Furthermore, this upper boundhas been proven to be tight up to constant factors in the exponent for resolution and polynomial calcu-lus [ALN16], and it follows from [LLMO09] that this also holds for Nullstellensatz. In the other direction,it has been shown for resolution and polynomial calculus that strong enough lower bounds on degree/widthimply lower bounds on size [IPS99, BW01]. This is known to be false for Nullstellensatz, and the pebblingformulas discussed in more detail later in this paper provide a counter-example [BCIP02].The size lower bounds in terms of degree/width in [IPS99, BW01] can be established by transformingrefutations in small size to refutations in small degree/width. This procedure blows up the size of the2 Introduction refutations exponentially, however. It is natural to ask whether such a blow-up is necessary or whetherit is just an artifact of the proof. More generally, given that a formula has proofs in small size and smalldegree/width, it is an interesting question whether both measures can be optimized simultaneously, orwhether there has to be a trade-off between the two.For resolution this question was finally answered in [Tha16], which established that there are indeedstrong trade-offs between size and width making the size blow-up in [BW01] unavoidable. For polynomialcalculus, the analogous question remains open.In this paper, we show that there are strong trade-offs between size and degree for Nullstellensatz.We do so by establishing a tight relation between Nullstellensatz refutations of pebbling formulas andreversible pebblings of the graphs underlying such formulas. In order to discuss this connection in moredetail, we first need to describe what reversible pebblings are. This brings us to our next topic.
In the pebble game first studied by Paterson and Hewitt [PH70], one places pebbles on the vertices of adirected acyclic graph (DAG) G according to the following rules: • If all (immediate) predecessors of an empty vertex v contain pebbles, a pebble may be placed on v . • A pebble may be removed from any vertex at any time.The game starts and ends with the graph being empty, and a pebble should be placed on the (unique) sinkof G at some point. The complexity measures to minimize are the total number of pebbles on G at anygiven time (the pebbling space ) and the number of moves (the pebbling time ).The pebble game has been used to study flowcharts and recursive schemata [PH70], register alloca-tion [Set75], time and space as Turing-machine resources [Coo74, HPV77], and algorithmic time-spacetrade-offs [Cha73, SS77, SS79, SS83, Tom78]. In the last two decades, pebble games have seen a revivalin the context of proof complexity (see, e.g., [Nor13]), and pebbling has also turned out to be useful forapplications in cryptography [DNW05, AS15]. An excellent overview of pebbling up to ca. 1980 is givenin [Pip80] and some more recent developments are covered in the upcoming survey [Nor20].Bennett [Ben89] introduced the reversible pebble game as part of a broader program [Ben73] aimedat eliminating or reducing energy dissipation during computation. Reversible pebbling has also been ofinterest in the context of quantum computing. For example, it was noted in [MSR +
19] that reversiblepebble games can be used to capture the problem of “uncomputing” intermediate values in quantumalgorithms.The reversible pebble game adds the requirement that the whole pebbling performed in reverse ordershould also be a correct pebbling, which means that the rules for pebble placement and removal becomesymmetric as follows: • If all predecessors of an empty vertex v contain pebbles, a pebble may be placed on v . • If all predecessors of a pebbled vertex v contain pebbles, the pebble on v may be removed.Reversible pebblings have been studied in [LV96, Kr´a04, KSS18] and have been used to prove time-space trade-offs in reversible simulation of irreversible computation in [LTV98, LMT00, Wil00, BTV01].In a different context, Potechin [Pot10] implicitly used reversible pebbling to obtain lower bounds inmonotone space complexity, with the connection made explicit in later works [CP14, FPRC13]. The pa-per [CLNV15] (to which this overview is indebted) studied the relative power of standard and reversiblepebblings with respect to space, and also established PSPACE -hardness results for estimating the mini-mum space required to pebble graphs (reversibly or not).
In this paper, we obtain an exactly tight correspondence between on the one hand reversible pebblings ofDAGs and on the other hand Nullstellensatz refutations of pebbling formulas over these DAGs. We show3
ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING that for any DAG G it holds that G can be reversibly pebbled in time t and space s if and only if there is aNullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s . This correspondenceholds regardless of the field in which the Nullstellensatz refutation is operating, and so, in particular,it follows that pebbling formulas have exactly the same complexity for Nullstellensatz regardless of theambient field.We then revisit the time-space trade-off literature for the standard pebble game, focusing on the papers[CS80, CS82, LT82]. The results in these papers do not immediately transfer to the reversible pebble game,and we are not fully able to match the tightness of the results for standard pebbling, but we neverthelessobtain strong time-space trade-off results for the reversible pebble game.This allows us to derive Nullstellensatz size-degree trade-offs from reversible pebbling time-spacetrade-offs as follows. Suppose that we have a DAG G such that:1. G can be reversibly pebbled in time t ≪ t .2. G can be reversibly pebbled in space s ≪ s .3. There is no reversible pebbling of G that simultaneously achieves time t and space s .Then for Nullstellensatz refutations of the pebbling formula Peb G over G (which will be formally definedshortly) we can deduce that:1. Nullstellensatz can refute Peb G in size t + 1 ≪ t + 1 .2. Nullstellensatz can also refute Peb G in degree s ≪ s .3. There is no Nullstellensatz refutation of Peb G that simultaneously achieves size t +1 and degree s .We prove four such trade-off results, which can be found in Section 4. The following theorem is oneexample of such a result (specifically, it is a simplified version of Theorem 4.1). Theorem 1.1.
There is a family of -CNF formulas { F n } ∞ n =1 of size Θ( n ) such that:1. There is a Nullstellensatz refutation of F n in degree s = O (cid:0) √ n log n (cid:1) .2. There is a Nullstellensatz refutation of F n of near-linear size and degree s = O (cid:0) √ n log n (cid:1) .
3. Any Nullstellensatz refutation of F n in degree at most √ n must have exponential size. It should be noted that this is not the first time proof complexity trade-off results have been obtainedfrom pebble games. Pebbling formulas were used in [Ben09, BN11] to obtain size-space trade-offs forresolution, and later in [BNT13] also for polynomial calculus. However, the current reductions betweenpebbling and Nullstellensatz are much stronger in that they go in both directions and are exact even up toadditive constants.With regard to Nullstellensatz, it was shown in [BCIP02] that Nullstellensatz degree is lower-boundedby standard pebbling price. This was strengthened in [dRMN + +
19] by constructing a more direct reduction, inspired by [GKRS18], that can simultaneouslycapture both time and space.
