Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.


Introduction
Finding a satisfying assignment for a propositional formula (SAT) is a central component for many computationally hard problems.Despite being older than 50 years and exponential time in the worst-case, the DPLL algorithm [DLL62, DP60,Rob65] is the core of essentially all high performance modern SAT-solvers.DPLL is a recursive boolean method: at each call one variable x of the formula F is chosen and the search recursively branches into the two cases obtained by setting x respectively to 1 and 0 in F. On UNSAT formulas DPLL performs the worst and it is well-known that the execution trace of the DPLL algorithm running on an unsatisfiable formula F is nothing more than a treelike refutation of F in the proof system of Resolution [Rob65] (Res).
Since SAT can be viewed as an optimization problem the question whether Integer Linear Programming (ILP) can be made feasible for satisfiability testing received a lot of attention and is considered among the most challenging problems in local search [SKM97,KS03].One proof system capturing ILP approaches to SAT is Cutting Planes, a system whose main rule implements the rounding (or Chvátal cut) approach to ILP.Cutting planes works with integer linear inequalities of the form ax ≤ b, with a, b integers, and, like resolution, is a sound and complete refutational proof system for CNF formulas: indeed a clause C = (x 1 ∨ . . .∨ x r ∨ ¬y 1 ∨ . . .∨ ¬y s ) can be written as the integer inequality y − x ≤ s − 1.
Beame et al. [BFI + 18], extended the idea of DPLL to a more general proof strategy based on ILP.Instead of branching only on a variable as in resolution, in this method one considers a pair (a, b), with a ∈ Z n and b ∈ Z, and branches limiting the search to the two half-planes: ax ≤ b − 1 and ax ≥ b.A path terminates when the LP defined by the inequalities in F and those forming the path is infeasible.This method can be made into a refutational treelike proof system for UNSAT CNF's called Stabbing planes (SP) ([BFI + 18]) and it turned out that it is polynomially equivalent to the treelike version of Res(CP), a proof system introduced by Krajíček [Kra98a] where clauses are disjunction of linear inequalities.Furthermore, Stabbing Planes captures the popular branch-and-cut ILP algorithms.
In this work we consider the complexity of proofs in SP focusing on the length, i.e. the number of queries in the proof; the depth (called also rank in [BFI + 18]), i.e. the length of the longest path in the proof tree; and the size, i.e. the bit size of all the coefficients appearing in the proof.
1.1.Previous works and motivations.After its introduction as a proof system in the work [BFI + 18] by Beame, Fleming, Impagliazzo, Kolokolova, Pankratov, Pitassi and Robere, Stabbing Planes received great attention.The quasipolynomial upper bound for the size of refuting Tseitin contradictions in SP given in [BFI + 18] was surprisingly extended to CP in the work of [DT20] of Dadush and Tiwari refuting a long-standing conjecture.Recently in [FGI + 21], Fleming, Göös, Impagliazzo, Pitassi, Robere, Tan and Wigderson were further developing the initial results proved in [BFI + 18] making important progress on the question whether all Stabbing Planes proofs can be somehow efficiently simulated by Cutting Planes.
Significant lower bounds for depth can be obtained for SP, using modern developments of a technique for CP based on communication complexity of search problems introduced by Impagliazzo, Pitassi, Urquhart in [IPU94]: in [BFI + 18] it is proven that size S and depth D SP refutations imply treelike Res(CP) proofs of size O(S) and width O(D); Kojevnikov [Koj07], improving the interpolation method introduced for Res(CP) by Krajíček [Kra98a], gave exponential lower bounds for treelike Res(CP) when the width of the clauses (i.e. the number of linear inequalities in a clause) is bounded by o(n/ log n).However [BFI + 18] shows that there are no n/ log n depth treelike Res(CP) proofs of the given formula at all.Hence these lower bounds are applicable only to very specific classes of formulas (whose hardness comes from boolean circuit hardness) and only to SP refutations of low depth.
Nevertheless SP appears to be a strong proof system.Firstly notice that the condition terminating a path in a proof is not a trivial contradiction like in resolution, but is the infeasibility of an LP, which is only a polynomial time verifiable condition.Hence linear size SP proofs might be already a strong class of SP proofs, since they can hide a polynomial growth into one final node whence to run the verification of the terminating condition.
