Lifting theorems in complexity theory are a method of transferring lower bounds in a weak computational model into lower bounds for a more powerful computational model, via function composition. There has been an explosion of lifting theorems in the last ten years, essentially reducing communication lower bounds to query complexity lower bounds.
Early papers that establish lifting theorems include Raz and McKenzie’s separation of the monotone NC hierarchy [RM99Separation] (by lifting decision tree complexity to deterministic communication complexity), and Sherstov’s pattern matrix method [Sherstov11Pattern] which lifts (approximate) polynomial degree to (approximate) matrix rank. Recent work has established query-to-communication lifting theorems in a variety of models, leading to the resolution of many longstanding open problems in many areas of computer science. Some examples include the resolution of open questions in communication complexity [GPW15Deterministic, GLMWZ15Rectangles, GKPW17QueryToCommunicationPtoNP, GJPW17Randomized, GPW18Landscape], monotone complexity [RPRC16Exponential, PR17StronglyExponential, PR18LiftingNS], proof complexity [HN12VirtueSuccinctProofs, GP18CommunicationLowerBounds, dRNV16LimitedInteraction, GGKS18MonotoneCircuit], extension complexity of linear and semidefinite programs [KMR17ApproximatingRectangles, GJW18ExtensionComplexity, LRS15LowerBoundsSDP], data structures [CKLM18Simulation] and finite model theory [BN16QuantifierDepth].
Lifting theorems have the following form: given functions (the “outer function”) and (the “gadget”), a lower bound for in a weak computational model implies a lower bound on in a stronger computational model. The most desirable lifting theorems are the most general ones. First, it should hold for any outer function, and ideally should be allowed to be a partial function or a relation (i.e., a search problem). Indeed, nearly all of the applications mentioned above require lifting where the outer function is a relation or a partial function. Secondly, it is often desirable that the gadget is as small as possible. The most general lifting theorems established so far, for example lifting theorems for deterministic and randomized communication complexity, require at least logarithmically-sized gadgets; if these theorems could be improved generically to hold for constant-sized gadgets then many of the current theorems would be vastly improved. Some notable examples where constant-sized gadgets are possible include Sherstov’s degree-to-rank lifting [Sherstov11Pattern], critical block-sensitivity lifting [GP18CommunicationLowerBounds, HN12VirtueSuccinctProofs], and lifting for monotone span programs [PR17StronglyExponential, PR18LiftingNS, Robere18Thesis].
1.1 A New Lifting Theorem
In this paper, we generalize a lifting theorem of Pitassi and Robere [PR18LiftingNS] to use any gadget that has nontrivial rank. This theorem takes a search problem associated with an unsatisfiable CNF, and lifts a lower bound on the Nullstellensatz degree of the CNF to a lower bound on a related communication problem.
More specifically, let be an unsatisfiable -CNF formula. The search problem associated with , , takes as input an assignment to the underlying variables, and outputs a clause that is falsified by the assignments. [PR18LiftingNS] prove that for any unsatisfiable , and for a sufficiently rich gadget , deterministic communication complexity lower bounds for the composed search problem follow from Nullstellensatz degree lower bounds for .111In fact the result is quite a bit stronger—it applies to Razborov’s rank measure [Razborov90Applications], which is a strict strengthening of deterministic communication complexity. We significantly improve this lifting theorem so that it holds for any gadget of large enough rank.
Let be a CNF over variables, let be any field, and let be any gadget of rank at least . Then the deterministic communication complexity of is at least , the Nullstellensatz degree of , as long as for some large enough constant .
An important special case of our generalized theorem is when the gadget is the equality function. In this work, we apply our theorem to resolve two open problems in proof complexity and circuit complexity. Both solutions depend crucially on the ability to use the equality gadget.
We note that lifting with the equality gadget has recently been the focus of another paper. Loff and Mukhopadhyay [LM19Lifting] observed that a lifting theorem for total functions with the equality gadget can be proven using a rank argument. Surprisingly, they also observed that it is not possible to lift query complexity to communication complexity for arbitrary relations! Concretely, [LM19Lifting] give an example of a relation with linear query complexity but whose composition with equality has only polylogarithmic communication complexity. Nonetheless, they are able to prove a lifting theorem for general relations using the equality gadget by replacing standard query complexity with a stronger complexity measure (namely, the -query complexity of the relation).
Unfortunately, we cannot use either of the lifting theorems of [LM19Lifting] for our applications. Specifically, in our applications we lift a search problem (and therefore cannot use their result for total functions), and this search problem has small -query complexity (and therefore we cannot use their lifting theorem for general relations). Indeed, this shows that our lifting theorem is incomparable to the results of [LM19Lifting], even when specialized to the equality gadget. We note that our theorem, too, bypasses the impossibility result of [LM19Lifting] by using a stronger complexity measure, which in our case is the Nullstellensatz degree.
