From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of CNF formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a"black-box"technique for proving space lower bounds via a"static"complexity measure that works against any resolution refutation---previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.


Introduction
A resolution proof for, or resolution refutation of, an unsatisfiable formula F in conjunctive normal form (CNF) is a sequence of disjunctive clauses (C 1 , C 2 , . . . , C τ ), where every clause C t is either a member of F or is logically implied by two previous clauses, and where the final clause is the contradictory empty clause ⊥ containing no literals. Resolution is arguably the most well-studied proof system in propositional proof complexity, and has served as a natural starting point in the quest to prove lower bounds for increasingly stronger proof systems on proof length/size (which for resolution is the number of clauses in a proof).
Resolution is also intimately connected to SAT solving in that it lies at the foundation of state-of-theart SAT solvers using so-called conflict-driven clause learning (CDCL). This connection has motivated the study of proof space as a second interesting complexity measure for resolution. The space usage at some step t in a proof is measured as the number of clauses occurring before C t that will be used to derive clauses after C t , and the space of a proof is obtained by taking the maximum over all steps t.
For both of these complexity measures, it turns out that a key role is played by the auxiliary measure of width, i.e., the size of a largest clause in the proof. In a celebrated result, Ben-Sasson and Wigderson [BW01] showed that there are short resolution refutations of a formula if and only if there are also (reasonably) narrow ones, and almost all known lower bounds on resolution length can be (re)derived using this connection. In 2003, Atserias and Dalmau (journal version in [AD08]) established that width also provides lower bounds on space, resolving a problem that had been open since the study of space complexity of propositional proofs was initiated in the late 1990s in [ABRW02,ET01]. This means that for space also, almost all known lower bounds can be rederived by using width lower bounds and appealing to [AD08]. This is not a two-way connection, however, in that formulas of almost worst-case space complexity may require only constant width as shown in [BN08].

Our Contributions
The starting point of our work is the lower bound on space in terms of width in [AD08]. This is a very elegant but also magical proof in that it translates the whole problem to Ehrenfeucht-Fraïssé games in finite model theory, and shows that resolution space and width correspond to strategies for two opposite players in such games. Unfortunately, this also means that one obtains essentially no insight into what is happening on the proof complexity side (other than that the bound on space in terms of width is true). It has remained an open problem to give a more explicit, proof complexity theoretic argument.
In this paper, we give a purely combinatorial proof in terms of simple syntactic manipulations of resolution refutations. To summarize in one sentence, we study the conjunctions of clauses in memory at each time step in a small-space refutation, negate these conjunctions and then expand them to conjunctive normal form again, and finally argue that the new sets of clauses listed in reverse order (essentially) constitute a small-width refutation of the same formula. 1 This new, simple proof also allows us to obtain a new technique for proving space lower bounds. This approach is reminiscent of [BW01] in that one defines a static "progress measure" on refutations and argues that when a refutation has made substantial progress it must have high complexity with respect to the proof complexity measure under study. Previous lower bounds on space have been inherently adaptive and in that sense less explicit.
One important motivation for our work was the hope that a simplified proof of the space-width inequality would serve as a stepping stone to resolving the analogous question for the polynomial calculus proof system. Here the the width of clauses corresponds to the degree of polynomials, space is measured as the total number of monomials of all polynomials currently in memory, and the problem is to determine whether space and degree in polynomial calculus are related in the same way as are space and width in resolution. A possible approach for attacking this question was proposed in [BG13]. In [FLM + 13] we obtained a result analogous to [BN08] that there are formulas of worst-case space complexity that require only constant degree. The question of whether degree lower bounds imply space lower bounds remains open, however, and other results in [FLM + 13] can be interpreted as implying that the techniques in [BG13] probably are not sufficient to resolve this question. Unfortunately, as discussed towards the end of this paper we also show that it appears unlikely that this problem can be addressed by methods similar to our proof of the corresponding inequality for resolution.

