aa r X i v : . [ c s . CC ] O c t Stabbing Planes
Paul BeameUniversity of Washington [email protected]
Noah Fleming ∗ University of Toronto noahfl[email protected]
Russell ImpagliazzoUniversity of California, San Diego [email protected]
Antonina Kolokolova † Memorial University of Newfoundland [email protected]
Denis Pankratov † University of Toronto [email protected]
Toniann Pitassi † University of Toronto [email protected]
Robert Robere † University of Toronto [email protected]
October 10, 2017
Abstract
We introduce and develop a new semi-algebraic proof system, called Stabbing Planesthat is in the style of DPLL-based modern SAT solvers. As with DPLL, there is onlyone rule: the current polytope can be subdivided by branching on an inequality and its“integer negation.” That is, we can (nondeterministically choose) a hyperplane ax ≥ b with integer coefficients, which partitions the polytope into three pieces: the points in thepolytope satisfying ax ≥ b , the points satisfying ax ≤ b − , and the middle slab b − While defined in terms of non-deterministic algorithms for the Tautology problem, proof com-plexity has also provided indispensible tools for understanding deterministic algorithms forsearch problems, and in particular, for Satisfiability algorithms. Many algorithms for searchcan be classified according to the types of reasoning they implicitly use for case-analysis andpruning unpromising branches. Particular families of search algorithms can be characterizedby formal proof systems ; the size of proofs in these formal proof system, the time of the non-deterministic algorithm, captures the time taken on the instance by an ideal implementation ofthe search algorithm. This allows us to factor understanding the power of search algorithmsof a given type into two questions:1. How powerful is the proof system? For which kinds of input are there small proofs?2. How close can actual implementations of the search method come to the ideal non-deterministic algorithm?As an illustrative example, let us recall the DPLL algorithm [9, 10], which is one of thesimplest algorithms for SAT and forms the basis of modern conflict-driven clause learningSAT solvers. Let F = C ∧ C ∧ · · · ∧ C m be a CNF formula over variables x , x , . . . , x n .A DPLL search tree for solving the SAT problem for F is constructed as follows. Begin bychoosing a variable x i (non-deterministically, or via some heuristic), and then recurse on theformulas F ↾ x i = 0 , F ↾ x i = 1 . If at any point we have found a satisfying assignment,the algorithm outputs SAT . Otherwise, if we have falsified every literal in some clause C , thenwe record the clause and halt the recursion. If every recursive branch ends up being labelledwith a clause and a falsifying assignment, then the original formula F is unsatisfiable and onecan take the tree as a proof of this fact; in fact, such a DPLL tree is equivalent to a tree-like Resolution refutation of the formula F .Modern SAT solvers still have a DPLL algorithm at the core (now with a highly tuned branching heuristic that chooses the “right” order for variables and assignments to recurse onin the search tree), but extends the basic recipe in two ways:1. Smart handling of unit clauses : if F contains a clause C with a single variable underthe current partial assignment, then we can immediately assign the variable so that theclause is satisfied.2. Clause learning to speed up search: if a partial assignment ρ falsifies a clause, then thealgorithm derives a new clause C ρ by a resolution proof that “explains” this conflict, andadds the new clause to the formula F .It is the synergy between these three mechanisms — branching heuristics, unit propagation,and clause learning — that results in the outstanding performance of modern SAT solvers. Inother words, while these algorithms are all formalizable in the same simple proof system, thesophistication of modern SAT-solvers comes from the attempt to algorithmically find smallproofs when they exist. In many ways, the simplicity of the proof system enables this sophis-tication in proof-search methods. 2n this work, we introduce a natural generalization of the DPLL-style branching algorithmto reasoning over integer-linear inequalities, formalized as a new semi-algebraic proof systemthat we call the Stabbing Planes ( SP ) system. We will give a more detailed description later,but intuitively, Stabbing Planes has the same branching structure as DPLL, but generalizesbranching on single variables to branching on linear inequalities over the variables. We feelthe closeness to DPLL makes Stabbing Planes a better starting point for understanding searchalgorithms based on linear inequalities, such as most integer linear programming(ILP) basedsolvers, than established proof systems using such inequalities such as Cutting Planes proofs.We compare the power of Stabbing Planes proofs to these other proof systems. We showthat one of these systems, Kraijicek’s system tree-like R ( CP ) , is polynomially equivalent toStabbing Planes (Theorem 4.2). However, the new formulation as Stabbing Planes proofsboth gives greater motivation to studying R ( CP ) and greatly clarifies the power of the proofsystem. Our main results about this system are:1. Stabbing planes has quasi-polynomial size and poly-log rank proofs of any tautologyprovable using linear algebra over a constant modulus. In particular, this is true for theTseitin graph tautologies, that are very frequently used to prove lower bounds for otherproof systems. (Theorem 3.1)2. Stabbing planes can simulate tree-like Cutting Planes proofs with only constant factorincreases in size and rank (Theorem 4.4)3. Stabbing planes can simulate general Cutting Planes proofs with a polynomial increasein size (Theorem 4.5)4. Lower bounds on real communication protocols imply rank lower bounds for StabbingPlanes proofs (Lemma 5.3)5. Stabbing planes proofs cannot be balanced (Theorem 5.7)Together, these show that Stabbing Planes is at least as strong as established proof systemsusing inequalities, and possibly much stronger. So the proof system combines strength as aproof system with a simple branching structure that raises the possibility of elegant algorithmsbased on this proof system.We now give a more precise description of the proof system. Let us formalize the systemin stages. First, observe that the setting is quite different: we are given a system A x ≥ b , A · x ≥ b , . . . , A m · x ≥ b m of integer-linear inequalities over real-valued variables x , x , . . . , x n (for simplicity we will always assume that the inequalities ≤ x i ≤ arepresent for each variable x i ), and we seek to prove that no { , } -solution exists. One canimmediately see that the basic DPLL algorithm immediately works in this setting with littlemodification: one can still query variables and assign them to { , } values; now we labelleaves of the search tree with any inequality a i · x ≥ b i in the system that is falsified by thesequence of assignments made on the path from the root to the leaf. If every leaf ends up beinglabelled with a falsified inequality, then the tree certifies that the system of inequalities has no { , } -solutions. 3owever, with the expanded domain we can consider the DPLL tree geometrically . To bemore specific, imagine replacing each { , } query to a variable x i in the decision tree withtwo “inequality queries” x i ≤ and x i ≥ . Each node u in the tree after this replacementis now naturally associated with a polyhedral set P u of points satisfying each of the the inputinequalities and each of the inequalities on the path to this node. Since we began with a DPLLrefutation, it is clear that for any leaf ℓ the polyhedral set P ℓ associated with the leaf is empty,as any { , } solution would have survived each of the inequalities queried on some path inthe tree and thus would exist in one of the polyhedral sets at the leaves.The stabbing planes system is then the natural generalization of the previous object: an SP refutation consists of a generalized DPLL refutation where each node is labelled with an arbitrary integral linear inequality Ax ≥ b (that is, the vector A and the parameter b are bothintegral), and the outgoing edges are labelled with the inequalities Ax ≥ b and Ax ≤ b − .Clearly, any integral vector x ∈ Z n will satisfy at least one of the inequalities labelling theoutgoing edges, and so if the polyhedral set at each leaf (again, obtained by intersecting theoriginal system with the inequalities on the path to the leaf) is empty then we have certifiedthat the original system of inequalities has no integral solutions. (See Figure 1 for a simpleexample.) The main innovation of Stabbing Planes is its simplicity: refutations are simply x + yx − yx + y ≤ x + y ≥ x − y ≥ x − y ≤ x + y ≤ x + y ≥ < x + y < x − y ≥ x − y ≤ < x − y < Figure 1: A partial SP refutation and the result on the unit square. The shaded areas are“removed” from the polytope, and we recurse on each side.decision trees that query linear inequalities. Note that the more obvious extension of DPLL tolinear inequalities would branch on Ax ≥ b and its actual negation, Ax < b . However withthis branching rule, we would have to add additional rules in order to have completeness. Bybranching on an inequality and its “integer negation”, we are able to get by with just one ruleanalogous to the resolution rule in DPLL.From the perspective of SAT solving, even though tree-like Resolution and the search for4atisfying assignments encapsulated by DPLL are equivalent, it is the search point of view ofDPLL that has led to the major advances in SAT algorithms now found in modern conflict-directed clause learning (CDCL) SAT solvers. A natural hypothesis is that it is much easier toinvent useful heuristics in the language of query-based algorithms, as opposed to algorithmsbased on the resolution rule. Stabbing Planes offers a similar benefit with respect to reasoningabout inequalities.With the exception of mixed integer programming (MIP) solvers (such as CPLEX [ ? ]),current solvers that, like Stabbing Planes, manipulate integer linear inequalities over Booleanvariables are generally built on the same backtracking-style