1.2 A Separation in Proof Complexity
The main application of our lifting theorem is the first separation in proof complexity between cutting planes proofs with high-weight versus low-weight coefficients. The cutting planes proof-system is a proof system that can be used to refute an unsatisfiable CNF by translating it into a system of integer inequalities and showing that this system has no integer solution. The latter is achieved by a sequence of steps that derive new integer inequalities from old ones, until we derive the inequality (which clearly has no solution). The efficiency of such a refutation is measured by its length (i.e., the number of steps) and its space (i.e., the maximal number of inequalities that have to be stored simultaneously during the derivation).
The standard variant of the cutting planes proof system, commonly denoted by CP, allows the inequalities to use coefficients of arbitrary size. However, it is also interesting to consider the variant in which the coefficients are polynomially bounded, which is commonly denoted by CP. This gives rise to the natural question of the relative power of CP vs. CP: are they polynomially equivalent or is there a super-polynomial length separation? This question appeared in [BC96CuttingPlanes] and remains stubbornly open to date. In this work we finally make progress by exhibiting a setting in which unbounded coefficients afford an exponential increase in proof power.
There is a family of CNF formulas of size that have cutting planes refutations of length and space , but for which any refutation in length and space with polynomially bounded coefficients must satisfy .
Our result is the first result in proof complexity demonstrating any situation where high-weight coefficients are more powerful than low-weight coefficients. In comparison, for computing Boolean functions, the relative power of high-weight and low-weight linear threshold functions has been understood for a long time. The greater-than function can be computed by high-weight threshold functions, but not by low-weight threshold functions, and weights of length polynomial in suffice [Muroga-book] for Boolean functions. For higher depth threshold formulas, it is known that depth- threshold formulas of high-weight can efficiently be computed by depth- threshold formulas of low-weight [Goldmann-threshold].
In contrast to our near-complete knowledge of high versus low weights for functions, almost nothing is known about the relative power of high versus low weights in the context of proof complexity. Buss and Clote [BC96CuttingPlanes], building on work by Cook, Coullard, and Turán [CCT87ComplexityCP], proved an analog of Muroga’s result for cutting planes, showing that weights of length polynomial in the length of the proof suffice. Quite remarkably, this result is not known to hold for other linear threshold proof systems: there is no nontrivial upper bound on the weights for more general linear threshold propositional proof systems (such as stabbing planes [BFIKPPR18Stabbing], and Krajíček’s threshold logic proof system [Krajicek95Frege] where one can additionally branch on linear threshold formulas). Prior to our result, there was no separation between high and low weights, for any linear threshold proof system.
1.3 A Separation in Circuit Complexity
A second application of our lifting theorem relates to monotone real circuits, which were introduced by Pudlák [Pudlak97LowerBounds]. A monotone real circuit is a generalization of monotone Boolean circuits where each gate is allowed to compute any non-decreasing real function of its inputs, but the inputs and output of the circuit are Boolean. A formula is a tree-like circuit, that is, every gate has fan-out one. The first (exponential) lower bound for monotone real circuits was proven already in [Pudlak97LowerBounds] by extending the lower bound for computing the clique-colouring function with monotone Boolean circuits [Razborov85LowerBounds, AB87Monotone]
. This lower bound, together with a generalization of the interpolation technique[Krajicek97Interpolation] which applied only to CP, was used by Pudlák to obtain the first exponential lower bounds for CP.
Shortly after monotone real circuits were introduced, there was an interest in understanding the power of monotone real computation in comparison to monotone Boolean computation. By extending techniques in [RM99Separation], Bonet et al. prove that there are functions with polynomial size monotone Boolean circuits that require monotone real formulas of exponential size [Johannsen98Lower, BEGJ00RelativeComplexity]. This illustrates the power of DAG-like computations in comparison to tree-like. In the other direction, we would like to know whether monotone real circuits are exponentially stronger than monotone Boolean circuits. Rosenbloom [Rosenbloom97Monotone] presented an elegant, simple proof that monotone real formulas are exponentially stronger than (even non-monotone) Boolean circuits, since slice functions can be computed by linear-size monotone real formulas, whereas by a counting argument we know that most slice functions require exponential size Boolean circuits.
The question of finding explicit functions that demonstrate that monotone real circuits are stronger than general Boolean circuits is much more challenging since it involves proving explicit lower bounds for Boolean circuits—a task that seems currently completely out of reach. A more tractable problem is that of finding explicit functions showing that monotone real circuits or formulas are stronger than monotone Boolean circuits or formulas, but prior to this work, no such separation was known either. We provide an explicit separation for monotone formulas, that is, we provide a family of explicit functions that can be computed with monotone real formulas of near-linear size but require exponential monotone Boolean formulas. This is the first explicit example that illustrates the strength of monotone real computation.