Outline of This Paper
The rest of this paper is organized as follows. After some brief preliminaries in Section 2, we present the new proof of the space-width inequality in [AD08] in Section 3. In Section 4 we showcase the new technique for space lower bounds by studying so-called Tseitin formulas. Section 5 explains why we believe it is unlikely that our methods will extend to polynomial calculus. Some concluding remarks are given in Section 6.

Preliminaries
Let us start by a brief review of the preliminaries. The following material is standard and can be found, e.g., in the survey [Nor13].
A literal over a Boolean variable x is either the variable x itself (a positive literal) or its negation that is denoted either as ¬x or x (a negative literal). We define x = x. A clause C = a 1 ∨ · · · ∨ a k is a disjunction of literals and a term T = a 1 ∧ · · · ∧ a k is a conjunction of literals. We denote the empty clause by ⊥ and the empty term by ∅. The logical negation of a clause C = a 1 ∨ · · · ∨ a k is the term a 1 ∧· · ·∧a k that consists of the negations of the literals in the clause. We will sometimes use the notation ¬C or C for the term corresponding to the negation of a clause and ¬T or T for the clause negating a term. A clause (term) is trivial if it contains both a variable and its negation. For the proof systems we study, trivial clauses and terms can always be eliminated without any loss of generality.
A clause C ′ subsumes clause C if every literal from C ′ also appears in C. A k-clause (k-term) is a clause (term) that contains at most k literals. A CNF formula F = C 1 ∧ · · · ∧ C m is a conjunction of clauses, and a DNF formula F = T 1 ∨ · · · ∨ T m is a disjunction of terms. A k-CNF formula (k-DNF formula) is a CNF formula (DNF formula) consisting of k-clauses (k-terms). We think of clauses, terms, and CNF formulas as sets: the order of elements is irrelevant and there are no repetitions.
Let us next describe a slight generalization of the resolution proof system by Krajíček [Kra01], who introduced the family of r-DNF resolution proof systems, denoted R(r), as an intermediate step between resolution and depth-2 Frege systems. An r-DNF resolution configuration C is a set of r-DNF formulas. An r-DNF resolution refutation of a CNF formula F is a sequence of configurations (C 0 , . . . , C τ ) such that C 0 = ∅, ⊥ ∈ C τ , and for 1 ≤ t ≤ τ we obtain C t from C t−1 by one of the following steps: where A is a clause in F (sometimes referred to as an axiom clause).
where D is inferred by one of the following rules (where G, H denote r-DNF formulas, T, T ′ denote r-terms, and a 1 , . . . , a r denote literals): , as long as |T ∪ T ′ | ≤ r.
for any r-DNF formula H.
For r = 1 we obtain the standard resolution proof system. In this case the only nontrivial inference rules are weakening and r-cut, where the former can be eliminated without loss of generality (but is sometimes convenient to have for technical purposes) and the latter simplifies to the resolution rule (2.1) We identify a resolution configuration C with the CNF formula C∈C C.
The length L(π) of an r-DNF resolution refutation π is the number of download and inference steps, and the space Sp(π) is the maximal number of r-DNF formulas in any configuration in π. We define the length L R(r) (F ⊢ ⊥) and the space Sp R(r) (F ⊢ ⊥) of refuting a formula F in r-DNF resolution by taking the minimum over all refutations F with respect to the relevant measure. We drop the proof system R(r) from this notation when it is clear from context.
For the resolution proof system, we also define the width W(π) of a resolution refutation π as the size of a largest clause in π, and taking the minimum over all resolution refutations we obtain the width W(F ⊢ ⊥) of refuting F . We remark that in the context of resolution the space measure defined above is sometimes referred to as clause space to distinguish it from other space measures studied for this proof system.

From Space to Width
In this section we present our new combinatorial proof that width is a lower bound for clause space in resolution. The formal statement of the theorem is as follows (in this article all CNF formulas are assumed to be non-trivial in that they do not contain the contradictory empty clause).

Theorem 3.1 ([AD08]).