Rank and depth in CP and SP.It is known that, contrary to the case of other proof systems like Frege, neither CP nor SP proofs can be balanced (see [BFI + 18]), in the sense that a depth-d proof can always be transformed into a size 2 O(d) proof.The depth of CP-proofs of a set of linear inequalities L is measured by the Chvátal rank of the associated polytope P . 1 It is known that rank in CP and depth in SP are separated, in the sense that Tseitin principles can be proved in depth O(log 2 n) depth in SP [BFI + 18], but are known to require rank Θ(n) to be refuted in CP [BOGH + 06].In this paper we further develop the study of proof depth for SP.
Rank lower bound techniques for Cutting Planes are essentially of two types.The main method is by reducing to the real communication complexity of certain search problem [IPU94].As such this method only works for classes of formulas lifted by certain gadgets capturing specific boolean functions.A second class of methods have been developed for Cutting Planes, which lower bound the rank measures of a polytope.In this setting, lower bounds are typically proven using a geometric method called protection lemmas [BOGH + 06].These methods were recently extended in [FGI + 21] also to the case of Semantic Cutting Planes.In principle this geometric method can be applied to any formula and not only to the lifted ones, furthermore for many formulas (such as the Tseitin formulas) it is known how to achieve Ω(n) rank lower bounds in CP via protection lemmas, while proving even ω(log n) lower bounds via real communication complexity is impossible, due to a known folklore upper bound.
Lower bounds for depth in Stabbing Planes, proved in [BFI + 18], are instead obtained only as a consequence of the real communication approach extended to Stabbing Planes.In this paper we introduce two geometric approaches to prove depth lower bounds in SP.
Specifically the results we know at present relating SP and CP are: (1) SP polynomially simulates CP (Theorem 4.5 in [BFI + 18]).Hence in particular the PHP m n can be refuted in SP by a proof of size O(n 2 ) ( [CCT87]).Furthermore it can be refuted by a O(log n) depth proof since polynomial size CP proofs, by Theorem 4.4 in [BFI + 18], can be balanced in SP. (1) a Ω(n/ log 2 n) lower bound for the formula Ts(G, w)  Kra98b] for CP of reducing shallow proofs of a formula F to efficient real communication protocols computing a related search problem and then proving that such efficient protocols cannot exist.
The only lower bounds techniques on the depth of Stabbing Planes proofs come from reductions to communication complexity, which is a lower bound technique for CP.This is also in contrast with other weaker proof systems such as Resolution and Cutting Planes, where we have direct combinatorial and geometric techniques for proving depth lower bounds.Direct lower bound techniques are valuable as they are tailored to the proof system and thus shed light on its behaviour and weaknesses, unlike semantic techniques such as reductions to monotone circuits or communication complexity, which prove lower bounds on more general objects (such as monotone circuits and real communication protocols).
In this work we address such problems.Our first steps in this direction were to set up methods working for truly combinatorial statements, like Ts(G, w) or PHP m n , which we know to be efficiently provable in SP, but on which we cannot use methods reducing to the complexity of boolean functions, like the ones based on communication complexity.
We present two new methods for proving depth lower bounds in SP which in fact are the consequence of proving length lower bounds that do not depend on the bit-size of the coefficients.
As applications of our two methods we respectively prove: (1) An exponential separation between the rank3 in CP and the depth in SP, using a new counting principle which we introduce and that we call the Simple Pigeon Principle SPHP.We prove that SPHP has O(1) rank in CP and requires Ω(log n) depth in SP.
Together with the results proving that Tseitin formulas requires Ω(n) rank lower bounds in CP ([BOGH + 06]) and O(log 2 n) upper bounds for the depth in SP ([BFI + 18]), this proves an incomparability between the two measures.
(2) An almost linear lower bound on the size of SP proofs of the PHP m n and for Tseitin Ts(G, ω) contradictions over the complete graph.These lower bounds immediately give optimal Ω(log n) lower bound for the depth of SP proofs of the corresponding principles.