There is an explicit family of functions over variables that can be computed by monotone real formulas of size but for which every monotone Boolean formula requires size .
Another motivation for studying lifting theorems with simple gadgets, and in particular the equality gadget, are connections with proving non-monotone formula size lower bounds. As noted earlier, lifting theorems have been extremely successful in proving monotone circuit lower bounds, and it has also been shown to be useful in some computational settings that are only “partially” monotone; notably monotone span programs [RPRC16Exponential, PR17StronglyExponential, PR18LiftingNS] and extended formulations [GJW18ExtensionComplexity, KMR17ApproximatingRectangles].
This raises the question of to what extent lifting techniques can help prove non-monotone lower bounds. The beautiful work by Karchmer, Raz and Wigderson [KRW95Superlogarithmic] initiated such an approach for separating P from —this opened up a line of research popularly known as the KRW conjecture. Intriguingly, steps towards resolving the KRW conjecture are closely connected to proving lifting theorems for the equality gadget. The first major progress was made in [EIRS] where lower bounds for the universal relation game are proven, which is an important special case of the KRW conjecture. Their result was recently improved in several papers [GMWW17TowardBetter, HW90Composition, KM18ImprovedComposition], and Dinur and Meir [Dinur-Meir] gave a new top-down proof of the state-of-the-art formula-size lower bounds via the KRW approach.
The connection to lifting using the equality gadget is obtained by observing that the KRW conjecture involves communication problems in which Alice and Bob are looking for a bit on which they differ—this is exactly an equality problem. Close examination of the results in [EIRS, HW90Composition] show that they are equivalent to proving lower bounds for the search problem associated with the pebbling formula when lifted with a -bit equality gadget on a particular graph [Pitassi16note]. Our proof of Theorem 4.1 actually establishes near-optimal lower bounds on the communication complexity of the pebbling formula lifted with equality for any graph, but where the size of the equality is not 1. Thus if our main theorem could be improved with one-bit equality gadgets this would imply the results of [EIRS, HW90Composition] as a direct corollary and with significantly better parameters.
1.4 Overview of Techniques
We conclude this section by giving a brief overview of our techniques, also trying to convey some of the simplicity of the proofs which we believe is an extra virtue of these results.
In order to prove their lifting theorem, Pitassi and Robere [PR18LiftingNS] defined a notion of a “good” gadget. They then showed that if we compose a polynomial with a good gadget , the rank of the resulting matrix is determined exactly by the non-zero coefficients of and the rank of . Their lifting theorem follows by using this correspondence to obtain bounds on the ranks of certain matrices, which in turn yield the required communication complexity lower bound.
In this work, we observe that every gadget can be turned into a good gadget using a simple transformation. This observation allows us to get an approximate bound on the rank of for any with nontrivial rank. While the correspondence we get in this way is only an approximation and not an exact correspondence as in [PR18LiftingNS], it turns out that this approximation is sufficient to prove the required lower bounds. We thus get a lifting theorem that works for every gadget with sufficiently large rank.
Cutting planes separation
The crux of our separation between CP and CP is the following observation: CP can encode a conjunction of linear equalities with a single equality, by using exponentially large coefficients. This allows CP refutations to obtain a significant saving in space when working with linear equalities. This saving is not available to CP, and this difference between the proof systems allows the separation.
In order to exploit this observation, one of our main innovations is to concoct the separating formula. To do this, we must come up with a candidate formula that can only be refuted by reasoning about a large conjunction of linear equalities, to show that cutting planes (CP) can efficiently refute it, and to show that low-weight cutting planes (CP) cannot.
To find such a candidate formula family we resort to pebbling formulas which have played a major role in many proof complexity trade-off results. Interestingly, pebbling formulas have short resolution proofs that reason in terms of large conjunctions of literals. When we lift such formulas with the equality gadget this proof can be simulated in cutting planes by using the large coefficients to encode many equalities with a single equality. This yields cutting planes refutation of any pebbling formula in quadratic length and constant space.
On the other hand we prove our time-space lower bound showing that any CP refutation requires large length or large space for the same formulas. To prove this lower bound, the first step is to instantiate the connection in [HN12VirtueSuccinctProofs] linking time/space bounds for many proof systems to communication complexity lower bounds for lifted search problems. This connection means that we can obtain the desired CP-lower bounds for our formulas by proving communication complexity lower bounds for the corresponding lifted search problem .
In order to prove the latter communication lower bounds, we prove lower bounds on the Nullstellensatz degree of , and then invoke our new lifting theorem to translate them into communication lower bounds for . To show the Nullstellensatz lower bounds, we prove the following lemma, which establishes an equivalence between Nullstellsatz degree and the reversible pebbling price, and may be interesting in its own right. (We remark that connections between Nullstellensatz degree and pebbling were previously shown in [BCIP02Homogenization]; however their result was not tight.)