Let F be a k-CNF formula and let π : F ⊢ ⊥ be a resolution refutation in space Sp(π) = s. Then there is a resolution refutation π ′ of F in width W(π ′ ) ≤ s + k − 3.
The proof idea is to take the refutation π in space s, negate the configurations one by one, rewrite them as equivalent sets of disjunctive clauses, and list these sets of clauses in reverse order. This forms the skeleton of the new refutation, where all clauses have width at most s. To see this, note that each configuration in the original refutation is the conjunction of at most s clauses. Therefore, the negation of such a configuration is a disjunction of at most s terms, which is equivalent (using distributivity) to a conjunction of clauses of width at most s. To obtain a legal resolution refutation, we need to fill in the gaps between adjacent sets of clauses. In this process the width increases slightly from s to s + k − 3.
Before presenting the full proof, we need some technical results. We start by giving a formal definition of what a negated configuration is.
Definition 3.2. The negated configuration neg(C) of a configuration C is defined by induction on the number of clauses in C: where we remove trivial and subsumed clauses from the final configuration.
Each clause of the original configuration contributes at most one literal to each clause of the negated configuration. Hence, the width of the new clauses must be small.

Observation 3.3. The width of any clause in the negated configuration neg(C) is at most |C|.
In the proof we will use a different characterization of negated configurations that is easier to work with.
Proposition 3.4. The negated configuration neg(C) is the set of all minimal (non-trivial) clauses C such that ¬C implies the configuration C. That is, Proof. Let us fix the configuration C and let D denote the set of all minimal clauses implying C. We prove that for each clause C ∈ neg(C) there is a clause C ′ ∈ D such that C ′ ⊆ C and vice versa. The proposition then follows because by definition neither D nor neg(C) contains subsumed clauses.
First, let C ∈ neg(C). By the definition of neg(C) we know that for every clause D ∈ C the clause C contains the negation of some literal from D. Hence, ¬C implies C as it is a conjunction of literals from each clause in C. By taking the minimal clause In the opposite direction, we want to show for any C ∈ D that C must contain a negation of some literal in D for every clause D ∈ C. Assume for the sake of contradiction that D ∈ C is a clause such that none of its literals has a negation appearing in C. Let α be a total truth value assignment that satisfies ¬C (such an assignment exists because C is non-trivial). By assumption, flipping the variables in α so that they falsify D cannot falsify ¬C. Therefore, we can find an assignment that satisfies ¬C but falsifies D ∈ C, which contradicts the definition of D. Hence, C must contain a negation of some literal in D for every D ∈ C and by the definition of neg(C) there is a C ′ ∈ neg(C) such that C ′ ⊆ C.
The following observation, which formalizes the main idea behind the concept of negated configurations, is an immediate consequence of Proposition 3.4.

Observation 3.5. An assignment satisfies a clause configuration C if and only if it falsifies the negated clause configuration neg(C). That is, C is logically equivalent to ¬neg(C).
Recall that what we want to do is to take a resolution refutation π = (C 0 , C 1 , . . . , C τ ) and argue that if π has small space complexity, then the reversed sequence of negated configurations π ′ = (neg(C τ ), neg(C τ −1 ), . . . , neg(C 0 )) has small width complexity. However, as noted above π ′ is not necessarily a legal resolution refutation. Hence, we need to show how to derive the clauses in each configuration of the negated refutation without increasing the width by too much. We do so by a case analysis over the derivation steps in the original refutation, i.e., axiom download, clause inference, and clause erasure. The following lemma shows that for inference and erasure steps all that is needed in the reverse direction is to apply weakening.
Proof. For any clause C in neg(C) it holds by Proposition 3.4 that ¬C C. Since C C ′ , this in turns implies that ¬C C ′ . Applying Proposition 3.4 again, we conclude that there exists a clause C ′ ⊆ C such that C ′ ∈ neg(C ′ ).
The only time in a refutation π = (C 0 , C 1 , . . . , C τ ) when it does not hold that C t−1 C t is when an axiom clause is downloaded at time t, and such derivation steps will require a bit more careful analysis. We provide such an analysis in the full proof of Theorem 3.1, which we are now ready to present.