After having discussed the necessary preliminaries in Section 2, we present the reductions between Null-stellensatz and reversible pebblings in Section 3. In Section 4, we prove time-space trade-offs for re-versible pebblings in order to obtain size-degree trade-offs for Nullstellensatz. Section 5 contains someconcluding remarks with suggestions for future directions of research.4
Preliminaries
All logarithms in this paper are base unless otherwise specified. For a positive integer n we write [ n ] todenote the set of integers { , , . . . , n } .A literal a over a Boolean variable x is either the variable x itself or its negation x (a positive or negative literal, respectively). A clause C = a ∨ · · · ∨ a k is a disjunction of literals. A k -clause is aclause that contains at most k literals. A formula F in conjunctive normal form (CNF) is a conjunctionof clauses F = C ∧ · · · ∧ C m . A k -CNF formula is a CNF formula consisting of k -clauses. We think ofclauses and CNF formulas as sets, so that the order of elements is irrelevant and there are no repetitions.A truth value assignment ρ to the variables of a CNF formula F is satisfying if every clause in F containsa literal that is true under ρ . Let F be any field and let ~x = { x , . . . , x n } be a set of variables. We identify a set of polynomials P = { p i ( ~x ) | i ∈ [ m ] } in the ring F [ ~x ] with the statement that all p i ( ~x ) have a common { , } -valuedroot. A Nullstellensatz refutation of this claim is a syntactic equality m X i =1 r i ( ~x ) · p i ( ~x ) + n X j =1 s j ( ~x ) · ( x j − x j ) = 1 , (2.1)where r i , s j are also polynomials in F [ ~x ] . We sometimes refer to the polynomials p i ( ~x ) as axioms and ( x j − x j ) as Boolean axioms.As discussed in the introduction, Nullstellensatz can be used as a proof system for CNF formulas bytranslating a clause C = W x ∈ P x ∨ W y ∈ N y to the polynomial p ( C ) = Q x ∈ P (1 − x ) · Q y ∈ N y andviewing Nullstellensatz refutations of { p ( C i ) | i ∈ [ m ] } as refutations of the CNF formula F = V mi =1 C i .The degree of a Nullstellensatz refutation (1.1) is max { deg( r i ( ~x ) · p i ( ~x )) , deg( s j ( ~x ) · ( x j − x j )) } .We define the size of a refutation (2.1) to be the total number of monomials encountered when all productsof polynomials are expanded out as linear combinations of monomials. To be more precise, let mSize ( p ) denote the number of monomials in a polynomial p written as a linear combination of monomials. Thenthe size of a Nullstellensatz refutation on the form (1.1) is m X i =1 mSize (cid:0) r i ( ~x ) (cid:1) · mSize (cid:0) p i ( ~x ) (cid:1) + n X j =1 · mSize (cid:0) s j ( ~x ) (cid:1) . (2.2)This is consistent with how size is defined for the “dynamic version” of Nullstellensatz known as polyno-mial calculus [CEI96, ABRW02], and also with the general size definitions for so-called algebraic andsemialgebraic proof systems in [ALN16, Ber18, AO19].We remark that this is not the only possible way of measuring size, however. It can be noted that thedefinition (2.2) is quite wasteful in that it forces us to represent the proof in a very inefficient way. Otherpapers in the semialgebraic proof complexity literature, such as [GHP02, KI06, DMR09], instead definesize in terms of the polynomials in isolation, more along the lines of m X i =1 (cid:0) mSize (cid:0) r i ( ~x ) (cid:1) + mSize (cid:0) p i ( ~x ) (cid:1)(cid:1) + n X j =1 (cid:0) mSize (cid:0) s j ( ~x ) (cid:1) + 2 (cid:1) , (2.3)or as the bit size or “any reasonable size” of the representation of all polynomials r i ( ~x ) , p i ( ~x ) , and s j ( ~x ) .In the end, the difference is not too important since the two measures (2.2) and (2.3) are at most asquare apart, and for size we typically want to distinguish between polynomial and superpolynomial. Inaddition, and more importantly, in this paper we will only deal with k -CNF formulas with k = O(1) ,and in this setting the two definitions are the same up to a constant factor k . Therefore, we will stick5 ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING with (2.2), which matches best how size is measured in the closely related proof systems resolution andpolynomial calculus, and which gives the cleanest statements of our results. When proving lower bounds for algebraic proof systems it is often convenient to consider a multilinear setting where refutations are presented in the ring F [ ~x ] / { x j − x j | j ∈ [ n ] } . Since the Boolean axioms x j − x j are no longer needed, the refutation (2.1) can be written simply as m X i =1 r i ( ~x ) · p i ( ~x ) = 1 , (2.4)where we assume that all results of multiplications are implicitly multilinearized. It is clear that anyrefutation on the form (2.1) remains valid after multilinearization, and so the size and degree measurescan only decrease in a multilinear setting. In this paper, we prove our lower bound in our reduction inthe multilinear setting and the upper bound in the non-multilinear setting, making the tightly matchingresults even stronger. Throughout this paper G = ( V, E ) denotes a directed acyclic graph (DAG) of constant fan-in with ver-tices V ( G ) = V and edges E ( G ) = E . For an edge ( u, v ) ∈ E we say that u is a predecessor of v and v a successor of u . We write pred G ( v ) to denote the sets of all predecessors of v , and drop the subscriptwhen the DAG is clear from context. Vertices with no predecessors/successors are called sources / sinks .Unless stated otherwise we will assume that all DAGs under consideration have a unique sink z .A pebble configuration on a DAG G = ( V, E ) is a subset of vertices P ⊆ V . A reversible pebblingstrategy for a DAG G with sink z , or a reversible pebbling of G for short, is a sequence of pebble config-urations P = ( P , P , . . . , P t ) such that P = P t = ∅ , z ∈ S ≤ t ≤ t P t , and such that each configurationcan be obtained from the previous one by one of the following rules:1. P i +1 = P i ∪ { v } for v / ∈ P i such that pred G ( v ) ⊆ P i (a pebble placement on v ).2. P i +1 = P i \ { v } for v ∈ P i such that pred G ( v ) ⊆ P i (a pebble removal from v ).The time of a pebbling P = ( P , . . . , P t ) is time ( P ) = t and the space is space ( P ) = max ≤ t ≤ t {| P t |} .We could also say that a reversible pebbling P = ( P , . . . , P t ) should be such that P = ∅ and z ∈ P t ,and define the time of such a pebbling to be t . This is so since once we have reached a configurationcontaining z we can simply run the pebbling backwards (because of reversibility) until we reach the emptyconfiguration again, and without loss of generality all time- and space-optimal reversible pebblings canbe turned into such pebblings. For simplicity, we will often take this viewpoint in what follows.For technical reasons it is sometimes important to distinguish between visiting pebblings , for which z ∈ P t , and persistent pebblings , which meet the more stringent requirement that z ∈ P t = { z } . (Itcan be noted that for the more relaxed standard pebble game discussed in the introductory section anypebbling can be assumed to be persistent without loss of generality.)Pebble games can be encoded in CNF by so-called pebbling formulas [BW01], or pebbling contra-dictions . Given a DAG G = ( V, E ) with a single sink z , we associate a variable x v with every vertex v and add clauses encoding that • the source vertices are all true; • if all immediate predecessors are true, then truth propagates to the successor; • but the sink is false. We refer the reader to Section 2.4 in [AH18] for a more detailed discussion of the definition of proof size in algebraic andsemialgebraic proof systems. Reversible Pebblings and Nullstellensatz Refutations zu vp q r (a)