infrastructure as DPLL and CDCLSAT solvers but maintaining information as integer linear inequalities as opposed to clausalforms. The solvers are known as pseudoBoolean solvers and have been the subject of consid-erable effort and development.PseudoBoolean solvers work very well at handling the kinds of symmetric counting prob-lems associated with, for example, the pigeonhole principle (PHP), which is known to behard for CDCL SAT solvers, as well as other problems where the input constraints are muchmore succinctly and naturally expressed in inequality rather than clausal form. Innovations inpseudoBoolean solvers include use of normal forms for expressing constraints, techniques togeneralize fast unit propogation and watched literals from DPLL to the analogue for integerlinear inequalities, as well as methods to learn from conflicts and simplify learned constraintswhen integer coefficients from derived inequalities get too large [ ? , ? ]. Despite all of this,even for the best pseudoBoolean solvers, the benefits of expressibility are usually not enoughcompensation for the added costs of manipulating and deriving new inequalities and they out-perform CDCL solvers only in very limited cases in practice [ ? ].A key limitation of these pseudoBoolean solvers, which likely constrains their effective-ness, is the fact that all branching is based on assigning values to individual variables; i.e.,dividing the problem by slabs parallel to one of the coordinate axes. Stabbing Planes elimi-nates this constraint on the search and allows one to apply a divide and conquer search basedon arbitrary integer linear constraints that are not necessarily aligned with one of these coordi-nate axes. This opens up the space of algorithmic ideas considerably and should allow futurepseudoBoolean solvers to take fuller advantage of the expressibility of integer linear con-straints. For example, a Stabbing Planes search could choose to branch on a linear inequalitythat is derived from the geometric properties of the rational hull of the current constraints by,say, splitting the volume, or by doing a balanced split at a polytope vertex, since properties ofthe rational hull can be determined efficiently. Such operations could be potentially be donein conjunction with solvers such as CPLEX to obtain the best of both kinds of approaches.Beyond the prospect of Stabbing Planes yielding improved backtracking search for pseu-doBoolean solvers, Stabbing Planes should allow the same kind of learning of inequalitiesfrom conflicts that is being done in existing pseudoBoolean solvers, and hence get the benefitsof both. In this work we do not focus on the theoretical benefits of learning from conflicts be-cause we already can show considerable theoretical benefit from the more general branchingalone and because the theoretical benefits of the restricted kinds of learned linear inequalitiesfrom conflicts available even in existing solvers are not at all clear.From a proof complexity perspective, the SP system turns out to be polynomially equiva-5ent to the semi-algebraic proof system tree-like R ( CP ) , introduced by Kraj´ıˇcek [23]. Roughlyspeaking one can think of R ( CP ) as a mutual generalization of both Cutting Planes and Reso-lution — the lines of an R ( CP ) proof are clauses of integer linear inequalities, and in a singlestep one can take two clauses and either apply a cutting-planes rule to a single inequality ineach clause or apply a resolution-style “cut”. However, even though SP turns out to be equiv-alent to a system already in the literature, this new perspective turns out to be quite useful:we show that SP has quasi-polynomial size refutations of the Tseitin principle, and also that SP can polynomially-simulate Cutting Planes (neither result was previously known to hold fortree-like R ( CP ) ).We also investigate the relationship between SP refutations and communication complex-ity. Given an unsatisfiable CNF F and any partition of the variables ( X, Y ) of F into two sets,one can consider the following two-party search problem Search X,Y ( F ) : Alice receives an as-signment to the X -variables, Bob receives an assignment to the Y -variables, and they mustcommunicate and output a falsified clause of F under their joint assignment. At this time all strong lower bound results for Cutting Planes refutations essentially follow from studying thecommunication complexity of Search X,Y ( F ) . For instance, it is known that: • A depth- d Cutting Planes refutation of F yields a d -round real communication protocolfor Search X,Y ( F ) [24]. • A length- L tree-like Cutting Planes refutation of F yields an O (log L ) -round real com-munication protocol for Search X,Y ( F ) [5, 24, 27]. • A length- L , space s Cutting Planes refutation of F yields an O ( s log L ) -round realcommunication protocol for Search X,Y ( F ) [11]. • A length- L Cutting Planes refutation of F yields a size L real communication game for Search X,Y ( F ) [20].Each of these results has been used to derive strong lower bounds on Cutting Planes refutationsby proving the corresponding lower bound against the search problem [5, 11, 13, 21, 24, 27].Furthermore, the above lower bound techniques hold even for the stronger semantic CuttingPlanes system (the lines of which are integer linear inequalities, and from two integer linearinequalities we are allowed to make any sound deduction over integer points) [12]. This makesthe known lower bounds much stronger, and it is quite surprising that all one needs to exploitfor strong lower bounds is that the lines are linear inequalities (rather than exploiting someweakness of the deduction rules). However, this strength also illustrates a weakness of currenttechniques, as once the lines of a proof system P become expressive enough, semantic prooftechniques (i.e. ones that work for the semantic version of the proof systems) completely breakdown since every tautology has a short semantic proof. Therefore, it is of key importanceto develop techniques which truly exploit the “syntax” of proof systems, and not just theexpressive power of the lines.Hence, it is somewhat remarkable that we are able to show that each of the simulationresults above still hold if we replace real communication protocols with SP refutations. Thatis, we show 6 A depth- d Cutting Planes refutation of F yields a depth- d SP refutation of F . • A length- L tree-like Cutting Planes refutation of F yields a depth O (log L ) SP refuta-tion of F . • A length- L , space s Cutting Planes refutation of F yields a depth O ( s log L ) SP -refutationof F . • A length- L Cutting Planes refutation of F yields a size O ( L ) SP -refutation of F .Since SP is a syntactic system this further motivates studying its depth- and size-complexity.We can use semantic techniques to get some lower bounds for SP : we show that a size- S anddepth- d SP refutation yields a real communication protocol with cost O ( d ) and for which theprotocol tree has size O ( S · n ) . This simulation yields new proofs of some depth lower boundsalready known in the literature; however, these lower bounds are complemented by showingthat neither SP refutations nor real communication protocols can be balanced. This shouldbe viewed in a positive light: the depth- and size-complexity problems are truly different for SP , and furthermore, one seemingly cannot obtain size lower bounds for SP by proving depthlower bounds for real communication protocols (in contrast to, say, tree-like Cutting Planes).In sum, SP appears to be a very good candidate for a proof system on the “boundary” wherecurrent techniques fail to prove strong size lower bounds.The rest of the paper is outlined as follows. After some preliminaries in Section 2, we givea simple refutation of the Tseitin problem in SP in Section 3. In Section 4, we prove a raftof simulation and equivalence results for SP — showing it is equivalent to R ( CP ) , relating itto Cutting Planes in various measures such as depth, length, and space, and showing how an SP proof yields a real communication protocol for the canonical search problem. Finally, inSection 5, we prove depth lower bounds for SP and some impossibility results for balancing. Before we define the new proof system formally, we need to make a few general definitionsthat are relevant to semi-algebraic proof systems.An integer linear inequality (or simply a linear inequality ) in the variables x = x , . . . , x n is Ax ≥ b, where A ∈ Z n and b ∈ Z . A system of linear inequalities F is unsatisfiable if thereis no Boolean assignment α ∈ { , } n which simultaneously satisfies every inequality in F .We sometimes refer to inequalities as lines and write L ≡ Ax ≥ b . The integer negation of aline L is the inequality ¬ L ≡ Ax ≤ b − .An unsatisfiable formula in a conjuctive normal form (CNF) defines an unsatisfiable sys-tem of linear inequalities F in a natural way. A clause W ki =1 x i ∨ W li =1 ¬ x i , is translated intothe inequality P ki =1 x i + P li =1 (1 − x i ) ≥ , and F is the set of translations of all clauses. Weassume that F always contains the axioms x i ≥ and − x i ≥ − for all variables x i , as weare interested in propositional proof systems for refuting unsatisfiable Boolean formulas.7 efinition 2.1. A propositional proof system P is a non-deterministic polynomial time Turingmachine (TM) deciding the language of unsatisfiable CNF formulas. Given an unsatisfiableCNF, the NP -certificate is called the proof or the refutation .To compare the strength of proof systems, one typically uses the notion of polynomialsimulation. Definition 2.2. Let P and P be two proof systems. We say that P polynomially simulates P if for every unsatisfiable formula F , the shortest refutation of F in P is at most polynomiallylonger than the shortest refutation in P . P is strictly stronger than P if P polynomiallysimulates P , but the converse does not hold. Finally, we say that P and P are incomparable if neither can polynomially simulate the other.We now describe the proof system Stabbing Planes, which will be the central object ofstudy of this paper. Definition 2.3. Let F be an unsatisfiable system of linear integral inequalities. A StabbingPlanes ( SP ) refutation of F is a threshold decision tree — a directed binary tree in which eachedge is labelled with a linear integral inequality. If the right outgoing edge of a certain node islabelled with Ax ≥ b , then the left outgoing edge has to be labelled with its integer negation, Ax ≤ b − . We refer to Ax (or the pair of inequalities Ax ≤ b − , Ax ≥ b ) as the query corresponding to the node. The slab corresponding to the query is { x ∗ ∈ R n | b − < Ax ∗