Proof of Theorem 3.1. Let π = (C 0 , C 1 , . . . , C τ ) be a resolution refutation of F in space s. For every configuration C t ∈ π, let D t denote the corresponding negated configuration neg(C t ). By assumption each C t has at most s clauses, and thus Observation 3.3 guarantees that D t has width at most s. We need to show how to transform the sequence π ′ = (D τ , D τ −1 , . . . , D 0 ) into a legal resolution refutation of width at most s + k − 3.
The initial configuration of the new refutation is D τ itself, which is empty by Definition 3.2. If C t+1 follows C t by inference or erasure, then we can derive any clause of D t from a clause of D t+1 by weakening, as proven in Lemma 3.6. If C t+1 follows C t by axiom download, then we can derive D t from D t+1 in width at most s + k − 3, as we show below. The last configuration D 0 includes the empty clause ⊥ by Definition 3.2, so the new refutation is complete.
It remains to take care of the case of axiom download. We claim that we can assume without loss of generality that prior to each axiom download step the space of the configuration C t is at most s − 2. Otherwise, immediately after the axiom download step the proof π needs to erase a clause in order to maintain the space bound s. By reordering the axiom download and clause erasure steps we get a valid refutation of F for which it holds that Sp(C t ) ≤ s − 2.
Suppose C t+1 = C t ∪ {A} for some axiom A = a 1 ∨ · · · ∨ a ℓ , with ℓ ≤ k. Consider now some clause C that is in the negated configuration D t and that does not belong to D t+1 . By Observation 3.3 To derive C from D t+1 we first download axiom A and then show how to derive C from the clauses in D t+1 ∪ {A}.
First, note that all clauses C a = C ∨ a are either contained in or are weakenings of clauses in D t+1 . This follows easily from Definition 3.2 as adding an axiom A to the configuration C t results in adding negations of literals from A to all clauses C ∈ D t . Hence, we can obtain C by the following derivation: When C is the empty clause, the width of this derivation is upper-bounded by W(A) ≤ k. Otherwise, it is upper bounded by W(C)+W(A)−1 ≤ s+k−3. Any resolution refutation has space at least 3 (unless the formula contains the empty clause itself), so the width of π ′ is upper-bounded by W(π ′ ) ≤ s+k −3.
The proof of Theorem 3.1 also works for r-DNF resolution, with some loss in parameters. Let us state this as a theorem and sketch the proof.
Theorem 3.7. Let F be a k-CNF formula and π : F ⊢ ⊥ be an r-DNF resolution refutation of F in space Sp(π) ≤ s. Then there exists a resolution refutation π ′ of F in width at most W(π ′ ) ≤ (s − 2)r + k − 1.
Proof sketch. We define the negated configuration neg R(r) (C) of an R(r)-configuration to be with trivial and subsumed clauses removed. It is easy to see that r-DNF configuration of space s gets transformed into a resolution configuration of width at most sr. We can prove an analogue of Proposition 3.4 for this definition of the negated configuration from which the analogue of Lemma 3.6 easily follows. The case of axiom download is the same as in the proof of Theorem 3.1 as axioms are clauses. Hence, running the negated refutation backwards we get a resolution refutation of F in width at most (s − 2)r + k − 1.

A Static Technique for Proving Space Lower Bounds
Looking at the proof complexity literature, the techniques used to prove lower bounds for resolution length and width (e.g., [BW01,CS88,Hak85,Urq87]) differ significantly from those used to prove resolution space lower bounds (e.g., [ABRW02,BG03,ET01]) in that the former are static or oblivious while the latter are dynamic.
Lower bounds on resolution length typically have the following general structure: if a refutation is too short, then we obtain a contradiction by applying a suitable random restriction (the length of the proof figures in by way of a union bound); so any refutation must be long. When proving lower bounds on resolution width, one defines a complexity measure and uses the properties of this measure to show that every refutation must contain a complex clause; in a second step one then argues that such a complex clause must be wide.