Pyramid of height 2. x p ∧ x q ∧ x r ∧ ( x p ∨ x q ∨ x u ) ∧ ( x q ∨ x r ∨ x v ) ∧ ( x u ∨ x v ∨ x z ) ∧ x z (b) Pebbling formula in CNF. x p − x q − x r − x p x q (1 − x u ) x q x r (1 − x v ) x u x v (1 − x z ) x z (c) Polynomial translation.
Figure 1:
Example pebbling contradiction for the pyramid graph of height . In short, the pebbling formula over G consists of the clauses x v ∨ W u ∈ pred ( v ) ¬ x u for all v ∈ V (note thatif v is a source pred ( v ) = ∅ ), and the clause ¬ x z .We encode this formula by a set of polynomials in the standard way. Given a set U ⊆ V , we denoteby x U the monomial Q u ∈ U x u (in particular, x ∅ = 1 ). For every vertex v ∈ V , we have the polynomial A v := (1 − x v ) · x pred( v ) , (2.5)and for the sink z we also have the polynomial A sink := x z . (2.6)See Figure 1 for an illustration, including how the CNF formula is translated to a set of polynomials. In this section, we prove the correspondence between the reversible pebbling game on a graph G andNullstellensatz refutation of the pebbling contradiction of G . Specifically, we prove the following result. Theorem 3.1.
Let G be a directed acyclic graph with a single sink, let φ be the corresponding pebblingcontradiction, and let F be a field. Then, there is a reversible pebbling strategy for G with time at most t and space at most s if and only if there is a Nullstellensatz refutation for φ over F of size at most t + 1 and degree at most s . Moreover, the same holds for multilinear Nullstellensatz refutations. We prove each of the directions of Theorem 3.1 separately in Sections 3.1 and 3.2 below.
We start by proving the “only if” direction of Theorem 3.1. Let P = ( P , . . . , P t ) (3.1)be an optimal reversible pebbling strategy for G . Let P t ′ be the first configuration in which there is apebble on the sink z . Without loss of generality, we may assume that t = 2 · t ′ : if the last t − t ′ steps weremore efficient than the first t ′ steps, we could have obtained a more efficient strategy by replacing the first t ′ steps with the (reverse of) the last t − t ′ steps, and vice versa.We use P to construct a Nullstellensatz refutation over F for the pebbling contradiction φ . To this end,we will first construct for each step i ∈ [ t ′ ] of P a Nullstellensatz derivation of the polynomial x P i − − x P i .The sum of all these polynomials is a telescoping sum, and is therefore equal to x P − x P t ′ = 1 − x P t ′ . (3.2)We will then transform this sum into a Nullstellensatz refutation by adding the polynomial x P t ′ = A sink · x P t ′ −{ z } . (3.3)7 ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING
We turn to constructing the aforementioned derivations. To this end, for every i ∈ [ t ′ ] , let v i ∈ V denote the vertex which was pebbled or unpebbled during the i -th step, i.e., during the transition from P i − to P i . Then, we know that in both configurations P i − and P i the predecessors of v i have pebbles on them,i.e., pred( v ) ⊆ P i − , P i . Let us denote by R i = P i − { v i } − pred( v i ) the set of other vertices that havepebbles during the i -th step. Finally, let p i be a number that equals to if v i was pebbled during the i -thstep, and equals to − if v i was unpebbled. Now, observe that x P i − − x P i = p i · x P i − (1 − x v i ) = p i · x R i A v i , (3.4)where the last step follows since the predecessors of v i are necessarily in P i − . Therefore, our finalrefutation for φ is t ′ X i =1 A v i · p i · x R i + A sink · x P t ′ −{ z } = x P t ′ + t ′ X i =1 x P i − − x P i = x P t ′ + ( x P − x P t ′ ) (3.5) = x P t ′ + (1 − x P t ′ ) = 1 . Note, in fact, it is a multilinear Nullstellensatz refutation, since it contains only multilinear monomialsand does not use the Boolean axioms. It remains to analyze its degree and size.For the degree, observe that every monomial in the proof is of the form x P i , and the degree of eachsuch monomial is exactly the number of pebbles used in the corresponding configuration. It follows thatthe maximal degree is exactly the space of the pebbling strategy P .Let us turn to considering the size. Observe that for each of the configurations P , . . . , P t ′ , the refuta-tion contains exactly two monomials: for all i ∈ [ t ′ − , one monomial for P i is generated in the i -th step,and another in the ( i + 1) -th step, and for the configuration P t ′ the second monomial is generated whenwe add A sink · x P t ′ −{ z } . In addition, the refutation contains exactly one monomial for the configuration P ,which is generated in the first step. Hence, the total number of monomials generated in the refutation isexactly · t ′ + 1 = t + 1 , as required. We turn to prove the “if” direction of Theorem 3.1. We note that it suffices to prove it for multilinearNullstellensatz refutations, since every standard Nullstellensatz refutation implies the existence of a mul-tilinear one with at most the same size and degree. Let X v ∈ V A v · Q v + A sink · Q sink = 1 (3.6)be a multilinear Nullstellensatz refutation of φ over F of degree s . We use this refutation to construct areversible pebbling strategy P for G .To this end, we construct a “configuration graph” C , whose vertices consist of all possible configura-tions of at most s pebbles on G (i.e., the vertices will be all subsets of V of size at most s ). The edges of C will be determined by the polynomials Q v of the refutation, and every edge { U , U } in C will constitutea legal move in the reversible pebbling game (i.e., it will be legal to move from U to U and vice versa).We will show that C contains a path from the empty configuration ∅ to a configuration U z that containsthe sink z , and our pebbling strategy will be generated by walking on this path from ∅ to U z and back.The edges of the configuration graph C are defined as follows: Let v ∈ V be a vertex of G , and let q bea monomial of Q v that does not contain x v . Let W ⊆ V be the set of vertices such that q = x W (such aset W exists since the refutation is multilinear). Then, we put an edge e q in C that connects W ∪ pred( v ) and W ∪ pred( v ) ∪ { v } (we allow parallel edges). It is easy to see that the edge e q connects configurationsof size at most s , and that it indeed constitutes a legal move in the reversible pebbling game. We notethat C is a bipartite graph: to see it, note that every edge e q connects a configuration of an odd size to aconfiguration of an even size. 8 Reversible Pebblings and Nullstellensatz Refutations
For the sake of the analysis, we assign the edge e q a weight in F that is equal to coefficient of q in Q v .We define the weight of a configuration U to be the sum of the weights of all the edges that touch U (where the addition is done in the field F ). We use the following technical claim, which we prove at theend of this section. Claim 3.2.