Let F be an unsatisfiable system of linear inequalities. A Cutting Planes ( CP ) refutation of F is a sequence of linear inequalities { L , . . . , L ℓ } such that L ℓ = 0 ≥ andeach L i is either an axiom ∈ F or is obtained from previous lines via one of the followinginference rules. Let α, β be positive integers. Ax ≥ a Bx ≥ b Linear Combination: ( αA + βB ) x ≥ αa + βb αAx ≥ b Division: Ax ≥ (cid:6) bα (cid:7) We refer to ℓ as the length of the refutation. The length of refuting F in CP is the minimumlength of a CP refutation of F .The directed acyclic graph (DAG) G = ( V, E ) associated with a CP refutation { L , . . . , L ℓ } is defined as follows. We have V = { L , . . . , L ℓ } and ( u, v ) ∈ E if and only if the line la-belling v was derived by an application of an inference rule involving the line labelling u .Without loss of generality, we may assume that there is only one vertex with out degree 0,which we call the root. The root of G is labelled with L ℓ and the leaves are labelled with theaxioms.The rank or depth of the refutation is the length of the longest root-to-leaf path in G . Therank of refuting an unsatisfiable system of linear inequalities F is the minimum rank of anyrefutation of F in the given proof system. Finally, tree-like CP is defined by restricting proofsto be such that the underlying graph G is a tree. On the Issue of Weights. A well-known theorem of Muroga shows that for any integerlinear inequality Ax ≥ b separating two subsets U , V ⊆ { , } n of / vectors, there is an-other linear integer inequality A ′ x ≥ b ′ separating the same set of vectors and also satisfying || A || ∞ ≤ poly ( n ) [ ? ]. In the same vein, Cook, Coullard, and Tur´an showed that any CuttingPlanes refutation of length ℓ can be transformed into a refutation of length at most polynomi-ally larger and in which each coefficient has magnitude poly ( ℓ,n ) [8]. This implies that the sizeof any CP proof (measured by the length of its encoding in bits) is polynomially related to its9ength, and thus we may study the length of cutting planes proofs without loss of generalityfor the purpose of upper and lower bounds.An analogous result is not known for SP and therefore we currently must make the dis-tinction between its size and length. Fortunately, all of our results hold in the best possiblescenario; our upper bounds are low weight (polynomial-length); the simulations are lengthpreserving, and our lower bounds hold for any weight. SP Refutations of Tseitin Formulas Tseitin contradictions are among the most well-studied unsatisfiable formulas in proof com-plexity, and are the quintessential formulas that are believed to be hard for CP [22]. Despitethe fact that exponential lower bounds for CP are known for many natural families of formulas(including recent lower bounds for random O (log n ) -CNF formulas), there are no nontriviallower bounds known for the Tseitin contradictions, and for good reason: the only known lowerbound method for CP reduces the problem of refuting a formula in CP to a monotone circuitproblem, for which the corresponding monotone circuit problem for Tseitin is easy.In this section, we demonstrate the power of Stabbing Planes by showing that there existsa shallow quasi-polynomial size SP refutation of the Tseitin formulas. This, together with oursimulation results from Section 4, strongly suggests that SP is strictly more powerful than CP .In addition, we immediately conclude that SP is provably more powerful than CP in terms ofdepth.Tseitin contradictions are any unsatisfiable family of mod-2 equations subject to the con-straint that every variable occurs in exactly two equations. An instance of Tseitin, denoted Tseitin( G, ℓ ) is defined by a connected undirected graph G = ( V, E ) and a node labelling ℓ ∈ { , } V of odd total weight: P v ∈ V ℓ v = 1 mod 2 . For each edge e ∈ E there is a variable x e in Tseitin( G, ℓ ) . Corresponding to each vertex v ∈ V is the equation X e ∋ v x e ≡ ℓ v mod 2 stating that the sum of the variables x e incident with v is ℓ v modulo . Summing up all of theseequations modulo , the left-hand-side sums to zero since every variable occurs exactly twice,but the right-hand-side is one, since the node labelling is odd, and therefore the equationsare unsatisfiable. When G has degree D , we can express Tseitin( G, ℓ ) as a D -CNF formulacontaining | V | · D − clauses.The obvious way to refute Tseitin( G, ℓ ) under an assignment x is to find a vertex w forwhich the corresponding vertex equation is falsified. This can be achieved by the followingdivide-and-conquer procedure, which maintains a set U ⊆ V such that w ∈ U . The processbegins by setting U = V . Then, V is partitioned arbitrarily into two sets V , V of roughlythe same size. Query x e for all edges e crossing the cut ( V , V ) , and suppose that the sumof all such x e is odd (the case when it is even is similar). We know that either P v ∈ V ℓ v or P v ∈ V ℓ v is even: if the first sum is even then the Tseitin formula restricted to V contains acontradiction, and otherwise the formula restricted to V contains a contradiction. In eithercase, we can remove roughly half of the graph and recurse.10y keeping track of a few more variables, we can repeat this procedure recursively until | U | = 1 . Since we reduce the size of U by half each time, this procedure results in therecursion depth logarithmic in | V | . It turns out that this procedure can be realized in StabbingPlanes, where recursion depth roughly corresponds to the depth of the refutation. This resultsin a quasi-polynomial size refutation. Theorem 3.1. Let G = ( V, E ) be an undirected graph, and let ℓ be a { , } vertex labellingwith odd total weight. Then Tseitin( G, ℓ ) has an SP refutation of size n O (log n + D/ log n ) andrank O ( D + log n ) , where n = | V | and D is the maximum degree in G .Proof. If U ⊆ V is a set of vertices, then let E ( U ) = { uv ∈ E | u, v ∈ U } , and Cut( U ) = (cid:8) uv ∈ E | u ∈ U, v ∈ U (cid:9) . Similarly, if U , U ⊆ V are disjoint then we let Cut( U , U ) = { uv ∈ E | u ∈ U , v ∈ U } . We construct the SP refutation recursively. During the recursionwe maintain a set U of current vertices (initially U = V ). At each recursive step, we split U into two halves U and U and query the total weight k of the edges crossing ( U , U ) via SP inequalities. Knowing k , a few additional queries allows us to determine which of U or U contains a contradiction, and we then recurse on the corresponding set of vertices.More formally, we construct a proof while maintaining the following invariant: for thecurrent subset of vertices U ⊆ V , we have queried linear inequalities implying that X e ∈ Cut( U ) x e = k for some ≤ k ≤ | Cut( U ) | such that k P v ∈ U ℓ v mod 2 . Note that this invariant ensuresthat our Tseitin instance restricted to the edges incident on U is unsatisfiable, since summingup all vertex constraints within U yields X e ∈ Cut( U ) x e + X e ∈ E ( U ) x e ≡ k X v ∈ U ℓ v (mod 2) . Initialization. Initially we have U = V and the invariant clearly holds. Recursive Step. Let U be the current set of vertices. By the invariant we know that P v ∈ Cut( U ) x e = k for some k P v ∈ U ℓ v mod 2 . Partition U into two halves U and U arbitrarily, subjectto | U | = ⌊| U | / ⌋ . We first determine the value of the edges going between U and U byquerying X e ∈ Cut( U ,U ) x e ≥ β for β = 1 , . . . , | Cut( U , U ) | . To each leaf of this tree we attach a second binary search treefor determining the value | Cut( U ) | by querying P e ∈ Cut( U ) x e ≥ γ for γ = 1 , . . . , | Cut( U ) | .After these queries, at each leaf of the “combined” tree we will have X e ∈ Cut( U ) x e = γ for some ≤ γ ≤ | Cut( U ) | , X e ∈ Cut( U ,U ) x e = β for some ≤ β ≤ | Cut( U , U ) | . | Cut( U ) | + | Cut( U ) | = | Cut( U ) | + 2 | Cut( U , U ) | , we will have X e ∈ Cut( U ) x e = δ for some ≤ δ ≤ | Cut( U ) | , where δ + γ = k + 2 β .For any leaf of this tree where δ > | Cut( U ) | , we can derive a contradiction by summingthe axioms − x e ≥ − for all e ∈ Cut( U ) with P e ∈ Cut( U ) x e ≥ δ . Otherwise, for theremaining leaves observe that δ + γ ≡ k X v ∈ U ℓ v + X v ∈ U ℓ v (mod 2) . Thus, exactly one of the following cases holds:1. If γ P v ∈ U ℓ ( v ) mod 2 , then recurse on U .2. Otherwise, δ P v ∈ U ℓ ( v ) mod 2 . Recurse on U . Termination. Our recursion terminates when U contains a single vertex v . By the invari-ant, we have derived P e ∈ Cut( v ) x e ≡ k ℓ ( v ) mod 2 for some ≤ k ≤ | Cut( { v } ) | . Theaxioms of Tseitin( G, ℓ ) rule out Boolean assignments to the variables x e for e ∈ Cut( { v } ) ,which contradict ℓ ( v ) ; these axioms do not prohibit incorrect fractional assignments. There-fore, to derive a contradiction, we still need to enforce that the variables x e for e ∈ Cut( { v } ) take { , } values. We achieve this by querying all variables x e for e ∈ Cut( { v } ) via SP inequalities x e ≥ , x e ≤ . This results in a complete binary tree of depth ≤ D . Clearly, ≥ is immediately obtained at the leaves that disagree with P e ∈ Cut( { v } ) x e = k . At theleaves that agree with P e ∈ Cut( { v } ) x e = k , the inequality ≥ immediately follows from theassignment to the edges incident to v and one of the axioms of Tseitin.Finally, we analyze the size and rank of the constructed SP proof. In each recursive step,we make O (( nD ) ) queries to determine weights of edges crossing the two cuts. Each recur-sive step is computed by a pair of binary trees, each of depth at most log( nD ) . Our recursionterminates in log n rounds because we halve the number of current vertices in each step. Oncethe recursion terminates, we query the variables corresponding to edges incident to the sin-gle