For any field and any directed acyclic graph the Nullstellensatz degree of is equal to the reversible pebbling price of .
We remark that due to known lower and upper bounds in query and proof complexity, this lemma immediately implies that Nullstellensatz degree coincides for (deterministic) decision tree and parity decision tree complexity. We record this here as a corollary, as it may be of independent interest, and provide its proof in Appendix C.
For any field and any directed acyclic graph , the Nullstellensatz degree over of , the decision tree depth of , and the parity decision tree depth of coincide and are equal to the reversible pebbling price of .
Using the above equivalence, we obtain near-linear Nullstellensatz degree refutations for a family of graphs with maximal pebbling price, which completes our time/space lower bound for CP. However, in order to separate CP and CP we require a very specific gadget and lifting theorem. Specifically, the gadget should be strong enough, so that lifting holds for deterministic communication complexity (which can efficiently simulate small time/space CP proofs), but on the other hand also weak enough, so that lifting does not hold for stronger communication models (randomized, real) that can efficiently compute high-weight inequalities. The reason that we are focusing on the equality gadget is that it hits this sweet spot—it requires large deterministic communication complexity, yet has short randomized protocols, and furthermore equalities can be represented with a single pair of inequalities.
Separation for monotone formulas
As was the case for the separation between CP and CP, to obtain a separation between monotone Boolean formulas and monotone real formulas we must find a function that has just the right level of hardness.
To obtain a size lower bound for monotone Boolean formulas we invoke the characterization of formula depth by communication complexity of the Karchmer–Wigderson game [KW90Monotone]. By choosing a function that has the same Karchmer–Wigderson game as the search problem of a lifted pebbling formula, we get a depth lower bound for monotone Boolean formulas from the communication lower bound of the search problem. Note that since monotone Boolean formulas can be balanced, a depth lower bound implies a size lower bound.
In the other direction, we would like to show that these functions are easy for real computation. Analogously to the Karchmer–Wigderson relation, it was shown in [HP18Note] that there is a correspondence between real DAG-like communication protocols (as defined in [Krajicek98Interpolation]) and monotone real circuits. Using this relation, a small monotone real circuit can be extracted from a short CP proof of the lifted pebbling formula. However, we would like to establish a monotone real formula upper bound. One way to achieve this is by finding small tree-like CP refutations of lifted pebbling formulas. The problem is that for many gadgets lifted pebbling formulas require exponentially long tree-like proofs. Nevertheless, for pebbling formulas lifted with the equality gadget we are able to exhibit a short semantic tree-like CP refutation, which via real communication yields a small monotone real formula.
1.5 Organization of This Paper
Section 2 contains formal definitions of concepts discussed above and some useful facts. Our main lifting theorem is proven in Section 3. Section 4 is devoted to proving our separation between high-weight and low-weight cutting planes. In Section 5 we prove the separation between monotone real and Boolean formulas. We conclude in Section 6 with some open problems.
In this section we review some background material from communication complexity and proof complexity.
2.1 Communication Complexity and Lifted Search Problems
Given a function , we denote by the function that takes as input independent instances of and applies to each of them separately. A total search problem is a relation such that for all there is an such that . Intuitively, represents the computational task in which we are given an input and would like to find an output that satisfies .
An important example of a search problem, which has proved to be very useful for proof complexity results, comes from unsatisfiable -CNF formulas. Given a -CNF formula over variables , the search problem takes as input an assignment and outputs a clause that is falsified by .
Given a search problem with a product input domain and a function , we define the composition in the natural way: if and only if . We remark that this composition notation extends naturally to functions: for instance, if is a function taking values in some field , for example, then the composition is a matrix over . Second, we remark that we will sometimes write instead of if is clear from context.
A communication search problem is a search problem with a bipartite input domain . A communication protocol for a search problem is a strategy for a collaborative game where two players Alice and Bob hold , respectively, and wish to output an such that while communicating as few bits as possible. Messages are sent sequentially until one player announces the answer and only depend on the input of one player and past messages. The cost of a protocol is the maximum number of bits sent over all inputs, and the communication complexity of a search problem, which we denote by , is the minimum cost over all protocols that solve . For more details on communication complexity, see, for instance, [KN97CommunicationCplx].
Given a CNF formula on variables and a Boolean function , we define a lifted formula as follows. For each variable of , we have new variables . For each clause we replace each literal or in by a CNF encoding of either or according to the sign of the literal. We then expand the resulting expression into a CNF, which we denote by , using de Morgan’s rules. The substituted formula is .
For the sake of an example, consider the clause , and we will substitute with the equality gadget on two bits. Formally, we replace with and with . We can encode a two-bit equality as the CNF formula
and a two-bit disequality as the CNF formula
So, in the clause , we would substitute for the CNF encoding of and with the CNF encoding of ; finally, we would convert the new formula to a CNF by distributing the top over the s from the new CNF encodings.