In contrast, most lower bound proofs for resolution space use an adversary argument. Assuming that the resolution derivation has small space, one constructs a satisfying assignment for each clause configuration. Such assignments are updated inductively as the derivation progresses, and one shows that the update is always possible given the assumption that the space is small. This in turn shows that the contradictory empty clause can never be reached, implying a space lower bound on refutations. The essential feature separating this kind of proofs from the ones above is that the satisfying assignments arising during the proof depend on the history of the derivation; in contrast, the complexity measures in width lower bounds are defined once and for all, as are the distributions of random restrictions in length lower bounds.
In this section we present a static lower bound on resolution space. Our proof combines the ideas of Section 3 and the complexity measure for clauses used in [BW01]. We define a complexity measure for configurations which can be used to prove space lower bounds along the lines of the width lower bounds mentioned above.
This approach works in general in that any complexity measure for clauses can be transformed into a complexity measure for configurations. This turns many width lower bound techniques into space lower bound ones (e.g., width lower bounds for random 3-CNF formulas.) In this section we give a concrete example of this for Tseitin formulas, which are a family of CNFs encoding a specific type of systems of linear equations; see Figure 1 for illustration.   When the degree of G is bounded by d, PARITY v,χ has at most 2 d−1 clauses, all of width at most d, and hence Ts(G, χ) is a d-CNF formula with at most 2 d−1 |V | clauses. We say that a set of vertices U has odd (even) charge if u∈U χ(u) is odd (even). A simple parity argument shows that when V (G) has odd charge, Ts(G, χ) is unsatisfiable. On the other hand, if G is connected then for each v ∈ V it is always possible to satisfy the constraints PARITY u,χ for all u = v.
The hardness of Tseitin formulas are governed by the expansion properties of the underlying graph. We next present a new technique to show that if a graph G is a good edge expander, then large space is needed to refute Ts(G, χ) in resolution. We remark that this was originally proven in [ABRW02,ET01] (and with slightly better parameters, as discussed below). For the rest of this section we fix a particular d-regular connected graph G and a function χ with respect to which V (G) has odd charge, and consider the corresponding Tseitin formula Ts(G, χ). The main tool used to prove Theorem 4.3 is a complexity measure for configurations. We show that if G is a good expander, then every refutation of Ts(G, χ) must have a configuration with intermediate measure. We conclude the proof by showing that the space of a configuration is at least its measure if the latter falls within a specific range of values.
We first define our configuration complexity measure for terms (i.e., configurations consisting of unit clauses), and then extend it to general configurations. In words, the term complexity measure is the smallest number of parity axioms of Ts(G, χ) that collectively contradict the term, and the configuration complexity measure is the maximum measure over all terms that imply the configuration.

Definition 4.4 (Configuration complexity measure). The term complexity measure
The configuration complexity measure µ(C) of a resolution configuration C is defined as µ(C) = max {ν(T ) : T C}. When C is contradictory we have µ(C) = 0.
Note that ν(T ) is a monotone decreasing function, since T ⊆ T ′ implies ν(T ) ≥ ν(T ′ ) by definition. Hence, we only need to look at minimal terms T for which T C in order to determine µ(C). These minimal terms are the negations of the clauses in neg(C) (compare Proposition 3.4). We now introduce the convenient concept of witness for the measure.

Definition 4.5 (Witness of measure).
A witness of the measure ν(T ) of the term T is a set of vertices V * for which ν(T ) = |V * | and T ∧ v∈V * PARITY v,χ ⊥. Similarly, for configurations C a witness for µ(C) is a term T * for which µ(C) = ν(T * ) and T * C.
There is a big gap between the measure of the initial and final configurations of a refutation, and we will see that the measure does not change much at each step. Hence, the refutation must pass through a configuration of intermediate measure. Formally, if G is connected then µ(∅) = |V |, because the empty term has measure |V |, and µ(C) = 0 when ⊥ ∈ C.