remaining vertex — this contributes D factor to size and increases depth by at most D .Overall, the SP proof has size n O (log n + D/ log n ) and rank O ( D + log n ) .Combining this upper bound with a know lower bound on the rank of refutations in CP forthe Tseitin formulas allows us to separate SP and CP in terms of rank. Corollary 3.2. SP is strictly stronger than CP with respect to proof rank.Proof. By Theorem 4.4, any Cutting Planes refutation of rank r can be converted into a SP refutation of rank O ( r ) . Buresh-Oppenheim et al. proved Ω( n ) lower bound on the rank ofCutting Planes of the Tseitin formulas on constant-degree expander graphs [6], while Theorem3.1 shows that SP can refute such Tseitin formulas in rank O (log n ) .12 Simulation Theorems In this section, we prove simulation theorems relating the SP proof system to other similarproof systems in the literature. We begin by showing that SP is polynomially equivalent to thetree-like R ( CP ) system (introduced by Krajicek in [23]), which can be thought of as tree-likeResolution with clauses of inequalities and allowing CP rules. Since tree-like R ( CP ) simu-lates tree-like CP , the natural question is whether SP (and consequently R ( CP ) ) can simulategeneral CP . We answer this question positively by providing two simulations. First of all, weobserve that SP can depth-simulate CP . This simulation, while preserving depth of the proof,can lead to an exponential increase in the size. Thus, by a different simulation we show that SP can size-simulate CP . This time around, while the simulation preserves the size of a CP refutation, it can significantly increase the depth. It is an interesting open question whetherthere is a simulation of CP by SP that can simultaneously preserve depth and size of CP refu-tations. To complete the picture, we note that general R ( CP ) can trivially simulate tree-like R ( CP ) (and consequently SP ) and CP . We also show that tree-like CP refutations can be effi-ciently converted into balanced (logarithmic-depth) SP refutations — this shows that tree-like CP refutations, which cannot in general be balanced, can be balanced in SP . Figure 2 is agraphical depiction of the relative strengths of various proof systems related to SP . CP SP = Tree - R ( CP ) Tree - CPR ( CP ) Figure 2: Relationships between proof systems considered here. SP ’s equivalence to tree-like R ( CP ) , as well as SP (and Tree - R ( CP ) ) simulating CP are the new relationships proved in thispaper.We then turn to the question of space-time simulations. Recall, that a proof system canbe thought of as a non-deterministic Turing machine. The notion of space of CP refutationsintuitively corresponds to the minimum size of the work tape of such a non-deterministicTuring machine that is required to carry out the computation. In this analogy, the notionof length of CP refutations corresponds to the running time of the given TM. We show thatgeneral CP refutations that use length ℓ and space s can be turned into depth O ( s log ℓ ) SP refutations. Thus, sufficiently strong lower bounds on the depth of SP refutations lead totime-space tradeoffs for CP . 13 quivalence of SP with tree-like R ( CP ) . Here we show the SP system is polynomiallyequivalent to the R ( CP ) proof system. We begin by formally defining the R ( CP ) proof system. Definition 4.1. The R ( CP ) proof system is a syntactic proof system defined as follows. Thelines of the R ( CP ) system are disjunctions of integer linear inequalities Γ = L ∨ L ∨ · · · ∨ L ℓ ,and the lines are equipped with the following deductive rules. Let Γ be an arbitrary disjunctionof integer linear inequalities, let Ax ≥ b, Cx ≥ d be arbitrary integer linear inequalities, andlet α, β be any positive integers. ( Ax ≥ b ) ∨ Γ ( Cx ≥ d ) ∨ Γ Linear Combination: ( αA + βC ) x ≥ ( αb + βd ) ∨ Γ ( αAx ≥ b ) ∨ Γ Division: ( Ax ≥ ⌈ b/α ⌉ ) ∨ Γ Axiom Introduction: ( Ax ≥ b ) ∨ ( Ax ≤ b − 1) Γ Weakening: ( Ax ≥ b ) ∨ Γ( Ax ≥ b ) ∨ Γ ( Ax ≤ b − ∨ Γ Cut: Γ (0 ≥ ∨ Γ Elimination: Γ An R ( CP ) proof is tree-like if the proof DAG is a tree. Theorem 4.2. Let C be an unsatisfiable CNF, and let C , C , . . . , C m be the representation of C as an integer-linear system of inequalities. For any SP refutation of C with size s and depth d there is a tree-like R ( CP ) refutation of C of size O ( s ( d + dm )) and width d + 1 .Proof. Let T be the SP refutation of C , and consider any path p in the tree T . Let L , L , . . . , L t be the sequence of inequalities on p . We first show how to derive the clause ¬ L ∨ ¬ L ∨ · · · ∨ ¬ L t efficiently in R ( CP ) from C . Begin by using the Axiom Introduction rule to introduce the lines L i ∨ ¬ L i for each i = 1 , , . . . , t . Then, for each i , repeatedly apply the Weakening rule to theline L i ∨ ¬ L i to deduce L i ∨ ¬ L ∨ ¬ L · · · ∨ ¬ L t , and then for each input line C i similarly apply the Weakening rule to deduce C i ∨ ¬ L ∨ ¬ L ∨ · · · ∨ ¬ L t . Since T is a Stabbing Planes refutation, there is a convex combination of L , L , . . . , L t , C , . . . , C m equalling ≥ . Furthermore, by Fact 2.4, we can assume without loss of gen-erality that this convex combination contains at most a linear number of these inequalities.Finally, because any convex combination can be performed in tree-like Cutting Planes by re-peatedly applying the “Linear Combination” rule, there is a linear size tree-like cutting planes14efutation of the system L , L , . . . , L t , C , . . . , C m . By simulating this proof in R ( CP ) on thelines L ∨ ¬ L ∨ · · · ∨ ¬ L t L ∨ ¬ L ∨ · · · ∨ ¬ L t ... L t ∨ ¬ L ∨ · · · ∨ ¬ L t C ∨ ¬ L ∨ · · · ∨ ¬ L t ... C m ∨ ¬ L ∨ · · · ∨ ¬ L t , (that is, by applying the appropriate cutting planes rules to the first inequality in each line), wecan deduce the line (0 ≥ ∨ ¬ L ∨ · · · ∨ ¬ L t , which can be simplified to ¬ L ∨ ¬ L ∨ · · · ∨ ¬ L t by the elimination rule. The proof of this path requires size O ( t + tm + t + m ) = O ( d + dm ) ,where d is the depth of the tree T , and clearly can be implemented as a tree-like proof.Now we are nearly finished. In parallel, for each path p of the tree T construct the corre-sponding clause in short tree-like R ( CP ) as above. Applying the cut rule to the paths appro-priately yields the empty clause in size O ( s ( d + dm )) .Next, we prove the converse. Theorem 4.3. Let C be an unsatisfiable CNF, and let C , C , . . . , C m be the representation of C as an integer linear system of equations. For any tree-like R ( CP ) proof of the disjunction ¬ C ∨ ¬ C ∨ · · · ∨ ¬ C m with size s and depth d there is an SP refutation of C of size at most s and rank at most d .Proof. Let R be the R ( CP ) proof of the disjunction, and we construct the SP refutation bystructural induction. Consider any leaf of the proof R which, by assumption, is labelled withan input axiom L ∨ ¬ L for some integer linear inequality L . It is easy to give a short SP refutation of L, ¬ L : query the inequality L, ¬ L and refute each side of the tree appropriately.By induction suppose that we have a tree-like R ( CP ) proof of a clause Γ . We break intocases depending on the last inference rule used to derive Γ . Case 1. Linear Combination. Write Γ as the line ( αA + βB ) x ≥ ( αa + βb ) ∨ ∆ , which was deduced from lines ( Ax ≥ a ) ∨ ∆ , ( Bx ≥ b ) ∨ ∆ 15y the Linear Combination rule. Let L A be the inequality Ax ≥ a , and L B the inequality Bx ≥ b . Begin the SP proof by making the query to the line L A and ¬ L A — on the branchlabelled with ¬ L A , by induction we can construct a refutation of the clause L A ∨ ∆ . Then,on the branch labelled with L A , branch on the inequalities L B and ¬ L B . Again, on the ¬ L B branch, we can apply induction to get an SP refutation of the clause L B ∨ ∆ . On the pathlabelled with L A and L B , we can immediately deduce a contradiction in stabbing planes from L A , L B , and ¬ ( αL A + βL B ) . Case 2. Division. Write Γ as ( Ax ≥ ⌈ a/α ⌉ ) ∨ ∆ , deduced from the line ( αAx ≥ a ) ∨ ∆ .Let L A ≡ αAx ≥ a . Begin the SP refutation by querying the inequalities L A and ¬ L A . Onthe branch labelled ¬ L A , we can inductively construct a refutation of the clause L A ∨ ∆ . Onthe other branch, labelled with L A , it is enough to observe that the intersection of L A , whichis αAx ≥ a , and the inequality Ax ≤ ⌈ a/α ⌉ − , provided as an axiom to SP , is empty. Case 3. Weakening. Write Γ as L A ∨ ∆ , deduced from ∆ . This case is easy — by induction ¬ ∆ has a short SP refutation, which implies that ¬ L A and ¬ ∆ has a short refutation. Case 4. Cut. Suppose the line Γ was deduced from lines L A ∨ Γ and ¬ L A ∨ Γ by the cutrule. This case is also straightforward — begin by querying the inequalities L A and ¬ L A .On the branch labelled with L A , we apply induction to construct a refutation of ¬ L A ∨ Γ ;symmetrically, on the branch labelled ¬ L A , apply induction to construct a refutation of L A ∨ Γ .At worst, each line of the R ( CP ) proof is replaced with two inequality queries, of whichat most two of the children are not immediately labelled with a convex combination equalling ≥ , and so the size of the resulting tree is at most s and the depth is at most d . Depth Simulation of Cutting Planes. Here we prove that there is a depth-preserving simu-lation of CP by SP . Theorem 4.4. For every Cutting Planes refutation of rank d , there is a SP refutation of thesame tautology with rank at most d . Moreover, if the CP refutation is tree-like of size s thenthe resulting SP refutation is of size O ( s ) and rank d .Proof. It is sufficient to prove the “moreover” part of the statement, since by recursive dou-bling, any CP refutation can be converted into a tree-like CP refutation where the rank remainsthe same, but the size may increase exponentially. Thus, from now on we assume that R CP is a size- s rank- d tree-like Cutting Planes refutation. We show that there is a size O ( s ) , rank r ≤ d SP refutation R SP of the same contradiction.Let G be the graph (tree) associated with R CP . The refutation R SP will be constructedfrom R CP by proceeding from the root of G to the leaves. In the process, we keep track ofa subtree T in G , which we are left to simulate, and an