While is not the same problem as , we can reduce the former to the latter. Specifically, suppose we are given a protocol for . Consider the following protocol for : Given an input , the protocol interprets as an input to . Now, assume that outputs on a clause of , which was obtained from a clause of . Then, the clause is a valid on , so outputs it. Let us record this observation.
for any unsatisfiable CNF and any Boolean gadget .
As a proof system, Nullstellensatz allows verifying that a set of polynomials does not have a common root, and it can also be used to refute CNF formulas by converting them into polynomials. It plays an important role in our lower bounds.
Let be a field, and let be an unsatisfiable system of polynomial equations in . A Nullstellensatz refutation of is a sequence of polynomials such that where the equality is syntactic. The degree of the refutation is ; the Nullstellensatz degree of , denoted , is the minimum degree of any Nullstellensatz refutation of .
Let be an unsatisfiable CNF formula over Boolean variables . We introduce a standard encoding of each clause as a polynomial equation. If is a clause then let denote the set of variables occurring positively in and denote the set of variables occurring negatively in ; with this notation we can write From define the polynomial
over formal variables . Observe that is satisfied (over assignments to ) if and only if the corresponding assignment satisfies . We abuse notation and let and note that the second set of polynomial equations restricts the inputs to values. The -Nullstellensatz degree of , denoted , is the Nullstellensatz degree of refuting .
How do we know that a Nullstellensatz refutation always exists? One can deduce this from Hilbert’s Nullstellensatz, but for our purposes it is enough to use a simpler version proved by Buss et al. (Theorem 5.2 in [BIKPRS97ProofCplx]): if is a system of polynomial equations over with no solutions, then there exists a Nullstellensatz refutation of .
2.3 Cutting Planes
The Cutting planes (CP) proof system was introduced in [CCT87ComplexityCP]
as a formalization of the integer linear programming algorithm in[Gomory63AlgorithmIntegerSolutions, Chvatal73EdmondPolytopes]. Cutting planes proofs give a formal method to deduce new linear inequalities from old that are sound over integer solutions
—that is, if some integral vectorsatisfies a set of linear inequalities , then will also satisfy any inequality deduced from by a sequence of cutting planes deductions. The allowed deductions in a cutting planes proof are the following:
where , , , , , and are all integers and .
In order to use cutting planes to refute unsatisfiable CNF formulas, we need to translate clauses to inequalities. It is easy to see how to do this by example: we translate the clause to the inequality , or, equivalently, if we collect all constant terms on the right-hand side. For refuting CNF formulas we equip cutting planes proofs with the following additional rules ensuring all variables take values:
The goal, then, is to prove unsatisfiability by deriving the inequality . This is possible if and only if there is no -assignment satifying all constraints.
As discussed in the introduction, we are interested in several natural parameters of cutting planes proof—length, space, and the sizes of the coefficients. So, we define a cutting planes refutation as a sequence of configurations (this is also known as the blackboard model). A configuration is a set of linear inequalities with integer coefficients, and a sequence of configurations is a cutting planes refutation of a formula if , contains the contradiction , and each configuration follows from either by adding an inequality in , by adding the result of one of the above inference rules where all the premises are in , or by removing an inequality present in . The length of a refutation is then defined to be the number of configurations ; the space222Formally, this is known as the line space. is , the maximum number of inequalities in a configuration; and the coefficient bit size is the maximum size in bits of a coefficient that appears in the refutation.
For any proof system, it is natural to ask what is the minimal amount of space needed to prove tautologies. Indeed, there has been much work in the literature studying this, and for proof systems such as resolution (e.g. [ET01SpaceBounds, ABRW02SpaceComplexity, BG03SpaceComplexity, BN08ShortProofs]) and polynomial calculus (e.g. [ABRW02SpaceComplexity, FLNRT15SpaceCplx, BG15Framework, BBGHMW17SpaceProofCplx]) it is known that there are unsatisfiable CNF formulas which unconditionally require large space to refute. In contrast (and quite surprisingly!) it was shown in [GPT15SpaceComplexityCP] that for cutting planes proofs, constant line space is always enough. The proof presented in [GPT15SpaceComplexityCP] does use coefficients of exponential magnitude, but the authors are not able to show that this is necessary—only that coefficients of at most constant magnitude are not sufficient.
Similarly, one can ask whether cutting planes refutations require large coefficients to realize the full power of the proof system. Towards this, define CP to be cutting planes proofs with polynomially-bounded coefficients or, in other words, a cutting planes refutation of a formula with variables is a CP refutation if the largest coefficient in has magnitude .