To study how the measure changes during the refutation, we look separately at what happens at each type of step. As in the proof of Theorem 3.1, we can deal with inference and clause erasure steps together, whereas axiom downloads require more work.
Proof. Let T * be a witness for µ(C). Then, T * C and, hence, we also have T * C ′ . Therefore, µ(C ′ ) ≥ ν(T * ), because µ(C ′ ) is equal to the maximum value of ν(T ) for terms T implying C ′ . As ν(T * ) is equal to µ(C), the bound µ(C ′ ) ≥ µ(C) follows. Ts(G, χ) and a graph G of bounded degree d,

Lemma 4.7. For a clause A in
Proof. Fix a witness T * for µ(C). Since µ(C) = ν(T * ), to prove the lemma we need to upper-bound the value ν(T * ) by d · µ(C ′ ) + 1.
For any literal a in A, we know that T * ∧ a implies C ′ because T * implies C and a implies A. Hence, it holds that µ(C ′ ) ≥ ν(T * ∧ a), and so it will be sufficient to relate ν(T * ) to the values ν(T * ∧ a).
To this end, we look at the set of vertices V * = a∈A V a ∪ {v A }, where each V a is a witness for the corresponding measure ν(T * ∧ a), and v A is the vertex such that A ∈ PARITY v A ,χ . Note that by definition it holds that |V a | = ν(T * ∧ a) for every a ∈ A and also that |V * | ≤ 1 + a∈A |V a |, which sum can in turn be bounded by d · µ(C ′ ) + 1 because A has at most d literals.
We conclude the proof by showing that T * ∧ v∈V * PARITY v,χ ⊥, which establishes that ν(T * ) ≤ |V * |. The implication holds because any assignment either falsifies the clause A, and so falsifies PARITY v A ,χ , or satisfies one of the literals a ∈ A. But then we have as a subformula T * ∧ v∈Va PARITY v,χ , which is unsatisfiable by the definition of V a when a is true. The bound ν(T * ) ≤ |V * | then follows, and so µ(C) ≤ |V * | ≤ d · µ(C ′ ) + 1.
The preceding results imply that every resolution refutation of the Tseitin formula has a configuration of intermediate complexity. This holds because every refutation starts with a configuration of measure |V | and needs to reach the configuration of measure 0, as noted above, while at each step the measure drops by a factor of at most 1/d by the lemmas we just proved. Let us state this formally as a corollary.
Corollary 4.8. For any resolution refutation π of a Tseitin formula Ts(G, χ) over a connected graph G of bounded degree d and any positive integer r ≤ |V | there exists a configuration C ∈ π such that the configuration complexity measure is bounded by r/d ≤ µ(C) ≤ r.
It remains to show that a configuration having intermediate measure must also have large space. This part of the proof relies on the graph being an expander. Lemma 4.9. Let G be an (s, δ)-edge expander graph. For every configuration C satisfying µ(C) ≤ s it holds that Sp(C) ≥ δ · µ(C).
Proof. To prove the lemma, we lower-bound the size of a minimal witness T * for µ(C) and then use the bound Sp(C) ≥ |T * |. This inequality follows by noting that at most one literal per clause in C is needed in the implying term T * .
Fix T * to be a minimal witness for µ(C) and let V * be a witness for ν(T * ). Note that |V * | = µ(C). We prove that T * must contain a variable for every edge in ∂(V * ). Towards contradiction, assume that T * does not contain some x e for an edge e in ∂(V * ), and let v e be a vertex in V * incident to e. Let α be an assignment that satisfies T * ∧ v∈V * \{ve} PARITY v,χ . Such an assignment must exist as otherwise V * would not be a witness for ν(T * ). We can modify α by changing the value of x e so that PARITY ve,χ 5 From Small Space to Small Degree in Polynomial Calculus? is satisfied. By the assumption, the new assignment α ′ still satisfies T * and v∈V * \{ve} PARITY v,χ as neither contains the variable x e . Thus, we have found an assignment satisfying T * ∧ v∈V * PARITY v,χ , which is a contradiction.