associated current node N in R SP ,which we are constructing. Along the way the following invariant will be maintained: atevery recursive step ( N, T ) such that T = G , if the root of the subtree T is labelled with theinequality Cx ≥ c , then the edge leading to N in R SP is labelled with Cx ≤ c − . Originally T = G and R SP contains only a single node N . Consider the subtree T at the current recursivestep in the proof, and we break into three cases.16 x ≥ a Bx ≥ bCx ≥ c CxAxN (1) BxN (2) N (3) ⊢ ≥ ≤ c − ≤ a − ≥ a ≤ b − ≥ b Figure 3: A tree-like CP refutation invoking the non-negative linear combination inferencerule and the the corresponding SP Refutation. Case 1. The root of T labelled with Cx ≥ c has two children labelled with Ax ≥ a and Bx ≥ b . Non-negative linear combinations are the only inferences in CP which take twopremises, therefore, Cx ≥ c is a non-negative linear combination of Ax ≥ a and Bx ≥ b .In the SP refutation R SP at the current node N , query Ax ≥ a . On the branch labelled with Ax ≥ a , query Bx ≥ b (see Figure 3). This sequence of queries results in three leaf nodes,labelled N (1) , N (2) , N (3) as in Figure 3. For the leaf node N (1) of the edge labelled with Ax ≤ a − in R SP let T (1) be the subtree rooted at the child of the root of T labelled with Ax ≥ a . Recurse on ( N (1) , T (1) ) . Similarly, for the leaf node N (2) of the pair of introducededges labelled with Ax ≥ a and Bx ≤ b − , let T (2) be the subtree rooted at the child of theroot of T labelled with Bx ≥ b . Recurse on ( N (2) , T (2) ) .For the final leaf node N (3) we can derive ≥ . To see this, observe that if the currentsubtree T is G (i.e. the base case) then the root node of T is labelled with ≥ . In thiscase, Ax ≥ a and Bx ≥ b are the premises used to derive ≥ by a non-negative linearcombination in R CP , and we can use this very non-negative combination at the leaf N (3) .Otherwise, the root of the current subtree T is labelled with some inequality Cx ≥ c . Bythe invariant, the edge leading to N in the refutation R SP we are constructing was labelledwith Cx ≤ c − . In the CP refutation R CP , Cx ≥ c was derived by a non-negative linearcombination of Ax ≥ a and Bx ≥ b . Therefore, a non-negative combination of Cx ≤ c − , Ax ≥ a , and Bx ≥ b derives ≥ , so, label N (3) with this combination. Case 2: The root of T , labelled with Ax ≥ a , has a single child derived by an application ofthe division rule from Bx ≥ b . Note that by the invariant, the edge leading to the current nodeis labelled with Ax ≤ a − . Thus, at this current node we query Bx ≥ b . Let N (1) be the leafnode labelled with Bx ≤ b − and N (2) the leaf labelled with Bx ≥ b . At N (1) , we let T (1) bethe subtree of T rooted at the child of the root of T and recurse on ( N (1) , T (1) ) . On the otherhand, at N (2) we can derive ≥ by a non-negative linear combination. This follows like so:by the division rule of CP the inequality Ax ≥ a is exactly the inequality Bd x ≥ (cid:6) bd (cid:7) for some d ∈ Z ≥ dividing all entries in B . Therefore, subtracting dAx ≤ d ( a − from Bx ≥ b , wederive ≥ b − ⌈ bd ⌉ + d ≥ , and so we label N (2) with this linear combination.17 ase 3: T is a single node — a leaf of the CP refutation labelled with some axiom Ax ≥ a .By the invariant, the edge leading to N in the SP refutation R SP is labelled Ax ≤ a − . Wederive ≥ at N by subtracting Ax ≤ a − from the axiom Ax ≥ a .To see that the SP refutation we have constructed has rank at most twice that of the tree-likeCutting Planes refutation, observe that non-negative linear combinations are the only inferencerule of Cutting Planes which this construction requires depth 2 to simulate, while all other rulesrequire depth 1.To measure the size, note that every CP rule with a single premise is simulated in SP by asingle query, where one of the outgoing edges of that query is immediately labelled with ≥ .Each rule with two premises (case 1) is simulated by two queries in the SP refutation, whereone of the three outgoing edges is immediately labelled with ≥ . Each of these queriesbranch only on the inequalities belonging to R CP . Therefore, the size of the SP refutation R SP is O ( s ) . Size Simulation of Cutting Planes. Next, we show that SP size-simulates CP . Theorem 4.5. SP polynomially simulates CP .Proof. Let R = { A x ≥ a , A x ≥ a , . . . , A m x ≥ a m } be a CP refutation of an unsatisfiableset of integer linear inequalities F . We construct a SP refutation of F line by line, following R . Our SP refutation is a tree where the right-most path is of length m + 1 with edges labelled A ≥ b , . . . , A m ≥ b m . The left child of node i ≤ m along this path is labelled with ≥ ,which is derived as a non-negative linear combination of A j x ≥ a j for j < i , A i x ≤ a i − ,and F . The last node in the path is also labelled with ≥ . See Figure 4. A x A x A m x ≥ ≥ ≥ ≥ ≤ a − ≥ a ≤ a − ≥ a ≤ a m − ≥ a m Figure 4: The SP simulation of a CP refutation.Since A m x ≥ a m ≡ ≥ , the last node is trivially labelled with ≥ . Thus, we onlyneed to show that the left child of every node can be legally labelled with ≥ .18. A i x ≥ a i is an axiom. We can derive ≥ by subtracting A i x ≤ a i − from A i x ≥ a i ∈ F .2. A i x ≥ a i is a non-negative combination of two previous inequalities, i.e., A i x ≥ a i is αA j x + βA j x ≥ αa j + βa j for some j , j < i and α, β ∈ Z ≥ . We can derive ≥ by subtracting A i x ≤ a i − from the non-negative linear combination of A j x ≥ a j and A j x ≥ a j used to derive A i x ≥ a i .3. A i x ≥ a i is obtained by an application of the division rule to A j x ≥ a j for some j < i ,i.e., A i x ≥ a i is A j c x ≥ l a j c m where c ∈ N divides every entry in A j . On the path to this node in our SP refutationwe queried A j x ≥ a j . Dividing this inequality by c and subtracting A i x ≤ a i − , weobtain ≥ a j /c − ( ⌈ a j /c ⌉ − . This gives us ≥ a j c + 1 − (cid:6) a j c (cid:7) . Since the right-handside is strictly positive this can be normalized to give ≥ . Tree-Like CP and Balanced SP . It is known that CP refutations cannot be balanced (i.e.size- s refutations being turned into size O ( s ) depth O (log s ) refutation) in CP . Here, weshow that CP proofs can be turned into balanced SP proofs. More specifically, we prove thefollowing. Theorem 4.6. Suppose there is a size s tree-like CP refutation of a set of linear integer in-equalities F . Then there is a size s depth O (log s ) SP refutation of F .Proof. The construction is recursive. Let T be the tree corresponding to P . Base case. | T | = O (1) , then we can use one of the previous simulation theorems to createan SP refutation of P . Recursive step. Let v be a node in T such that the subtree T v rooted at v satisfies | T | / ≤| T v | ≤ | T | / , which must exist since the size measure is additive. Let Bx ≥ b be the linein P corresponding to the node v . Our SP simulation starts by querying Bx . We continuerecursively depending on the outcome of the query. If Bx ≥ b then we apply the recursiveconstruction to T \ T v , where we can think of Bx ≥ b as a new axiom. Otherwise, if Bx ≤ b − (which contradicts the the input set of inequalities F ) we apply the recursive construction to T v . The size is clearly preserved, and the depth of the proof becomes logarithmic, since weare reducing the size of the proof to be simulated by a constant factor on each branch of aquery. Bounded Depth and Space CP yields Balanced SP Proofs In order to talk about the spaceof refuting a given CNF, one needs to generalize the notion of lines to the notion of configura-tions. Configurations model the state of the working tape of the non-deterministic TM in thedefinition of a proof system. More formally, we have the following.19 efinition 4.7. Let P be a proof system, we consider a refutation of an unsatisfiable formula F as a sequence of configurations D , . . . , D k , where each configuration D i is a set of linesof P such that D k = ∅ and each D i follows D i − by one of the derivation steps. Let D ′ i − is D i − \ { L i − , , . . . L i − ,p } where { L i − , , . . . L i − ,p } is an arbitrary (possibly empty) subset ofinequalities contained in D i − . • Axiom Download : D i = D ′ i − ∪ { L j } where L j is one of the lines in F or one of theaxioms of the proof system Π . • Inference : D i = D ′ i − ∪ { L } for some line L inferred by one of the inference rules of Π applied to the set of inequalities in D i − .The space of a refutation R = D , . . . , D k is Space ( R ) = max i ∈ [ k ] | D i | and the space ofrefuting F in a proof system P is Space ( F ) = min R - refutation of F Space ( R ) . Next, we show that strong depth lower bounds on SP refutations can lead to time-spacetradeoffs for CP . Theorem 4.8. Suppose that we have a length ℓ , space s CP refutation of an unsatisfiable setof linear integral inequalities. Then there is a depth O ( s log ℓ ) SP refutation of the same setof linear integral inequalities.Proof. Let D , . . . , D ℓ be configurations of the given CP refutation. Then D ℓ = { ≥ } .The proof follows by applying the following main observation with i = ℓ to obtain an SP treewhere each leaf is labelled with ≥ . Main observation. Let D , . . . , D ℓ be configurations in a CP refutation such that | D i | ≤ s for i ∈ [ ℓ ] . For every i there is a SP tree of queries of depth O ( s log i ) such that there is exactlyone path along which we know all L ∈ D i and other paths end in leaves labelled ≥ .In the rest, we prove the main observation. The construction is recursive. Let c ≥ be thehidden constant from the big-Oh notation (it will be clear from the proof that it exists). Base case. D is a singleton set containing an axiom. Take the tree to be a single node. Itclearly satisfies the statement of the lemma. Recursive step. Assume for simplicity that i is divisible by . The SP tree starts with acomplete binary tree in which every line from D i/ is queried. The depth of this tree is ≤ s .Exactly one path P is labelled with lines from D i/ . All other paths contain at least one labelthat is the negation of a line from D i/ . To finish the construction, we treat these two casesseparately.In the case that a path contains a negation of a line