The question of how CP relates to unrestricted cutting planes has been raised in several papers, e.g., [BPR97LowerBoundsCP, BEGJ00RelativeComplexity]. This question was studied already in [BC96CuttingPlanes], where it was proven that any cutting planes refutation in length can be transformed into a refutation with lines having coefficents of magnitude (here the asymptotic notation hides a mild dependence on the size of the coefficients in the input). The authors write, however, that their original goal had been to show that coefficients of only polynomial magnitude would be enough, i.e., that CP would be as powerful as cutting planes except possibly for a polynomial loss, but that they had to leave this as an open problem. To the best of our knowledge, there has not been a single example of any unsatisfiable formula where CP could potentially perform much worse than general (high-weight) cutting planes.
Finally, as observed in [BPS07LS, HN12VirtueSuccinctProofs], we can use an efficient cutting planes refutation of a formula to solve by an efficient communication protocol. Since the first configuration is always true and the last configuration is always false, the players can simulate a binary search by evaluating the truth value of a configuration according to their joint assignment and find a true configuration followed by a false configuration. It is not hard to see that the inequality being added corresponds to a clause in and it is a valid answer to .
Lemma 2.2 ([HN12VirtueSuccinctProofs]).
If there is a cutting planes refutation of in length , line space , and coefficient bit size , then there is a deterministic communication protocol for of cost .
3 Rank Lifting from Any Gadget
In this section we discuss our new lifting theorem, restated next.333In fact, we prove a somewhat more general theorem (see Theorem A.1 in Appendix A for details). We also remark that this theorem in fact holds for a stronger communication measure (Razborov’s rank measure [Razborov90Applications]), and so implies lower bounds for other models—see Appendix A for details.
Let be any unsatisfiable -CNF on variables and let be any field. For any Boolean valued gadget with we have
This generalizes a recent lifting theorem from [PR18LiftingNS], which only allowed certain “good” gadgets. The main technical step of that proof showed that “good” gadgets can be used to lift the degree of multilinear polynomials to the rank of matrices. In this section, we improve this, showing that any gadget with non-trivial rank can be used to lift polynomial degree to rank. Given this result, Theorem 3.1 is proved by reproducing the proof of [PR18LiftingNS] with a tighter analysis. With this in mind, in this section we will prove our new lifting argument for degree to rank, and then relegate the rest of the proof of Theorem 3.1 to Appendix A.
Let us now make these arguments formal. We start by recalling the definition of a “good” gadget of [PR18LiftingNS].
Definition 3.2 (Definition 3.1 in [PR18LiftingNS]).
Let be a field. A gadget is good if for any matrices of the same size we have
where denotes the all-s matrix.
In [PR18LiftingNS] it is shown that good gadgets are useful because they lift degree to rank when composed with multilinear polynomials.
Theorem 3.3 (Theorem 1.2 in [PR18LiftingNS]).
Let be any field, and let be a multilinear polynomial over . For any good gadget we have
In the present work, we show that a gadget being good is not strictly necessary to obtain the above lifting from degree to rank. In fact, composing with any gadget lifts degree to rank!
Let be any multilinear polynomial and let be any non-zero gadget with . Then
We remark that the lower bound in the theorem can be sharpened to if the gadget is not full rank. While the previous theorem does not require the gadget to be good, the notion of a good gadget will still play a key role in the proof. The general idea is that every gadget with non-trivial rank can be transformed into a good gadget with a slight modification. With this in mind, en-route to proving Theorem 3.4 we give the following characterization of good gadgets which may be of independent interest.
A gadget is good if and only if the all-s vector is not in the row or column space of .
3.1 Proof of Lemma 3.5
We begin by proving Lemma 3.5, which is by a simple linear-algebraic argument. Given a matrix over a field, let denote the row-space of and let denote the column-space of . The following characterization of when rank is additive will be crucial.
Theorem 3.6 ([MS72Rank]).
For any matrices of the same size over any field, if and only if .
The previous theorem formalizes the intuition that rank should be additive if and only if the corresponding linear operators act on disjoint parts of the vector space. Using the previous theorem we deduce the following general statement, from which Lemma 3.5 immediately follows.
Let be matrices over any fixed field of the same size. The following are equivalent:
For all matrices of the same size,
By choosing we instantly deduce (2) from (1). To prove the converse, we use Theorem 3.6. Let be matrices such that . Then by Theorem 3.6 it follows that there is a non-zero vector in the intersection of either the row- or column-spaces of and . Suppose that there is a non-zero vector , and we prove that there is a non-zero vector in implying . (A symmetric argument will apply to the row spaces.)
Assume that and are dimensional matrices, and that and are dimensional matrices. Let be the length non-zero vector in the column spaces of both and , and suppose without loss of generality that . It follows that there are length vectors such that . Write
where are vectors of length for each .