The preceding lemma and Corollary 4.8 together imply Theorem 4.3, because by Corollary 4.8 there is a configuration with measure between s/d and s, and this configuration has space at least δs/d by Lemma 4.9.
We want to point out that Theorem 4.3 gives inferior results compared to a direct application of Theorem 3.1 to known width lower bounds. The bounds that we get are worse by a multiplicative factor of 1/d. One might have hoped to remove this multiplicative factor by improving the bound in Lemma 4.7, but this is not possible because this lemma is tight.
To see this, suppose that the graph G is a d-star: it consists of a center v which is connected to d petals u 1 , . . . , u d by the edges e 1 , . . . , e d , the charge of the center is χ(v) = 1, and the charges of the petals are χ(u 1 ) = · · · = χ(u d ) = 0. Let A ∈ PARITY v,χ be the axiom A = x e 1 ∨ · · · ∨ x e d . Taking C = ∅ and C ′ = {A}, we have that µ(C) = d + 1 while µ(C ′ ) = 1. The latter equality holds because every minimal term implying A is of the form x e i , a term which is contradicted by the single axiom x e i ∈ PARITY u i ,χ . Hence, we have an example where d · µ(C ′ ) + 1 = µ(C), which shows that Lemma 4.7 is tight.

From Small Space to Small Degree in Polynomial Calculus?
An intriguing question is whether an analogue of the bound in Theorem 3.1 holds also for the stronger algebraic proof system polynomial calculus introduced in [CEI96]. In this context, it is more relevant to discuss the variant of this system presented in [ABRW02], known as polynomial calculus (with) resolution or PCR, which we briefly describe below.
In a PCR derivation, configurations are sets of polynomials in F[x, x, y, y, . . .], where x and x are different formal variables. Each polynomial P appearing in a configuration corresponds to the assertion P = 0. The proof system contains axioms x 2 − x and x + x − 1, which restrict the values of the variables to {0, 1}, and enforce the complementarity of x and x. A literal has truth value true if it is equal to 0, and truth value false if it is equal to 1. Each clause C is translated to a monomial m with the property that m = 0 if and only if C is satisfied. For example, the clause x ∨ y ∨ z is translated to the monomial xyz. There are two inference rules, linear combination p q αp+βq and multiplication p xp , where p and q are (previously derived) polynomials, the coefficients α, β are elements of F, and x is any variable (with or without bar). These rules are sound in the sense that if the antecedent polynomials evaluate to zero under some assignment, then so does the consequent polynomial. A CNF formula F is refuted in PCR by deriving the constant term 1 from the (monomials corresponding to the) clauses in F .
The size, degree and monomial space measures are analogues of length, width and clause space in resolution (counting monomials instead of clauses). PCR can simulate resolution refutations efficiently with respect to all of these measures.
Let us now discuss why the method we use to prove Theorem 3.1 is unlikely to generalize to PCR. An example of formulas that seem hard to deal with in this way are so-called pebbling contradictions, which we briefly describe next.
Pebbling contradictions are defined in terms of directed acyclic graphs (DAGs) G = (V, E) with bounded fan-in, where vertices with no incoming edges are called sources and vertices without outgoing edges sinks. Assume G has a unique sink z and associate a variable V to each vertex v ∈ V . Then the pebbling contradiction over G consists of the following clauses: • for each source vertex s, a clause s (source axioms), Res(1, 4) 6.
See Figure 2 for an illustration. Ben-Sasson [Ben09] showed that pebbling contradictions exhibit spacewidth trade-offs in resolution in that they can always be refuted in constant width as well as in constant space but that there are graphs for which optimizing one of these measures necessarily causes essentially worst-case linear behaviour for the other measure. There are two natural ways to refute pebbling contradictions in resolution. One approach is to go "bottom-up" from sources to sinks in topological order, and derive for each vertex v ∈ V (G) the unit clause v using the pebbling axiom for v and the unit clauses for its predecessors. When the refutation reaches z it derives a contradiction with the sink axiom z. See Figure 3(a) for an example. This refutation can always be carried out in constant width but for some graphs requires large space.