Let U ⊆ V be a configuration in C that does not contain the sink z . If U is empty, then itsweight is . Otherwise, its weight is .We now show how to construct the required pebbling strategy P for G . To this end, we first prove thatthere is a path in C from the empty configuration to a configuration that contains the sink z . Suppose forthe sake of contradiction that this is not the case, and let H be the connected component of C that containsthe empty configuration. Our assumption says that none of the configurations in H contains z .The connected component H is bipartite since C is bipartite. Without loss of generality, assume thatthe empty configuration is in the left-hand side of H . Clearly, the sum of the weights of the configurationson the left-hand side should be equal to the corresponding sum on the right-hand side, since they are bothequal to the sum of the weights of the edges in H . However, the sum of the weights of the configurationson the right-hand side is (since all these weights are by Claim 3.2), while the sum of the weights ofthe left-hand side is (again, by Claim 3.2). We reached a contradiction, and therefore H must containsome configuration U z that contains the sink z .Next, let ∅ = P , P , . . . , P t ′ = U z be a path from the empty configuration to U z . Our reversiblepebbling strategy for G is P = ( P , . . . , P t ′ − , P t ′ , P t ′ − , . . . , P ) . (3.7)This is a legal pebbling strategy since, as noted above, every edge of C constitutes a legal move of thereversible pebbling game. The strategy P uses space s , since all the configurations in C contain at most s pebbles by definition. The time of P is t = 2 · t ′ . It therefore remains to show that the size of theNullstellensatz refutation from Equation 3.6 is at least t + 1 .To this end, note that every edge e q in the path corresponds to some monomial q in some polyno-mial Q v . When the monomial q is multiplied by the axiom A v , it generates two monomials in the proof:the monomial q · x pred( v ) and the monomial q · x pred( v ) · x v . Hence, the Nullstellensatz refutation containsat least · t ′ monomials that correspond to edges from the path. In addition, the product A sink · Q sink mustcontains at least one monomial, since the refutation must use the sink axiom A sink (because φ without thisaxiom is not a contradiction). It follows that the refutation contains at least · t ′ + 1 = t + 1 monomials,as required. We conclude this section by proving Claim 3.2. Proof of Claim 3.2.
We start by introducing some terminology. First, observe that a monomial m may begenerated multiple times in the refutation of Equation 3.6, and we refer to each time it is generated as an occurrence of m . We say that an occurrence of m is generated by a monomial q v of Q v if it is generatedby the product A v · q v . Throughout the proof, we identify a configuration U with the monomial x U .We first prove the claim for the non-empty case. Let U ⊆ V be a non-empty configuration. Wewould like to prove the weight of U is . Recall that the weight of U is the sum of the coefficients ofthe occurrences of U that are generated by monomials q v that do not contain the corresponding vertex v .Observe that Equation 3.6 implies that the sum of the coefficients of all the occurrences of U is : thecoefficient of U on the right-hand side is , and it must be equal to the coefficient of U on the left-handside, which is the sum of the coefficients of all the occurrences.To complete the proof, we argue that every monomial q v that does contain the vertex v contributes to that sum. Let q v be a monomial of Q v that contains the vertex v and generates an occurrence of U . Let α be the coefficient of q . Then, it must hold that A v · q v = x pred( v ) · q v − x v · x pred( v ) · q v = x pred( v ) · q v − x pred( v ) · q v (3.8) = α · x U − α · x U , ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING where the second equality holds since we q v contains v and we are working with a multilinear refutation,and the third equality holds since we assumed that q v generates an occurrence of U . It follows that q v generates two occurrences of U , one with coefficient α and one with coefficient − α , and therefore itcontributes to the sum of the coefficients of all the occurrences of U .We have shown that the sum of the coefficients of all the occurrences of U is , and that the occurrencesgenerated by monomials q v that contain v contribute to this sum, and therefore the sum of coefficientsof occurrences that are generated by monomials q v that do not contain v must be , as required. In thecase that U is the empty configuration, the proof is identical, except that the sum of the coefficients of alloccurrences is , since the coefficient of ∅ is on the right hand side of Equation 3.6. In this section we prove Nullstellensatz refutation size-degree trade-offs for different degree regimes. Letus first recall what is known with regards to degree and size. In what follows, a Nullstellensatz refutationof a CNF formula F refers to a Nullstellensatz refutation of the translation of F to polynomials. Asmentioned in the introduction, if a CNF formula over n variables can be refuted in degree d then it canbe refuted in simultaneous degree d and size n O( d ) . However, for Nullstellensatz it is not the case thatstrong enough degree lower bounds imply size lower bounds.A natural question is whether for any given function d ( n ) there is a family of CNF formulas { F n } ∞ n =1 of size Θ( n ) such that1. F n has a Nullstellensatz refutation d ( n ) ;2. F n has a Nullstellensatz refutation of (close to) linear size and degree d ( n ) ≫ d ( n ) ;3. Any Nullstellensatz refutation of F n in degree only slightly below d ( n ) must have size nearly n d ( n ) .We present explicit constructions of formulas providing such trade-offs in several different parameterregimes. We first show that there are formulas that require exponential size in Nullstellensatz if the degreeis bounded by some polynomial function, but if we allow slightly larger degree there is a nearly linearsize proof. Theorem 4.1.