(3) An almost linear lower bound for the size and Ω(log n) lower bound of the depth for the the Linear Ordering Principle LOP n .(4) Finally, we prove a superlinear lower bound for the size of SP proofs of Ts(G, ω), when G is a n × n grid graph H n .In turn this implies an Ω(log n) lower bound for the depth of SP proofs of Ts(H n , ω).Proofs of depth O(log 2 n) for Ts(H n , ω) are given in [BFI + 18].
Our results are derived from the following initial geometrical observation: let S be a space of admissible points in {0, 1, 1/2} n satisfying a given unsatisfiable system of integer linear inequalities F(x 1 , . . ., x n ).In a SP proof for F, at each branch Q = (a, b) the set of points in the slab(Q) = {s ∈ S : b − 1 < ax < b} does not survive.At the end of the proof on the leaves, where we have infeasible LP's, no point in S can survive the proof.So it is sufficient to find conditions such that, under the assumption that a proof of F is "small", even one point of S survives the proof.In pursuing this approach we use two methods.
The antichain method.Here we use a well-known bound based on Sperner's Theorem [CCT09,vLW01] to upper bound the number of points in the slabs where the set of non-zero coefficients is sufficiently large.Trading between the number of such slabs and the number of points ruled out from the space S of admissible points, we obtain the lower bound.
We initially present the method and the Ω(log n) lower bound on a set of unsatisfiable integer linear inequalities -the Simple Pigeonhole Principle (SPHP) -capturing the core of the counting argument used to prove the PHP efficiently in CP.Since SPHP n has rank 1 CP proofs, it entails a strong separation between CP rank and SP depth.We then apply the method to PHP m n and to Ts(K n , ω).The covering method.The antichain method appears too weak to prove size and depth lower bounds on Ts(G, w), when G is for example a grid or a pyramid.To solve this case, we consider another approach that we call the covering method: we reduce the problem of proving that one point in S survives from all the slab(Q) in a small proof of F, to the problem that a set of polynomials which essentially covers the boolean cube {0, 1} n requires at least √ n polynomials, which is a well-known problem faced by Alon and Füredi in [AF93] and by Linial and Radhakrishnan in [LR05].For this reduction to work we have to find a high dimensional projection of S covering the boolean cube and defined on variables effectively appearing in the proof.We prove that cycles of distance at least 2 in G work properly to this aim on Ts(G, ω).Since the grid H n has many such cycles, we can obtain the lower bound on Ts(H n , ω).The use of Linial and Radhakrishnan's result is not new in proof complexity.Part and Tzameret in [PT21], independently of us, were using this result in a similar way to us to prove size lower bounds in the proof system Res(⊕) over integers which handles clauses over linear equations, and not relying on integer linear inequalities and geometrical reasoning.
We remark that while we were writing this version of the paper, Yehuda and Yehudayoff in [YY21a] slightly improved the results of [LR05] with the consequence, noticed in their paper too, that our size lower bounds for Ts(G, ω) over a grid graph is in fact superlinear.
The paper is organized as follows: We give the preliminary definitions in the next section and then we move to other sections: one on the lower bounds by the antichain method and the other on lower bounds by the covering method.The antichain method is presented on the formulas SPHP, PHP m n , Ts(K n , ω) and LOP n .The covering method is presented for the formulas Ts(G, ω) where G is a grid graph.

Proof systems.
Here we recall the definition of the Stabbing Planes proof system from [BFI + 18].
Definition 2.1.A linear integer inequality in the variables x 1 , . . ., x n is an expression of the form n i=1 a i x i ≥ b, where each a i and b are integral.A set of such inequalities is said to be unsatisfiable if there are no 0/1 assignments to the x variables satisfying each inequality simultaneously.
Note that we reserve the term infeasible, in contrast to unsatisfiable, for (real or rational) linear programs.Definition 2.2.Fix some variables x 1 , . . ., x n .A Stabbing Planes (SP) proof of a set of integer linear inequalities F is a binary tree T , with each node labeled with a query (a, b) with a ∈ Z n , b ∈ Z. Out of each node we have an edge labeled with ax ≥ b and the other labeled with its integer negation ax ≤ b − 1.Each leaf ℓ is labeled with a LP system P ℓ made by a nonnegative linear combination of inequalities from F and the inequalities labelling the edges on the path from the root of T to the leaf ℓ.