The other approach is a "top-down" refutation due to [Ben09] where one starts with the sink axiom z and derives clauses of the form v 1 ∨ · · · ∨ v ℓ . A new clause is derived by replacing any vertex v i in the old one by all its predecessors, i.e., by resolving with the pebbling axiom for v i . Since G is acyclic we can repeat this process until we get to the sources, for which the negated literals can be resolved away using source axioms. This refutation is illustrated in Figure 3(b). It is not hard to see that it can be performed in constant clause space, but it might require large width.
A careful study now reveals that the transformation of configurations in our proof of Theorem 3.1 maps either of the two refutations describe above into the other one. Instead of providing a formal argument, we encourage the reader to compute the tranformations of the refutations in Figures 3(a)  and 3(b), observing that the axioms are downloaded in opposite order in the two derivations. This observation is the main reason why our proof does not seem to generalize to PCR, as we now explain.
In PCR, we can represent any conjunction of literals a 1 ∧· · ·∧a r as the binomial 1− i a i . Using this encoding with the bottom-up approach yields a third refutation, which has constant space but possibly large degree: the fact that a set of vertices U "are true" can be stored as the high-degree binomial 1 − v∈U v instead of as a collection of low-degree monomials {v | v ∈ U }. Hence, there are constant space PCR refutations of pebbling contradictions in both the bottom-up and the top-down directions. This in turn means that if our proof method were to work for PCR, we would need to find constant degree refutations in both directions. For the top-down case it seems unlikely that such a refutation exists, since clauses of the form v∈U v cannot be represented as low-degree polynomials.

Concluding Remarks
In this work, we present an alternative, completely elementary, proof of the result by Atserias and Dalmau [AD08] that space is an upper bound on width in resolution. Our construction gives a syntactic way to convert a small-space resolution refutation into a refutation in small width. We also exhibit a new "black-box" approach for proving space lower bounds that works by defining a progress measurè a la Ben-Sasson and Wigderson [BW01] and showing that when a refutation has made medium progress towards a contradiction it must be using a lot of space. We believe that these techniques shed interesting new light on resolution space complexity and hope that they will serve to increase our understanding of this notoriously tricky complexity measure.
As an example of a question about resolution space that still remains open, suppose we are given a k-CNF formula that is guaranteed to be refutable in constant space. By [AD08] it is also refutable in constant width, and a simple counting argument then shows that exhaustive search in small width will find a polynomial-length resolution refutation. But is there any way of obtaining such a short refutation from a refutation in small space that is more explicit than doing exhaustive search? And can we obtain a short refutation without blowing up the space by more than, say, a constant factor? Known length-space trade-off results for resolution in [BBI12,BN11,BNT13,Nor09] do not answer this question as they do not apply to this range of parameters. Unfortunately, our new proof of the space-width inequality cannot be used to resolve this question either, since in the worst case the resolution refutation we obtain might be as bad as the one found by exhaustive search of small-width refutations (or even worse, due to repetition of clauses). This would seem to be inherent-a recent result [ALN14] shows that there are formulas refutable in space and width s where the shortest refutation has length n Ω(s) , i.e., matching the exhaustive search upper bound up to a (small) constant factor in the exponent.
An even more intriguing question is how the space and degree measures are related in polynomial calculus, as discussed in Section 5. For most relations between length, space, and width in resolution, it turns out that they carry over with little or no modification to size, space, and degree, respectively, in polynomial calculus. So can it be that it also holds that space yields upper bounds on degree in polynomial calculus? Or could perhaps even the stronger claim hold that polynomial calculus space is an upper bound on resolution width? These questions remain wide open, but in the recent paper [FLM + 13] we made some limited progress by showing that if a formula requires large resolution width, then the "XORified version" of the formula requires large polynomial calculus space. We refer to the introductory section of [FLM + 13] for a more detailed discussion of these issues.
sharing his proof of the theorem that space upper-bounds width, which is very similar to ours although expressed in a different language.