There is a family of explicitly constructible unsatisfiable -CNF formulas { F n } ∞ n =1 of size Θ( n ) such that:1. There is a Nullstellensatz refutation of F n in degree d = O (cid:0) √ n log n (cid:1) .2. For any constant ǫ > , there is a Nullstellensatz refutation of F n of size O( n ǫ ) and degree d = O (cid:0) d · √ n (cid:1) = O (cid:0) √ n log n (cid:1) .
3. There exists a constant
K > such that any Nullstellensatz refutation of F n in degree at most d = Kd / log n = O (cid:0) √ n (cid:1) must have size (cid:0) √ n (cid:1) ! . We also analyse a family of formulas that can be refuted in close to logarithmic degree and show thateven if we allow up to a certain polynomial degree, the Nullstellensatz size required is superpolynomial.
Theorem 4.2.
Let δ > be an arbitrarily small positive constant and let g ( n ) be any arbitrarily slowlygrowing monotone function ω (1) = g ( n ) ≤ n / . Then there is a family of explicitly constructibleunsatisfiable -CNF formulas { F n } ∞ n =1 of size Θ( n ) such that:1. There is a Nullstellensatz refutation of F n in degree d = g ( n ) log( n ) .2. For any constant ǫ > , there is a Nullstellensatz refutation of F n of size O( n ǫ ) and degree d = O (cid:0) d · n / /g ( n ) (cid:1) = O (cid:0) n / log n/g ( n ) (cid:1) . Nullstellensatz Trade-offs from Reversible Pebbling3. Any Nullstellensatz refutation of F n in degree at most d = O (cid:0) d /n δ log n (cid:1) = O (cid:0) n / − δ /g ( n ) (cid:1) must have size superpolynomial in n . Still in the small-degree regime, we present a very robust trade-off in the sense that superpolynomialsize lower bound holds for degree from log ( n ) to n/ log( n ) . Theorem 4.3.
There is a family of explicitly constructible unsatisfiable -CNF formulas { F n } ∞ n =1 of size Θ( n ) such that:1. There is a Nullstellensatz refutation of F n in degree d = O(log n ) .2. For any constant δ > , there is a Nullstellensatz refutation of F n of size O( n ) and degree d = O( d · n/ log − δ n ) = O( n/ log − δ n ) .
3. There exists a constant
K > such that any Nullstellensatz refutation of F n in degree at most d = Kd / log δ n = O ( n/ log n ) must have size n Ω(log log n ) . Finally, we study a family of formulas that have Nullstellensatz refutation of quadratic size and thatpresent a smooth size-degree trade-off.
Theorem 4.4.
There is a family of explicitly constructible unsatisfiable -CNF formulas { F n } ∞ n =1 of size Θ( n ) such that any Nullstellensatz refutation of F n that optimizes size given degree constraint d = n Θ(1) (and less than n ) has size Θ (cid:0) n /d (cid:1) . We prove these results by obtaining the analogous time-space trade-offs for reversible pebbling andthen applying the tight correspondence between size and degree in Nullstellensatz and time and space inreversible pebbling.
Our strategy for proving reversible pebbling trade-offs will be to analyse standard pebbling trade-offs.Clearly lower bounds from standard pebbling transfer to reversible pebbling; the next theorem showshow, in a limited sense, we can also transfer upper bounds . It is based on a reversible simulation ofirreversible computation proposed by [Ben89] and analysed precisely in [LS90].
Theorem 4.5 ([Ben89, LS90]).
Let G be an arbitrary DAG and suppose G can be pebbled (in the standardway) using s pebbles in time t ≥ s . Then for any ǫ > , G can be reversibly pebbled in time t ǫ /s ǫ using ǫ (2 /ǫ − s log( t/s ) pebbles. We also use the following general proposition, which allows upper bounding the reversible pebblingprice of a graph by using its depth and maximum in-degree.
Proposition 4.6.
Any DAG with maximum indegree ℓ and depth d has a persistent reversible pebblingstrategy in space at most dℓ + 1 .Proof. We will use the fact that if a graph has a persistent reversible strategy in space s then it has avisiting reversible strategy in space s .The proof is by induction on the depth. For d = 0 we can clearly persistently reversibly pebble thegraph with pebble. For d ≥ , we persistently reversibly pebble all but one of the (that is, at most ℓ − )immediate predecessors of the sink one at a time. By the induction hypothesis, this can be done with atmost ℓ − d − ℓ + 1 = dℓ − pebbles. At this point there are at most ℓ − predecessors of the sinkwhich are pebbled and no other pebbles on the graph. Let v be the only non-pebbled predecessor of thesink. We do a visiting reversible pebbling of v until a pebble is placed on v . We now pebble the sink andthen reverse the visiting pebbling of v until the subtree rooted at v has no pebbles on it. By the inductionhypothesis, this can be done with at most ℓ + ( d − ℓ + 1 = dℓ + 1 pebbles. All that is left to do is to toremove the ℓ − pebbles which are on predecessors of the sink. Again by the induction hypothesis, thiscan be done with ℓ + ( d − ℓ + 1 pebbles. 11 ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING
The first family of graphs for which we present reversible pebbling trade-offs consists of the so-calledCarlson-Savage graphs, which are illustrated in Figure 2 and are defined as follows.
Definition 4.7 (Carlson-Savage graph [CS80, CS82, Nor12]).