If F is an unsatisfiable set of integer linear inequalities, T is a Stabbing Planes (SP) refutation of F if all the LP's P ℓ on the leaves of T are infeasible.
Definition 2.3.The slab corresponding to a query Q = (a, b) is the set slab(Q) = {x ∈ R n : b − 1 < ax < b} satisfying neither of the associated inequalities.
Since each leaf in a SP refutation is labelled by an infeasible LP, throughout this paper we will actually use the following geometric observation on SP proofs T : the set of points in R n must all be ruled out by a query somewhere in T .In particular this will be true for those points in R n which satisfy a set of integer linear inequalities F and which we call feasible points for F.
Fact 2.4.The slabs associated with a SP refutation must cover the feasible points of F. That is, The length of a SP refutation is the number of queries in the proof tree.The depth of a SP refutation T is the longest root-to-leaf path in T .The size (respectively depth) of refuting F in SP is the minimum size (respectively depth) over all SP refutations of F. We call bit-size of a SP refutation T the total number of bits needed to represent every inequality in the refutation.A CP refutation is treelike if the directed acyclic graph underlying the proof is a tree.The length of a CP refutation is the number of inequalities in the sequence.The depth is the length of the longest path from the root to a leaf (sink) in the graph.The rank of a CP proof is the maximal number of rounding rules used in a path of the proof graph.The size of a CP refutation is the bit-size to represent all the inequalities in the proof.

2.2.
Restrictions.Let V = {x 1 , . . ., x n } be a set of n variables and let ax ≤ b be a linear integer inequality.We say that a variable x i appears in, or is mentioned by a query Q = (a, b) if a i ̸ = 0 and does not appear otherwise.
A restriction ρ is a function ρ : D → {0, 1}, D ⊆ V .A restriction acts on a half-plane ax ≤ b setting the x i 's according to ρ.Notice that the variables x i ∈ D do not appear in the restricted half-plane.
By T ↾ ρ we mean to apply the restriction ρ to all the queries in a SP proof T .The tree T ↾ ρ defines a new SP proof: if some Q↾ ρ reduces to 0 ≤ −b, for some b ≥ 1, then that node becomes a leaf in T ↾ ρ .Otherwise in T ↾ ρ we simply branch on Q↾ ρ .Of course the solution space defined by the linear inequalities labelling a path in T ↾ ρ is a subset of the solution space defined by the corresponding path in T .Hence the leaves of T ↾ ρ define an infeasible LP.
We work with linear integer inequalities which are a translation of families of CNFs F. Hence when we write F↾ ρ we mean the applications of the restriction ρ to the set of linear integer inequalities defining F.

The antichain method
This method is based on Sperner's theorem.Using it we can prove depth lower bounds in SP for PHP m n and for Tseitin contradictions Ts(K n , ω) over the complete graph.To motivate and explain the main definitions, we use as an example a simplification of the PHP m n , the Simplified Pigeonhole principle SPHP n , which has some interest since (as we will show) it exponentially separates CP rank from SP depth.
3.1.Simplified Pigeonhole Principle.As mentioned in the Introduction, the SPHP n intends to capture the core of the counting argument used to efficiently refute the PHP in CP.
Definition 3.1.The SPHP n is the following unsatisfiable family of inequalities: Lemma 3.2.SPHP n has a rank 1 CP refutation, for n ≥ 3. 1:8 Proof.Let S := n i=1 x i (so we have S ≥ 2).We fix some i ∈ [n] and sum x i + x j ≤ 1 over all j ∈ [n] \ {i} to find S + (n − 2)x i ≤ n − 1.We add this to −S ≤ −2 to get x i ≤ n − 3 n − 2 which becomes x i ≤ 0 after a single cut.We do this for every i and find S ≤ 0 -a contradiction when combined with the axiom S ≥ 2.
It is easy to see that SPHP n has depth O(log n) proofs in SP, either by a direct proof or appealing to the polynomial size proofs in CP of the PHP m n ([CCT87]) and then using the Theorem 4.4 in [BFI + 18] informally stating that "CP proofs can be balanced in SP".
Corollary 3.3.The SPHP n has SP refutations of depth O(log n).
We will prove that this bound is tight.