Let denote the first row of and denote the first row of ; note they are both vectors of length . Define the length- vectors
Then, by definition, for each we have , and the vector is non-zero since by assumption. Thus and the column spaces of and intersect at a non-zero vector. ∎
3.2 Proof of Theorem 3.4
Let be any field, and let be any gadget with . For any matrices of the same size we have
where is the all-s matrix.
Assume without loss of generality that and let . Thinking of as a matrix, let be any column vector of . If we zero the entries of in , then the remaining matrix cannot have full rank, implying that some row-vector of the remaining matrix will become linearly dependent. Let be the matrix consisting of the column and row of , and let be the matrix obtained by zeroing out and in . Observe , and also since contains an all- row and an all- column it is good by Lemma 3.5 (as any linear combination of rows/columns of must contain a zero coordinate).
Now, observe that
where the inequality follows since adding a rank- matrix can decrease the rank by at most . Since consists of a single non-zero row and column we have ; by the construction of we have . Using these facts and the fact that is good, we have
With the lemma in hand we can prove Theorem 3.4.
Proof of Theorem 3.4.
by induction on , the number of variables.
Observe that the inequality is trivially true if . Assume , and let . Write for multilinear polynomials . Note that it clearly holds that . From the claim we have by induction that
For the upper bound, by subadditivity of rank we have
Apply the above induction argument using this inequality mutatis mutandis. ∎
4 Application: Separating Cutting Planes Systems
In this section we prove a new separation between high-weight and low-weight cutting planes proofs in the bounded-space regime.
There is a family of -CNF formulas over variables and clauses that have CP refutations in length and line space , but for which any CP refutation in length and line space must satisfy .
By the results of [GPT15SpaceComplexityCP], any unsatisfiable CNF formula has a cutting planes refutation in constant line space, albeit with exponential length and exponentially large coefficients. In Theorem 4.1 we show that the length of such a refutation can be reduced to polynomial for certain formulas, described next.
At a high level, we prove Theorem 4.1 using the reversible pebble game. Given any DAG with a unique sink node , the reversible pebble game [Bennett89TimeSpaceReversible] is a single-player game that is played with a set of pebbles on . Initially the graph is empty, and at each step the player can either place or remove a pebble on a vertex whose predecessors already have pebbles (in particular the player can always place or remove a pebble on a source). The goal of the game is to place a pebble on the sink while using as few pebbles as possible. The reversible pebbling price of a graph, denoted , is the minimum number of pebbles required to place a pebble on the sink.
The family of formulas witnessing Theorem 4.1 are pebbling formulas composed with the equality gadget. Intuitively, the pebbling formula [BW01ShortProofs] associated with is a formula that claims that it is impossible to place a pebble on the sink (using any number of pebbles). Since it is always possible to place a pebble by using some amount of pebbles, this formula is clearly a contradiction.
Formally, the pebbling formula is the following CNF formula. For each vertex there is a variable (intuitively, should take the value “true” if and only if it is possible to place a pebble on using any number of pebbles). The variables are constrained by the following clauses.
a clause for each source vertex (i.e., we can always place a pebble on any source),
a clause for each non-source vertex with predecessors (i.e., if we can place a pebble on the predecessors of , then we can place a pebble on ), and
a clause for the sink (i.e., it is impossible to place a pebble on ).
Proving Theorem 4.1 factors into two tasks: a lower bound and an upper bound. By applying our lifting theorem from the previous section, the lower bound will follow immediately from a good lower bound on the Nullstellensatz degree of pebbling formulas. In order to prove lower bounds on the Nullstellensatz degree, we show in Section 4.1 that over every field, the Nullstellensatz degree required to refute is exactly the reversible pebbling price of . We then use it together with our lifting theorem to prove the time-space tradeoff for bounded-coefficient cutting planes refutations of in Section 4.2 for any high-rank gadget . Finally, in Section 4.3 we prove the upper bound by presenting a short and constant-space refutation of in cutting planes with unbounded coefficients.
4.1 Nullstellensatz Degree of Pebbling Formulas
In this section we prove that the Nullstellensatz degree of the pebbling formula of a graph equals the reversible pebbling price of .
For any field and any graph , .
We crucially use the following dual characterization of Nullstellensatz degree by designs [Buss98LowerBoundsNS].
Let be a field, let be a positive integer, and let be an unsatisfiable system of polynomial equations over . A -design for is a linear functional on the space of polynomials satisfying the following axioms:
For all and all polynomials such that , we have .
Clearly, if we have a candidate degree- Nullstellensatz refutation , then applying a -design to both sides of the refutation yields , a contradiction. Thus, if a -design exists for a system of polynomials then there cannot be a Nullstellensatz refutation of degree . Remarkably, a converse holds for systems of polynomials over .
Theorem 4.4 (Theorems 3, 4 in [Buss98LowerBoundsNS]).
Let be a field and let be a system of polynomial equations over containing the Boolean equations for all . Then does not have a degree- Nullstellensatz refutation if and only if it has a -design.