from D i/ , we attach to its leaf the SP tree we obtain recursively by running our construction on D , . . . , D i/ . We know that theattached tree has all, but one, leaves labelled ≥ . For the one leaf that is not labelled with ≥ , we know all lines from D i/ on the path to that leaf. Since we attached this tree to apath along which we know the negation of a line from D i/ , we can immediately label this leafby ≥ . The overall depth that we get in this case is s for the initial tree and cs log( i/ forthe tree obtained recursively, i.e., s + cs (log i − 1) = s + cs log i − cs ≤ cs log i , since c ≥ .20n case of the path P note that D i/ , . . . , D i can be viewed as configurations of a refu-tation of the original set of unsatisfiable linear integral inequalities together with D i/ treatedas additional axioms. In our SP construction, at the leaf of P we also know all inequali-ties D i/ , and so can treat them as axioms. Thus, we can apply the recursive construction to D i/ , . . . , D i . The calculation of the overall depth is exactly the same as in the previous case.The big-Oh is needed to take care of rounding issues when i is not divisible by 2. In this section, we prove near-optimal lower bounds on SP rank via reductions to randomizedand real communication complexity. We then tackle the harder problem of proving unre-stricted superpolynomial size lower bounds for SP . Although we are unable to prove suchlower bounds we explain why current approaches fail. Essentially all lower bounds for CP have been obtained by reducing to a communication complexity problem; in the case of tree-like CP , the reduction is to the communication complexity of a corresponding search problem.For more general dag-like CP , the reduction is to the size of “communication games” [13, 21](communication games are a dag-like model of communication that gives an equivalence be-tween communication size and monotone circuit size, analogous to the famous equivalencebetween communication depth and monotone formula size). Although tree-like CP proofscannot be balanced in general, communication protocols (both deterministic and randomized)can be balanced, and thus tree-like CP lower bounds follow from communication complexitylower bounds. Similarly, we show that it is not possible to balance SP refutations, and thus wecannot in general obtain size lower bounds directly from rank lower bounds. Moreover, weshow that SP refutations imply real communication protocols , and unlike ordinary commu-nication protocols, we show that real protocols cannot in general be balanced. This rules outproving length lower bounds on SP refutations from (real) communication complexity lowerbounds. SP Refutations Imply Communication Protocols. Real communication protocols were in-troduced by Kraj´ı˘cek [24]. In this model, the players are allowed to communicate by sendingreal-valued functions of their inputs to a referee who announces their comparison. Definition 5.1. A real communication protocol is a full binary tree in which every non-leafnode v is labeled with a pair of functions a v : X → R , b v : Y → R , and the leaves are labelledwith elements in Z . Two players, Alice and Bob, receive inputs from X × Y , with Alicereceiving x ∈ X and Bob receiving y ∈ Y . Beginning at the root, the players traverse the treeas follows: at each node, they send real values a v ( x ) and b v ( y ) to a “referee” who returns (toboth of them) a bit indicating the result of the comparison a v ( x ) ≥ b v ( y ) ; the players proceedto the left child if a v ≥ b v , and to the right child otherwise. Once they reach a leaf, theprotocol halts, and the players output the value in Z labelling the leaf; it computes a function f : X × Y → Z in the natural way.The cost of a real protocol is the depth of the tree, or equivalently the maximum numberof rounds of communications with the referee over any input ( x, y ) , and the size is the number21f nodes in the protocol. Similarly, the cost (size) of computing a function f is the smallestcost (size) real protocol computing f .Kraj´ı˘cek showed that from a low-rank CP refutation of an unsatisfiable CNF, one canobtain a real communication protocol for solving a related search problem [24]. We describethis search problem next. Definition 5.2. Let F = C ∧ C ∧ · · · ∧ C m be an unsatisfiable CNF and ( X, Y ) be a partitionof the variables. The relation Search X,Y ( F ) ⊆ { , } X × { , } Y × [ m ] consists of all triples ( x, y, i ) such that the total assignment z = ( x, y ) to all of the variables of C i falsifies the clause i . The search problem is the natural interpretation of a refutation in the setting of commu-nication. Indeed, essentially every lower bound for CP has been proved by reducing to thecommunication complexity of the search problem. In a similar manner, we show that SP refu-tations can be turned into both randomized and real protocols for the search problem whichpreserve the rank of the refutation. Lemma 5.3. Let F be an unsatisfiable CNF formula and ( X, Y ) be any partition of thevariables. Any SP refutation of F of rank r implies a real communication protocol of cost O ( r + log n ) and an O ( r log n + log n ) randomized bounded-error protocol for solving Search X,Y ( F ) .Proof. Let P be a rank r SP refutation of F and let ( x, y ) ∈ { , } | X | × { , } | Y | be anyassignment to the variables of F ; Alice is given x and Bob is given y . Our protocol willtraverse from the root of P to the leaves in search of a clause that is falsified by ( x, y ) . At eachnode in P that the players visit, labeled with a query ( Ax ≤ b − , Ax ≥ b ) , they will evaluatetheir input on Ax ≥ b . Because the assignment ( x, y ) is integral, it will satisfy exactly one ofthe inequalities Ax ≤ b − , Ax ≥ b, (1)and will falsify the other. Alice and Bob will then continue to the child of the current nodewhich is reached by traversing the edge labeled with the inequality in (1) that is satisfied by ( x, y ) . This protocol will continue for at most r iterations until the players reach a leaf of therefutation P .Every leaf of the protocol is labelled with a convex combination of the axioms of F alongwith the inequalities labelling the path leading to this leaf, which evaluates to ≥ . Becausewe have maintained that ( x, y ) satisfies each of the inequalities along the path leading to thisleaf, it must falsify one of the axioms of F used in the convex combination. If this was notthe case, the polytope formed by the inequalities in the convex combination would contain afeasible point, and by Fact 2.4, a convex combination equalling ≥ would not exist.Once at a leaf, Alice and Bob can communicate in at most O (log n ) rounds to find theclause of F that is falsified under ( x, y ) . Using Fact 2.4, we may assume that this is aconvex combination of ℓ ≤ n + 2 linear inequalities. Denote these inequalities by A x ≥ b , . . . , A ℓ x ≥ b ℓ , and their coefficients in the convex combination be c , . . . , c ℓ . Alice and22ob will repeat the following procedure, maintaining that the current inequality is always lessthan or equal to it’s constant term. Divide the set of inequalities into two halves, and definethe threshold function L := ℓ/ X i =1 c i ( A i x − b i ) − ℓ X j = ℓ/ c j ( A j x − b j ) ≥ . The players will run the protocol for deciding L on assignment ( x, y ) . If ( x, y ) falsifies L ,they recurse on the subset of inequalities A x ≥ b , . . . , A ℓ/ x ≥ b ℓ/ , otherwise, they recurseon the other half.Since the original convex combination evaluated to ≥ , at least one of P ℓ/ i =1 c i ( A i x − b i ) and P ℓj = ℓ/ c j ( A j x − b j ) must be strictly less than on the assignment ( x, y ) . Repeating thisprocess will converge to an inequality which is violated on ( x, y ) . Because we have ensuredthat the only inequalities that ( x, y ) falsifies in this convex combination are axioms of F , thisprocess will solve the search problem. Moreover, because this convex combination containsat most n + 2 linear inequalities, this process will terminate in O (log n ) rounds.In the model of real communication, any linear inequality Ax ≥ b can be evaluated in asingle bit of real communication; Alice sends Ax to the referee, while Bob sends b − Ay (Here, y is treated as a vector of length n having 0s in coordinates corresponding to x , and similarlyfor x ). Therefore, this leads to a O ( r + log n ) real communication protocol for Search X,Y ( F ) .Next, we adapt the above procedure to produce a randomized communication protocol byshowing that any inequality in the SP refutation can be computed in low communication. To dothis, Alice and Bob run the ε -error O (log m + log ε − ) -bit protocol of Nisan [ ? ] for decidingan m -bit linear inequality. By the well-known result of Muroga [26], any inequality on n Boolean variables only requires coefficients representable by O ( n log n ) bits (recall that Aliceand Bob’s input will always be a Boolean assignment and so this suffices). Because there are atmost r + log n inequalities evaluated along any root-to-leaf path in the refutation, the protocolis repeated at most r + log n many times. By a union bound, we require ε < c/ (log n + r ) ,where c is some constant bound on the error that we allow. Therefore, every inequality can becomputed in O (log n +log r ) many bits to compute, giving a O ( r log n +log n ) bounded-errorrandomized protocol for Search X,Y ( F ) . Rank Lower Bounds For SP . To take advantage of Lemma 5.3, we need to find somecandidate formulas on which to prove rank lower bounds and then study the search problemobtained from applying this transformation. We do so for both the Tseitin formulas and avariant of the pebbling contradictions, a reformulation of the classical black pebbling gamesas an unsatisfiable -CNF formula, originally introduced by Ben-Sasson et al. [3, 4].The black pebbling game can be phrased as a contradictory -CNF as follows: Let G bea DAG with a set of source nodes S ⊆ V ( G ) (having fanin 0), a unique sink node t (withfanin 2 and fanout 0), and the remaining nodes each having fanin exactly 2. The pebblingcontradictions Peb G consists of the following n + 1 clauses over variables v ∈ V ( G ) :23 sink