3.2.Sperner's Theorem.Let a ∈ R n .The width w(a) of a is the number of non-zero coordinates in a.The width of a query (a, b) is w(a), and the width of a SP refutation is the minimum width of its queries.
We consider the following extension of Sperner's theorem. .
Proof.Define I a = {i ∈ [n] : a i ̸ = 0}.Let ⪯ be the partial order over W Ia where x ⪯ y if x i ≤ y i for all i with a i > 0 and x i ≥ y i for the remaining i with a i < 0. Clearly the set of solutions to as = b forms an antichain under ⪯.Noting that ⪯ is isomorphic to the typical pointwise ordering on W Ia , we appeal to Theorem 3.5 to upper bound the number of solutions in W Ia by k w(a) √ w(a) , each of which corresponds to at most k n−w(a) vectors in W n .
3.3.Large admissibility.A (n, W )-word s is admissible for an unsatisfiable set of integer linear inequalities F over n variables if s satisfies all constraints of F. A set of (n, W )-words is admissible for F if all its elements are admissible.Let A(F, W ) be the set of all admissible (n, W )-words for F. The interesting sets W for an unsatisfiable set of integer linear inequalities F are those such that almost all (n, W )-words are admissible for F. We will apply our method on sets of integer linear inequalities which are a translation of unsatisfiable CNF's generated over a given domain.Typically these formulas on a size n domain have a number of variables which is not exactly n but a function of n, ν(n) ≥ n.Hence for the rest of this section we consider F := {F n } n∈N as a family of sets of unsatisfiable integer linear inequalities, where F n has ν(n) ≥ n variables.We call F an unsatisfiable family.
Consider then the following definition (recalling that we denote k = |W |): Notice that, because of the o notation, Definition 3.7 might be not necessarily true for all n ∈ N, but only starting from some n F .Definition 3.8.Given some almost full family F (over ν(n) variables) we let n F be the natural number with ≤ 2 for all n ≥ n F .
As an example we prove SPHP is almost full (notice that in the case of SPHP n , ν(n) = n).
Proof.Fix W = {0, 1/2} so that k = |W | = 2. Let U be the set of all (n, W )-words with at least four coordinates set to 1/2.U is admissible for SPHP n since inequalities x i + x j ≤ 1 are always satisfied for any value in W and inequalities x 1 + . . .+ x n ≥ 2 are satisfied by all points in U which contain at least four 1/2s.By a simple counting argument, in U there are Proof.We estimate the rate at which the slabs of the queries in T rule out admissible points in U .Let ℓ be the least common multiple of the denominators in W . Every (n, W )-word x falling in the slab of some query (a, b) satisfies one of ℓ equations ax = b + i/ℓ, 1 ≤ i < ℓ (as a is integral).Note that as |W | is a constant independent of n, so is ℓ.Since all the queries in T have width at least w, according to Lemma 3.6, each query in T rules out at most ℓ • k ν(n) √ w admissible points.By Fact 2.4 no point survives at the leaves, in particular the admissible points.Then it must be that We finish by noting that, by the assumption n ≥ n F , and then by Definition 3.8, we have 2 ≥ 3.4.Main theorem.We focus on restrictions ρ that after applied on an unsatisfiable family F = {F n } n∈N , reduce the set F to another set in the same family.
Definition 3.11.Let F = {F n } n∈N be an unsatisfiable family and c a positive constant.
F is c-self-reducible if for any set V of variables, with |V | = v < n/c, there is a restriction ρ with domain V ′ ⊇ V , such that F n ↾ ρ = F n−cv (up to renaming of variables).
Let us motivate the definition with an example.
p s 1 ∈ E}, V ′ ) and update S 2 = S 1 \ (V ′ ∪ {s 1 }).(Note that V ′ could possibly be emptyfor example, if the polynomial x e = 1/2 appears in T , where e ∈ s 1 .In this case however we still have |L 1 | ≥ |M 1 | 0.52 .If V ′ is not empty we have the same bound due to Theorem 4.3.)If S 2 is nonempty we repeat with any s 2 ∈ S 2 , and so on.We now show that as promised the left hand sides of these pairs partition a subset of T , which will give us the first inequality in Equation (4.2).Every polynomial p with p s i ∈ L i has every v t mentioned by p s i removed from S j for all j ≥ i, so the only way p could reappear in some later L j is if p s j ∈ T s j , where v s j does not appear in p s i .Let µ e , e ∈ s j be the coefficients of p in front of the four edges of s j .The coefficient in front of v s j in p s i is just e∈s j µ e .As v s j failed to appear this sum is 0 and p does not have the odd coefficient sum it would need to appear in T s j .