The two-parameter graph family Γ( c, r ) ,for c, r ∈ N + , is defined by induction over r . The base case Γ( c, is a DAG consisting of two sources s , s and c sinks γ , . . . , γ c with directed edges ( s i , γ j ) , for i = 1 , and j = 1 , . . . , c , i.e., edges fromboth sources to all sinks. The graph Γ( c, r + 1) has c sinks and is built from the following components: • c disjoint copies Π (1) r , . . . , Π ( c ) r of a pyramid graph of height r . • one copy of Γ( c, r ) . • c disjoint and identical line graphs called spines , where each spine is composed of r sections , andevery section contains c vertices.The above components are connected as follows: In every section of every spine, each of the first c vertices has an incoming edge from the sink of one of the first c pyramids, and each of the last c verticeshas an incoming edge from one of the sinks of Γ( c, r ) (where different vertices in the same section areconnected to different sinks).Note that Γ( c, r ) has multiple sinks. We define a (reversible) pebbling of a multi-sink graph to be a(reversible) pebbling that places pebbles on each sink at some point (the pebbles do not need to be presentin the last configuration). Let Γ ′ ( c, r ) be the single-sink subgraph of Γ( c, r ) consisting of all vertices thatreach the first sink of Γ( c, r ) . Since all sinks are symmetric, pebbling Γ ′ ( c, r ) and pebbling Γ( c, r ) arealmost equivalent. Proposition 4.8.
Any (reversible) pebbling P of Γ( c, r ) induces a (reversible) pebbling P ′ of Γ ′ ( c, r ) inat most the same space and the same time. From any (reversible) pebbling P ′ of Γ ′ ( c, r ) we can obtain(reversible) pebbling P of Γ( c, r ) by (reversibly) pebbling each sink of Γ( c, r ) one at a time, that is,simulating P ′ c times, once for each sink. Note that space ( P ) = space ( P ′ ) and time ( P ) = c · time ( P ′ ) . Carlson and Savage proved the following properties of this graph.
Lemma 4.9 ([CS82]).
The graphs Γ( c, r ) are of size Θ (cid:0) cr + c r (cid:1) , have in-degree , and have standardpebbling price r + 2 . Theorem 4.10 ([CS82]).
Suppose that P is a standard pebbling of Γ( c, r ) in space less than ( r + 2) + s for < s ≤ c − . Then time ( P ) ≥ (cid:18) c − ss + 1 (cid:19) r · r ! . This lower bound holds for space up to c + r − . By allowing only a constant factor more pebbles itis possible to pebble the graph (in the standard way) in linear time. Lemma 4.11 ([Nor12]).
The graphs Γ( c, r ) have standard pebbling strategies in simultaneous space O( c + r ) and time linear in the size of the graphs. The standard pebbling price upper bound does not carry over to reversible pebbling because the linegraph requires more pebbles in reversible pebbling than in standard pebbling. However, we can adapt thestandard pebbling strategy to reversible pebbling using the following fact on the line graph.
Proposition 4.12 ([LV96]).
The visiting reversible pebbling price of the line graph on n vertices is ⌈ log( n + 1) ⌉ and the persistent reversible pebbling price is ⌊ log( n − ⌋ + 2 . Using this result, we get the following upper bound (which is slightly stronger then what we wouldget by applying Theorem 4.5). 12
Nullstellensatz Trade-offs from Reversible Pebbling z γ z γ z γ Π (1) r Π (2) r Π (3) r Γ(3 , r ) Figure 2:
Inductive definition of Carlson-Savage graph
Γ(3 , r + 1) with spines and sinks. ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING
Lemma 4.13.
The graphs Γ( c, r ) have reversible pebbling price at most r (log( cr ) + 3) .Proof. The proof is by induction on r . Clearly, Γ( c, can be reversibly pebbled with pebbles. Inorder to pebble any sink of Γ( c, r ) , we can reversibly pebble the corresponding spine with the space-efficient strategy for reversibly pebbling a line graph (as per Proposition 4.12) and every time we need toplace or remove a pebble on a given vertex of the spine, we reversibly pebble the subgraph that reachesthis vertex. By Proposition 4.6, pyramids of depth r − can be reversibly pebbled with r −
1) + 1 pebbles. Therefore, by induction on r we get that the reversible pebbling price of Γ( c, r ) is at most max { ( r − cr ) + 3) , r −
1) + 1 } + log(2 cr ) + 2 ≤ ( r − cr ) + 3) + log( cr ) + 3 .In order to obtain nearly linear time reversible pebbling, we apply Theorem 4.5 to Lemma 4.11. Lemma 4.14.
For any ǫ > , the graphs Γ( c, r ) have reversible pebbling strategies in simultaneousspace O( ǫ /ǫ ( c + r ) log( cr )) and time O( n ǫ ) (where n denotes the number of vertices). We can now choose different values for the parameters c and r and obtain graphs with trade-offs indifferent space regimes. The first family of graphs we consider are those that exhibit exponential timelower-bounds. Theorem 4.15.
There are explicitly constructible families of single-sink DAGs { G n } ∞ n =1 of size Θ( n ) and maximum in-degree such that:1. The graph G n has reversible pebbling price s = O (cid:0) √ n log n (cid:1) .2. For any constant ǫ > , there is a reversible pebbling of G n in time O( n ǫ ) and space s = O (cid:0) s · √ n (cid:1) = O (cid:0) √ n log n (cid:1) .
3. There is a constant
K > such that any standard pebbling of G n in space at most s = Ks log n = O (cid:0) √ n (cid:1) must take time at least (cid:0) √ n (cid:1) ! .Proof. Let G n be the single-sink subgraph of Γ( c ( n ) , r ( n )) consisting of all vertices that reach the firstsink, for c ( n ) = √ n and r ( n ) = √ n .By Lemma 4.9, G n has Θ( n ) vertices and by Proposition 4.8, items 1–3 follow from Lemma 4.13,Lemma 4.14 and Theorem 4.10, respectively.Given Theorem 3.1 which proves the tight correspondence between reversible pebbling and Nullstel-lensatz refutations, Theorem 4.1 follow from Theorem 4.15.It is also interesting to consider families of graphs that can be reversibly pebbled in very small space,close to the logarithmic lower bound on the number of pebbles required to reversibly pebble a single-sinkDAG. In this small-space regime, we cannot expect exponential time lower bounds, but we can still obtainsuperpolynomial ones. Theorem 4.16.