axiom: a single clause ¬ t , • source axioms: a clause s for every source s ∈ S . • pebbling axioms: a clause ¬ u ∨ ¬ v ∨ w for every w ∈ V \ S with immediate children u, v .Unfortunately, both the pebbling contradictions and the Tseitin formulas have short SP refuta-tions. In particular, for any graph G , the polytope formed by the constraints of Peb G is emptyand therefore a nonnegative combination of the constraints yielding ≥ exists, this is a validrank-1 SP refutation. For Tseitin, this follows from the poly-logarithmic rank upper bound inTheorem 3.1. Therefore, we modify these formulas to make them harder to solve.A standard technique for amplifying the hardness of computing some function f : X n →Z is by lifting that function. This is done by obscuring the input variables by replacing themeach by a small function g : A → X known as a gadget , which must be evaluated beforelearning the input to the original function. For an input α ∈ A n , the function f lifted bygadget g is then ( f ◦ g n )( α ) = ( g ( α ) , . . . , g ( α n )) . The intuition is that this lifted function f ◦ g n should be much harder than the original becausethe players must first evaluate the gadget g ( α i ) to learn each bit of the actual input to thefunction f . Furthermore, intuition says that if the gadget is sufficiently difficult to compute,then the model will be reduced to using much more rudimentary methods to evaluate the liftedfunction.The standard hard-to-compute gadget is the pointer or index gadget, IND ℓ : [ ℓ ] × { , } ℓ →{ , } . For an input ( x, y ) ∈ [ ℓ ] ×{ , } ℓ , x is a log ℓ -bit string encoding a pointer into the ℓ -bitstring y ∈ { , } ℓ . The output of IND ℓ ( x, y ) is y [ x ] , the x -th bit in the string y . This is mostoften applied in communication complexity, where typically the variable partition between theplayers is that Alice is given x ∈ [ ℓ ] and Bob is given y ∈ { , } ℓ . In any standard model ofcommunication, for this partition of the variables, it is difficult to imagine any communicationprotocol which could compute the index gadget with significant advantage over the trivialprotocol; sending every bit of Alice’s pointer x to Bob.Raz and McKenzie formalized this intuition, in what has become known as a lifting the-orem [18, 29]. They show that deterministic communication protocols cannot compute anyfunction f lifted by the index gadget significantly better than simply mimicking a decisiontree computing f , and performing the trivial protocol for evaluating the index gadget everytime a bit of the input to f is needed.Lifting theorems for real communication were originally proved by Bonet et al. [ ? ] basedon the techniques of Raz-McKenzie. Their theorem lifts lower bounds on the decision treecomplexity of a function f to lower bounds on the cost of real communication protocols com-puting f ◦ IND nℓ . The decision tree complexity DT ( f ) of a function f is simply the minimumdepth need by any decision tree to compute f . We use a simplified lifting theorem for realcommunication by de Rezende et al. [11], which we state next. Theorem 5.4. (de Rezende et al. [11]) Let f be a function with domain { , } n and let ℓ = n .If there is a real communication protocol of cost c that solves f ◦ IND nℓ where Alice is given ∈ [ ℓ ] n and Bob is given y ∈ { , } nℓ , then there is a decision tree solving f using O ( c/ log ℓ ) queries. Our goal is now to is to combine this theorem with Lemma 5.3 in order to prove lowerbounds on the rank of SP refutations of Peb G ◦ IND nℓ . Syntactically speaking though, Peb G ◦ IND nℓ is not a valid input to our proof system. Therefore, we must show that the lifted functioncan indeed be phrased as a small CNF formula. The following encoding is due to Beame etal. [ ? ]:Let F = C ∨ . . . ∨ C m be a CNF formula over variables x , . . . , x n . The CNF representing F ◦ IND nℓ is defined on new sets of variables y i,j and z i,j for all i ∈ [ n ] and j ∈ [ ℓ ] . This CNFhas the following set of clauses • Pointer clauses: for each i ∈ [ n ] , a clause y i, ∨ . . . ∨ y i,ℓ . • F -clauses: for each clause C i ∈ F , where C i = y i ∨ . . . ∨ y i k ∨ ¬ x i k +1 ∨ . . . ∨ ¬ x i s and for every ( j , . . . , j n ) ∈ [ ℓ ] n , a clause ( y i ,j → z i ,j ) ∨ . . . ∨ ( y i k ,j k → z i k ,j k ) ∨ ( y i k +1 ,j k +1 → ¬ z i k +1 ,j k +1 ) ∨ . . . ∨ ( y i s ,j s → ¬ z i s ,j s ) . We will abuse notation and use F ◦ IND nℓ to denote the function, as well as it’s CNF formula-tion, and use context to differentiate between the two.A final subtlety that should be mentioned is that applying Theorem 5.4 to an SP refutationof Peb G ◦ IND nℓ yields a protocol for Search X,Y (Peb G ◦ IND nℓ ) which is not in the correct formto apply Theorem 5.4 ( Search X,Y (Peb G ◦ IND nℓ ) is a function of a lifted function, whereasTheorem 5.4 can only be applied to lifted functions). Luckily, this is not a significant issue;Huynh et al. [ ? ] show that, for any unsatisfiable CNF F , any real communication protocolfor Search X,Y ( F ◦ IND nℓ ) , where X = [ ℓ ] n and Y = { , } nℓ , implies a real communicationprotocol for Search X,Y ( F ) ◦ IND nℓ with the same parameters.It is now straightforward to obtain lower bounds on the rank of SP refutations. For thelifted pebbling formulas, SP rank lower bounds follow from combining Lemma 5.3 and The-orem 5.4 with a lower bound on the complexity of decision trees solving Peb G proved by deRezende et al. [11]. Theorem 5.5. There exists a graph G of indegree 2 on n vertices such that the unsatisfiableCNF formula Peb G ◦ IND nℓ , for ℓ = n , on n ( ℓ + log ℓ ) variables requires rank Ω( √ n log n ) to refute in SP .Proof. Consider the pebbling formulas. de Rezende et al. [11] showed the existence of agraph G on n vertices with indegree 2 such that the decision tree complexity of outputtinga falsified clause of the Peb G formulas is Ω( p n/ log n ) . Applying the real communicationlifting theorem (Theorem 5.4) and combining this with the fact that shallow SP refutationsgive efficient protocols for the associated search problem (Lemma 5.3), proves the desired Ω( √ n log n ) lower bound on the rank of SP refutations of Peb G ◦ IND nℓ .25inally, a similar technique can be applied to obtain a lower bound on the rank of SP refu-tations for a lifted variant of the Tseitin formulas. This follows from the lower bound on therandomized communication complexity of the search problem for the Tseitin formulas liftedby a small constant-size gadget, which was obtained by G¨o¨os and Pitassi [17]. In particular,they use the versatile gadget , VER : Z × Z → { , } , which is defines as VER ( x, y ) = 1 ⇐⇒ x + y (mod 4) ∈ { , } . Theorem 5.6. (G¨o¨os and Pitassi [17]) There exists a constant-degree graph G on n verticessuch that, if ℓ is any { , } vertex labelling with odd total weight and ( X, Y ) is any partitionof the variables, any bounded-error randomized communication protocol for Search X,Y (Tseitin( G, ℓ ) ◦ VER n ) on O ( n ) variables, requires Ω( n/ log n ) bits of communi-cation. Furthermore, G¨o¨os and Pitassi showed how, for any CNF formula F , the composed func-tion F ◦ VER n can be encoded as a CNF formula: For each variable z i in the F , definenew variables vars ( z i ) = { x i, , x i, , x i, , x i, } , where the ( x i, , x i, ) and ( x i, , x i, ) should bethought of as binary encodings of the pair of inputs in Z × Z to VER . In particular, let α bea truth assignment and α ↾ vars ( z i ) be its restriction to the variable set vars ( z i ) . We will abusenotation and interpret VER ( α ↾ vars ( z i ) ) as applying the gadget VER to the pair of elements in Z × Z that α ↾ vars ( z i ) is the binary encoding of.Let C be a clause in F and assume for simplicity that C = z ∨ ¬ z ; it will be clear thatthis definition will generalize to any arbitrary clause. For every truth assignment α ∈ { , } n such that VER ( α ↾ vars ( z ) ) = 0 and VER ( α ↾ vars ( z ) ) = 1 , define a new clause C α = _ i =1 ¬ x α ,i ! _ _ i =1 ¬ x α ,i ! , where x αi,j = x i,j if α ↾ x i,j = 1 and x αi,j = ¬ x i,j otherwise. These clauses simply state that theoutput of the gadgets which correspond to the variables occurring in the clause C cannot bea falsifying assignment to C . The CNF representation of F ◦ VER n is the conjunction of theclauses C α for every C ∈ F ◦ VER n .Observe that if F contains m clauses, each of width at most w , then the CNF representationof F ◦ VER n contains at most m · w clauses. The width of every clause in the Tseitin formulasare bounded by d , where d is the maximal degree of any vertex in the underlying graph. Usingthis fact, we are able to obtain near-optimal lower bounds on the rank of SP refutations bycombining Theorem 5.6 with Lemma 5.4 in the same manner as the proof of Theorem 5.5. Theorem 5.7. There a constant-degree graph G on n vertices such that if ℓ is any { , } vertexlabelling with odd total weight, the CNF formula Tseitin( G, ℓ ) ◦ VER n , on O ( n ) variablesand clauses, requires SP refutations of rank Ω( n/ log n ) . The lower bound from Theorem 5.7 should be contrasted with the logarithmic-rank SP upper bound on Tseitin from Theorem 3.1. 