With this characterization in hand we prove Lemma 4.2.
Proof of Lemma 4.2.
Let be a DAG, and consider the pebbling formula . Following the standard translation of CNF formulas into unsatisfiable systems of polynomial equations, we express with the following equations:
- Source Equations.
The equation for each source vertex .
- Sink Equations.
The equation for the sink vertex .
- Neighbour Equations.
The equation for each internal vertex .
- Boolean Equations.
The equation for each vertex .
We prove that a -design for the above system exists if and only if , and this implies the lemma. Let be a -design for the system. First, note that since the Boolean axioms are satisfied and since is linear, it follows that is completely specified by its value on multilinear monomials (with this notation note that ). Moreover, must satisfy the following properties:
- Empty Set Axiom.
- Source Axioms.
for every source and every with .
- Neighbour Axioms.
for every non-source vertex and every with .
- Sink Axiom.
for the sink and every with .
We may assume without loss of generality that for any set with .
Given a set of vertices of , we think of as the reversible pebbling configuration in which there are pebbles on the vertices in and there are no pebbles on any other vertex. We say that a configuration is reachable from a configuration if there is a sequence of legal reversible pebbling moves that changes to while using at most pebbles at any given point.
Now, we claim that the only way to satisfy the first three axioms is to set for every configuration that is reachable from . To see why, observe that those axioms are satisfiable if and only if the empty configuration is assigned the value , any configuration containing the sink is labelled , and for any two configurations with at most pebbles that are mutually reachable via a single reversible pebbling move. Hence, this setting of is the only one we need to consider.
Finally, observe that this specification of a design satisfies the sink axiom if and only if , since the sink is reachable from using pebbles but not with less (by the definition of ). Therefore, a -design for exists if and only if , as required. ∎
4.2 Time-Space Lower Bounds for Low-Weight Refutations
In this section we prove the lower bound part of the time-space trade-off for CP.
There is a family of graphs with vertices and constant degree, such that every CP refutation of in length and line space must have .
Our plan is to lift a pebbling formula that is hard with respect to Nullstellensatz degree, and as we just proved it is enough to find a family of graphs whose reversible pebbling price is large. Paul et al. [PTC76SpaceBounds] provide such a family (and in fact prove their hardness in the stronger standard pebbling model).
There is a family of graphs with vertices, constant degree, and for which .
We combine these graphs with our lifting theorem as follows.
There is a family of graphs with vertices and constant degree, such that .
This allows us to use our lifting theorem, Theorem 3.1, with an equality gadget of arity , and obtain that the lifted search problem requires deterministic communication
As we noted in Observation 2.1, this implies that the search problem of the lifted formula also requires deterministic communication
Since we collected our last ingredient, let us finish the proof.
Proof of Lemma 4.5.
Since the size of the lifted formula is , the coefficients of a CP refutation are bounded by a polynomial of in magnitude, and hence by in length. Substituting the value of in (4.4) we obtain that
as we wanted to show. ∎
4.3 Time-Space Upper Bounds for High Weight Refutations
We now prove Theorem 4.8, showing that cutting planes proofs with large coefficients can efficiently refute pebbling formulas composed with equality gadgets in constant line space. Let denote the equality gadget on bits.
Let be any constant-width pebbling formula. There is a cutting planes refutation of in length and space .
We also use the following lemma, which is a “derivational” analogue of the recent result of [GPT15SpaceComplexityCP] showing that any set of unsatisfiable integer linear inequalities has a cutting planes refutation in constant space. As the techniques are essentially the same we leave the proof to Appendix B.
Lemma 4.9 (Space Lemma).
Let be a set of integer linear inequalities over variables that implies a clause . Then there is a cutting planes derivation of from in length and space .
Let us begin by outlining the high level proof idea. We would like to refute the lifted formula using constant space. Consider first the unlifted formula . The natural way to refute it is the following: Let be a topological ordering of the vertices of . The refutation will go over the vertices in this order, each time deriving the equation that says that the variable must take the value “true” by using the equations that were derived earlier for the predecessors of . Eventually, the refutation will derive the equation that says that the sink must take the value “true”, which contradicts the axiom that says that the sink must be false.
Going back to the lifted formula , we construct a refutation using the same strategy, except that now the equation of is replaced with the equations
The main obstacle is that if we implement this refutation in the naive way, we will have to store all the equations simultaneously, yielding a refutation of space . The key idea of our proof is that CP can encode the conjunction of many equations using a single equation. We can therefore use this encoding in our refutation to store at any given point all the equations that were derived so far in a single equation. The implementation yields a refutation of constant space, as required.
To see how we can encode multiple equations using a single equation, consider the following example. Suppose we wish to encode the equations
where all the variables take values in . Then, it is easy to see that those equations are equivalent to the equation