Let δ > be an arbitrarily small positive constant and let g ( n ) be any arbitrarily slowlygrowing monotone function ω (1) = g ( n ) ≤ n / . Then there is a family of explicitly constructiblesingle-sink DAGs { G n } ∞ n =1 of size Θ( n ) and maximum in-degree such that:1. The graph G n has reversible pebbling price s ≤ g ( n ) log( n ) .2. For any constant ǫ > , there is a reversible pebbling of G n in time O( n ǫ ) and space s = O (cid:0) s · n / /g ( n ) (cid:1) = O (cid:0) n / log n/g ( n ) (cid:1) . Nullstellensatz Trade-offs from Reversible Pebbling3. Any standard pebbling of G n in space at most s = O (cid:0) s /n δ log n (cid:1) = O (cid:0) n / − δ /g ( n ) (cid:1) requires time superpolynomial in n .Proof. The proof is analogous to that of Theorem 4.16 with parameters r ( n ) = g ( n ) and c ( n ) = n / /g ( n ) .By applying Theorem 3.1 to the above result we obtain Theorem 4.2. Remark 4.17.
We note that in the second items of both the foregoing theorems, we could have reducedthe time of the reversible pebbling to O( n ) by applying Theorem 4.5 with ǫ = O (cid:16) n (cid:17) . Thiswould have come at a cost of an extra logarithmic factor in the corresponding space bounds. Lengauer and Tarjan [LT82] studied robust superpolynomial trade-offs for standard pebbling and showedthat there are graphs that have standard pebbling price
O(log n ) , but that any standard pebbling in spaceup to Kn/ log n , for some constant K , requires superpolynomial time. For reversible pebbling we getalmost the same result for the same family of graphs. Theorem 4.18.
There are explicitly constructible families of single-sink DAGs { G n } ∞ n =1 of size Θ( n ) and maximum in-degree such that:1. The graph G n has reversible pebbling price s = O(log n ) .2. For any constant δ > , there is a reversible pebbling of G n in time O( n ) and space s = O( s · n/ log − δ n ) = O( n/ log − δ n ) .
3. There exists a constant
K > such that any standard pebbling P n of G n using at most pebbles s = Ks log δ n = O ( n/ log n ) requires time n Ω(log log n ) . Note that, together with Theorem 3.1, this implies Theorem 4.3. Now in order to prove this theoremwe must first introduce the family of graphs we will consider.
Definition 4.19 (Superconcentrator [Val75]).
A directed acyclic graph G is an m -superconcentrator ifit has m sources S = { s , . . . , s m } , m sinks Z = { z , . . . , z m } , and for any subsets S ′ and Z ′ of sourcesand sinks of size (cid:12)(cid:12) S ′ (cid:12)(cid:12) = (cid:12)(cid:12) Z ′ (cid:12)(cid:12) = ℓ it holds that there are ℓ vertex-disjoint paths between S ′ and Z ′ in G .Pippenger [Pip77] proved that there are superconcentrators of linear size and logarithmic depth, andGabber and Galil [GG81] gave the first explicit construction. For concreteness, we will consider theexplicit construction by Alon and Capalbo [AC03] which has better parameters. Theorem 4.20 ([AC03]).
For all integers k ≥ , there are explicitly constructible m -superconcentratorsfor m = O(2 k ) with O( m ) edges, depth O(log m ) , and maximum indegree O(1) . It is easy to see that we can modify these superconcentrators so that the maximum indegree is bysubstituting each vertex with indegree δ > by a binary tree with δ leafs. Note that this only increase thesize and the depth by constant factors. Corollary 4.21.
There are m -superconcentrators with O( m ) vertices, maximum indegree and depth O(log m ) . ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING
Figure 3:
A bit-reversal permutation graph.
Given an m -superconcentrator G m , we define a stack of r superconcentrators G m to be r disjointcopies of G m where each sink of the i th copy is connected to a different source of the i + 1 st copy for i ∈ [ r − . Since we want single-sink DAGs, we add a binary tree with m leafs and depth ⌈ log m ⌉ , andconnect each sink of the r th copy of G m to a different leaf of the tree. Lengauer and Tarjan [LT82] provedthe following theorem for stacks of superconcentrators. Theorem 4.22 ([LT82]).
Let Φ( m, r ) denote a stack of r (explicitly constructible) linear-size m -super-concentrator with bounded indegree and depth log m . Then the following holds:1. The standard pebbling price of Φ( m, r ) is O( r log m ) .2. There is a linear-time standard pebbling strategy P for Φ( m, r ) with space ( P ) = O( m ) .3. If P is a standard pebbling strategy for Φ( m, r ) in space s ≤ m/ , then time ( P ) ≥ m · (cid:0) rm s (cid:1) r . With this result in hand we can now proceed to prove Theorem 4.18.
Proof of Theorem 4.18.
Let G n be a stack of log n linear-size ( n/ log n ) -superconcentrators as per Corol-lary 4.21. Note that G n has Θ( n ) vertices, indegree and depth O(log n ) . By Proposition 4.6 we havethat G n can be reversibly pebbled with O(log n ) pebbles (proving item 1).By item 2 in Theorem 4.22 and by choosing ǫ = 1 / ( δ log log n ) in Theorem 4.5 we conclude that G n can be reversibly pebbled in simultaneous time O (cid:0) n /δ (cid:1) and space O (cid:0) n/ ( δ log − δ n ) (cid:1) , from whichitem 2 follows. Finally, item 3 in the theorem follows from item 3 in Theorem 4.22. Another family of graphs that has been studied in the context of standard pebbling trade-offs is that ofpermutation graphs.
Definition 4.23.
Given a permutation σ ∈ S ([ n ]) , the permutation graph G ( σ ) consists of two lines ( x , . . . , x n ) (the bottom line ) and ( y , . . . , y n ) ( top line ) which are connected as follows: for every ≤ i ≤ n , there is an edge from x i to y σ ( i ) .Lengauer and Tarjan [LT82] proved that permutation graphs present the following smooth trade-offwhen instantiated with the permutation that reverses the binary representation of the index i (see Figure 3for an illustration). Theorem 4.24 ([LT82]).