26 P Refutations Cannot Be Balanced. Optimistically, one could hope that the length andrank of SP refutations may be closely related and therefore that we could leverage these rankbounds to obtain lower bounds on the length of SP refutations. We answer this question neg-atively, showing that there exists a contradictory CNF which admits short refutations, but forwhich these refutations must be almost maximally deep. That is, we show that SP refutationscannot be balanced; an SP refutation of length S does not imply one of rank O (log S ) . Thisshows that in SP the rank of refutations is a distinct complexity measure from the length.In order to obtain time-space tradeoffs, de Rezende et al. [11] proved Resolution upperbounds on the lifted pebbling contradictions. Combining this upper bound (which can besimulated efficiently in SP ) with the lower bound from Theorem 5.5 exhibits a formula thatrequires small size, but near-maximal rank to refute in SP . Theorem 5.8. SP refutations cannot be balanced.Proof. Suppose that a SP refutation of length S implied the existence of a SP refutation ofthe same formula of rank O (log S ) . Let G be the graph from Theorem 5.5 on n vertices,and let Peb G be the pebbling contradictions defined on this graph. It follows immediatelyfrom Lemmas 7.2 and 7.3 from de Rezende et al. [11] that for any graph of indegree 2 on n vertices, that there is a Resolution refutation of Peb G ◦ IND nℓ of length O ( nℓ ) . Since SP can p -simulate Cutting Planes (and therefore Resolution), this implies a poly ( n ) upper bound onthe same formula in SP . Under the assumption that SP refutation can be balanced, this wouldimply a SP refutation of depth O (log n ) of Peb G ◦ IND nℓ , contradicting the lower bound fromCorollary 5.5.Although it is a well-known fact that tree-like Cutting Planes refutations cannot be bal-anced, Impagliazzo et al. [ ? ] show that the randomized communication protocols for thesearch problem obtained from CP refutations can be balanced. Using this fact, they showthat a length S tree-like Cutting Planes proof implies a depth O (log S ) protocol for the searchproblem. This implies that communication cost lower bounds for the search problem implylength lower bounds for tree-like Cutting Planes refutations.Optimistically one could hope that a similar approach could be applied to SP refutations.This is reinforced by the fact that the real communication protocols for the search problemobtained from SP refutations (Lemma 5.3) maintains the same topology as the refutation.That is, the cost and size of the resulting protocol are approximately equal to the rank andlength of the proof (unfortunately, this is not the case for the randomized protocols obtainedfrom SP refutations). Therefore, one might hope that even though SP cannot be balanced, thecorresponding real communication protocols for the search problem can be balanced. Thus,lower bounds on the rank of real-communication protocols for the search problem would implylower bounds on the size of SP proofs. Corollary 5.9. Any SP refutation of length S and rank r of an unsatisfiable formula F impliesa real communication protocol of size O ( S · n ) and cost O ( r +log n ) for solving Search X,Y ( F ) .Proof. This follows from observing that the protocol obtained in Lemma 5.3 also preservesthe topology (and therefore both the rank and the length) of the refutation.27 eal Communication Protocols Cannot Be Balanced. Analogous to Theorem 5.8 (show-ing that SP proofs cannot be balanced) in this section we will show that real communicationprotocols cannot be balanced. This should be contrasted with other standard models of com-munication such as randomized and deterministic, which can be balanced. In particular, weexhibit a function which has a real communication protocol of small size, but for which everyreal protocol must have high cost. Towards this end, we first prove lower bounds on the realcommunication complexity of the famous set disjointness function.The set disjointness function DISJ n is the canonical NP cc -complete problem. Each playeris given an n -bit string, interpreted as indicator vectors for an underlying set of n elements,and they are asked to determine whether their sets are disjoint. That is, the players aim tosolve the function DISJ n ( x, y ) = _ i ∈ [ n ] ( x i ∧ y i ) . To our knowledge, the only known technique for obtaining lower bounds on the real com-munication of any problem are via lifting theorems, reducing the task of proving lower boundson lifted functions to the decision tree complexity of the un-lifted function. Although DISJ n can be seen as a lifted function (the OR n function lifted by the two-bit AND gadget), thesereal communication lifting theorems work only for super-constant sized gadgets, and thereforecannot be applied directly to DISJ n . We circumvent this difficulty by exploiting the fact that DISJ n is NP cc -complete. To do so, we find a lifted function in NP cc to which our simulationtheorems can be applied. Consider the n -bit OR n function composed with the index gadget, OR ◦ IND nℓ , for some ℓ defined later. Lemma 5.10. OR n ◦ IND nℓ ∈ NP cc , for any ℓ ≤ polylog ( n ) .Proof. First, observe that the index gadget IND ℓ ( x i , y i ) , for a single bit i of the input to the OR n function, can be computed by a brute force protocol in log ℓ bits of communication.Alice simply sends to Bob the log ℓ bits of her input x i = x i, , . . . , x i, log ℓ . Bob is then able toevaluate IND ℓ ( x i , y i ) .Now, consider the following NP cc protocol for OR n ◦ IND nℓ : Alice and Bob are given as aproof, a log n -bit string indicating the index i ∈ [ n ] of the OR n function where IND ℓ ( x i , y i ) =1 . They then perform the brute force protocol to evaluate IND ℓ ( x i , y i ) and verify that theoutcome is indeed 1. In total, this requires log ℓ + log n + 1 = polylog ( n ) bits of NP cc -communication.To obtain lower bounds on OR n ◦ IND nℓ , we appeal to the real communication simulationtheorem (Theorem 5.4), reducing communication lower bounds for OR n ◦ IND nℓ on the liftedproblem to the well-known linear decision tree lower bounds on the OR n function. Lemma 5.11. Let ℓ = n . The cost of any real communication protocol computing OR n ◦ IND nℓ is Ω( n log ℓ ) . roof. Combining the Ω( n ) lower bound on the decision tree complexity of computing the OR n function with the simulation theorem of de Rezende et al. [11] proves the result. Theorem 5.12. The cost of any real communication protocol computing DISJ n is Ω(( n log n ) / ) .Proof. Let ℓ = n . Consider the following reduction from OR ◦ IND nℓ to an instance of setdisjointness. By Lemma 5.10, the NP cc -complexity of OR n ◦ IND nℓ is log ℓ + log n + 1 . Thatis, there exists a cover of the 1s of the communication matrix of OR n ◦ IND nℓ with at most nℓ rectangles. Enumerating the 1-rectangles gives us an instance of set disjointness: on input ( x, y ) to OR n ◦ IND nℓ , Alice constructs the nℓ -bit string which is the indicator vector I x ( x ) of the set of 1 rectangles which x belongs to, similarly Bob constructs I y ( y ) the same for y .Thus OR n ◦ IND nℓ ( x, y ) = 1 iff DISJ nℓ ( I x ( x ) , I y ( y )) = 1 .Therefore, the lower bound of Ω( n log ℓ ) on the cost of computing OR n ◦ IND nℓ , fromLemma 5.11, implies a lower bound of Ω(( n log ℓ ) / ) on the cost of any real communicationprotocol computing DISJ n . Corollary 5.13. Real communication protocols cannot be balanced.Proof. We begin by giving a cost n , size n + 1 real communication protocol for DISJ n = ∨ ni =1 ( x i ∧ y i ) . Sequentially from i = 1 , . . . , n , Alice and Bob solve x i ∧ y i . To do this, Alicesends x i to the referee and Bob sending − y i . Observe that x i ∧ y i = 1 iff x i ≥ − y i . Thisprotocol contains exactly n nodes, one for each query to x i ∧ y i for i ∈ [ n ] , one for each nodeannouncing that x i = y i = 1 and a single node announcing that x ∩ y = ∅ .Suppose by contradiction that one could balance real communication protocols. The size n +1 protocol would therefore imply a cost log(2 n +1) real protocol for DISJ n , contradictingthe lower bound from Theorem 5.12. This paper introduces and develops the Stabbing Planes proof system as a natural extension ofDPLL and pseudoBoolean solvers to handle a more expressive set of queries. Although it isequivalent to a tree-like version of a system already in the literature, this new perspective turnsout to be quite useful for proving upper bounds. This paper is only a preliminary explorationof the SP proof system and leaves open many interesting problems from both a theoretical aswell as a practical perspective.1. As mentioned in the preliminaries, we do not have an analog to Cook et al. [8] forStabbing Planes and so it is unknown whether for SP refutations, the length and size(number of bits) can be treated as the same measure. That is, is it possible to provethat any SP refutation of length l can be simulated by an SP planes refutation of size poly ( l, n ) ?2. We have shown that that Cutting Planes refutations of small rank can be simulated byStabbing Planes refutation of small rank, and that Cutting Planes refutations of small29ize can also be simulated by SP refutations of small size. Can we simulate both rankand size efficiently? That is, can any CP refutation of rank r and size s be simulated bya SP refutation of rank poly ( r ) and size poly ( s ) ?3. Prove superpolynomial lower bounds for SP . Kraj´ı˘cek [ ? ] gave exponential lowerbounds on the length of R ( CP ) refutations when both the width of the clauses, andthe size of the coeficients appearing in the inequalities are sufficiently bounded. Thiswas later improved by Kojevnikov [ ? ] to remove the restriction on the size of the coef-ficients for tree-like R ( CP ) . In particular, to obtain any lower bound at all, the width ofthe clauses appearing in the R ( CP ) refutations must be bounded by o ( n/ log n ) . FromTheorem 4.2, a size S and rank D SP refutation implies an R ( CP ) proof of size O ( S ) and width O ( D ) . Therefore, this result is also a size lower bound for bounded-depth SP . Unfortunately, it appears that these techniques are fundamentally limited to be ap-plicable only to SP refutations with low depth, and so new techniques seem needed toovercome this barrier.4. As mentioned in the introduction, we feel that SP has potential, in combination withstate-of-the-art algorithms for SAT, for improved performance on certain hard instances,or possibly to solve harder problems such as maxSAT or counting satisfying assign-ments, possibly in conjunction with solvers such as CPLEX. The upper bound on theTseitin example illustrates the kind of reasoning that SP is capable of: arbitrarily split-ting the solution space into sub-problems based on some measure of progress. Thisopens up the space of algorithmic ideas for solvers and should allow one to take fulleradvantage of the expressibility of integer linear inequalities. For example, since geo-metric properties of the rational hull formed by the set of constraints can be determinedefficiently, an SP -based solver could branch on linear inequalities representing somegeometric properties of the rational hull. 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