Conclusions and acknowledgements
The Ω(log n) depth lower bound for Ts(H n , ω) is not optimal since [BFI + 18] proved an O(log 2 n) upper bound for Ts(G, ω), for any bounded-degree G.Even to apply the covering method to prove a depth Ω(log 2 n) lower bound on Ts(K n , ω) (notice that it would imply a superpolynomial length lower bound), the polynomial covering of the boolean cube should be improved to work on general cubes.To this end the algebraic method used in [LR05] should be improved to work with generalizations of multilinear polynomials.
We use essential covering of the Boolean cube to prove size lower bounds in SP.However our lower bounds are quite weak, in fact almost linear.It would be very interesting to understand whether the essential covering technique can prove stronger size lower bounds in SP.Notice that any polytope in [0, 1] n can be covered by n hyperplanes.But the polytopes produced by the Stabbing Planes procedure are more specific and in fact they might require a weaker form of covering.For example a recursive covering, where the slabs on one branch do not affect the points on a different independent branch.Exploring this and similar ideas might eventually lead to improve our lower bounds.
While finishing the writing of this manuscript we learned about [FGI + 21] from Noah Fleming.We would like to thank him for answering some questions on his paper [BFI + 18], and sending us the manuscript [FGI + 21] and for comments on a preliminary version of this work.
We are grateful also to several anonymous referees on both the conference and journal versions of this paper.
2 (2) Beame et al. in [BFI + 18] proved the surprising result that the class of Tseitin contradictions Ts(G, ω) over any graph G of maximum degree D, with an odd charging ω, can be refuted in SP in size quasipolynomial in |G| and depth O(log 2 |G| + D). (3) Fleming et al. in [FGI + 21] proved that a size S (and maximal coefficient size C) SP refutation of a unsatisfiable formula F over n variables can be converted into a CP refutation of F of size S(Cn) log S .However in this case the depth of the proof may potentially blow-up as well.Depth lower bounds for SP are proved in [BFI + 18]: Definition 2.5[CCT87].The Cutting Planes (CP) proof system is equipped with boolean axioms and two inference rules:Boolean Axioms Linear Combination Rounding x≥0 −x≥−1 ax≥c bx≥d αax+βbx≥αc+βd αax≥b ax≥⌈b/α⌉ where α, β, b ∈ Z + and a, b ∈ Z n .A CP refutation of some unsatisfiable set of integer linear inequalities is a derivation of 0 ≥ 1 by the aforementioned inference rules from the inequalities in F.
Theorem 3.5[MR08,CCT09].Fix any t ≥ 2, t ∈ N.For all f ∈ N, with the pointwise ordering of [t] f , any antichain has size at most t f 6 π(t 2 −1)f (1 + o(1)).We will use the simplified bound that any antichain A has size |A| ≤ t f √ f .Lemma 3.6.Let a ∈ Z n and |W | = k ≥ 2. The number of (n, W )-words s such that as = b, where b ∈ Q, is at most k n √ w(a) words.Hence the claim.Lemma 3.10.Let F = {F n } n∈N be an almost full unsatisfiable family, where F n has ν(n) variables.Further let T be a SP refutation of F of minimal width ω.If n ≥ n F then |T | = Ω( √ w).
• VER n , composing Ts(G, ω) (over an expander graph G) with the gadget function VER n (see Theorem 5.7 in [BFI + 18] for details); and (2) a Ω( √ n log n) lower bound for the formula Peb(G) • IND n l over n 5 + n log n variables obtained by lifting a pebbling formula Peb(G) over a graph with high pebbling number, with a pointer function gadget IND n l (see Theorem 5.5. in [BFI + 18] for details).Similar to size, these depth lower bounds are applicable only to very specific classes of formulas.In fact they are obtained by extending to SP the technique introduced in [IPU94,