Let G be a bit-reversal permutation graph on n vertices. For any ≤ s ≤ n ,there is a standard pebbling in space s and time O (cid:0) n /s (cid:1) . Moreover, any standard pebbling P in space s satisfies time ( P ) = Ω (cid:0) n /s (cid:1) . We show that these graphs also present a smooth reversible pebbling trade-off and, in particular, for s = n Θ(1) and s ≤ n , any reversible pebbling P in space s satisfies time ( P n ) = Ω (cid:0) n /s (cid:1) and there arematching upper bounds. To this end, we use the following proposition.16 Concluding Remarks
Proposition 4.25.
For every natural number k , the line graph over n vertices can be reversibly pebbledusing k · n /k pebbles in time k · n .Proof. Observe that the line graph over n can be pebbled (in the standard way) using pebbles in time n . The proposition follows now by applying Theorem 4.5 with ǫ = k/ log( n ) .Using Proposition 4.25, we obtain the following result. Theorem 4.26.
Let G n be a bit-reversal permutation graph on n vertices. Then G n satisfies the followingproperties:1. The reversible pebbling price of G n is at most n + 2 .2. If s satisfies n ≤ s ≤ n and k is such that s = 4 kn /k , then there is a reversible strategy insimultaneous space s and time O (cid:0) k k · n /s (cid:1) .3. Any standard pebbling P n of G n satisfies time ( P n ) = Ω (cid:0) n / space ( P n ) (cid:1) .Proof. The upper bounds (item 1 and item 2) hold for any permutation graph.For item 1, we can simulate a reversible pebbling of the top line that uses space at most log n + 1 (asper Proposition 4.12), and every time we need a pebble on a vertex v of the bottom line in order to placeor remove a pebble on the top line, we reversibly pebble the bottom line until v is pebbled (can be donewith log n + 1 pebbles), make the move on the top line, and then unpebble the bottom line.To obtain item 2, we consider a two stage strategy. In the first phase, we place n /k pebbles spacedequally apart on the bottom line. We refer to these pebbles as fixed pebbles, since they will remain onthe graph until the sink is pebbled. In the second phase, we simulate a reversible pebbling on the top linewith kn /k pebbles and every time we need a pebble on a vertex v on the bottom line to make a moveon the top line, we reversibly pebble v (with k − n /k pebbles) from the nearest fixed pebble, makethe move on the top line, and then unpebble the segment on the bottom line.All that is left to show is that this can be done within the space budget of kn /k in time O(2 k · n /s ) .For the first phase, we reversibly pebble n /k segments of length m = n − /k . By Proposition 4.25,each of the segments can be reversibly pebbled using k − n /k = 2( k − m k − pebbles in time k − n − /k . Since every segment must be pebbled and then unpebbled, the total time for the first phaseis · k − n − /k · n /k = 2 k n , and the total number of pebbles used is less than kn /k : n /k for thefixed pebbles and k − n /k for pebbling each segment.We turn to analyze the second phase. By Proposition 4.25, the top line can be reversibly pebbled insimultaneous space kn /k and time k n . For each move in the top line, we need to pebble and unpebble asegment of length at most n − /k . As argued before, this can be done in simultaneous space k − n /k and time · k − n − /k . Therefore, at any point in the pebbling strategy there are at most kn /k pebbleson the bottom line and at most kn /k pebbles on the top line, and the total time of the pebbling is at most k n + 2 k n − /k ≤ k k n /s .Finally, item 3 follows from the standard pebbling lower bound in Theorem 4.24.From Theorem 4.26 we obtain the following corollary that, together with Theorem 3.1, implies The-orem 4.4. Corollary 4.27.
Any reversible pebbling strategy P n for G n that optimizes time given space constraint n Θ(1) (and less than n ) exhibits a trade-off time ( P n ) = Θ (cid:0) n / space ( P n ) (cid:1) . In this paper we prove that size and degree of Nullstellensatz refutations in any field of pebbling formulasare exactly captured by time and space of the reversible pebble game on the underlying graph. This allowsus to prove a number of strong size-degree trade-offs for Nullstellensatz. To the best of our understandingno such results have been known previously. 17
ULLSTELLENSATZ SIZE-DEGREE TRADE-OFFS FROM REVERSIBLE PEBBLING
The most obvious, and also most interesting, open question is whether there are also size-degreetrade-offs for the stronger polynomial calculus proof system. Such trade-offs cannot be exhibited by thepebbling formulas considered in this work, since such formulas have small-size low-degree polynomialcalculus refutations, but the formulas exhibiting size-width trade-offs for resolution [Tha16] appear to benatural candidates. Another interesting question is whether the tight relation between Nullstellensatz and reversible peb-bling could make it possible to prove even sharper trade-offs for size versus degree in Nullstellensatz,where just a small constant drop in the degree would lead to an exponential blow-up in size. Such resultsfor pebbling time versus space are known for the standard pebble game, e.g., in [GLT80]. It is conceivablethat a similar idea could be applied to the reversible pebbling reductions in [CLNV15], but it is not ob-vious whether just adding a small amount of space makes it possible to carry out the reversible pebblingtime-efficiently enough.Finally, it can be noted that our results crucially depend on that we are in a setting with variablesonly for positive literals. For polynomial calculus it is quite common to consider the stronger settingwith “twin variables” for negated literals (as in the generalization of polynomial calculus in [CEI96] to polynomial calculus resolution in [ABRW02]). It would be nice to generalize our size-degree trade-offsfor Nullstellensatz to this setting, but it seems that some additional ideas are needed to make this work.
Acknowledgements
We are grateful for many interesting discussions about matters pebbling-related (and not-so-pebbling-related) with Arkadev Chattopadhyay, Toniann Pitassi, and Marc Vinyals.This work was mostly carried out while the authors were visiting the Simons Institute for the Theory ofComputing in association with the DIMACS/Simons Collaboration on Lower Bounds in ComputationalComplexity, which is conducted with support from the National Science Foundation.Or Meir was supported by the Israel Science Foundation (grant No. 1445/16). Robert Robere wassupported by NSERC, and also conducted part of this work at DIMACS with support from the NationalScience Foundation under grant number CCF-1445755. Susanna F. de Rezende and Jakob Nordstr¨omwere supported by the
Knut and Alice Wallenberg grant KAW 2016.0066
Approximation and Proof Com-plexity . Jakob Nordstr¨om was also supported by the Swedish Research Council grants 621-2012-5645and 2016-00782.
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