On Compiling Structured CNFs to OBDDs

We present new results on the size of OBDD representations of structurally characterized classes of CNF formulas. First, we prove that variable convex formulas (that is, formulas with incidence graphs that are convex with respect to the set of variables) have polynomial OBDD size. Second, we prove an exponential lower bound on the OBDD size of a family of CNF formulas with incidence graphs of bounded degree. We obtain the first result by identifying a simple sufficient condition—which we call the few subterms property—for a class of CNF formulas to have polynomial OBDD size, and show that variable convex formulas satisfy this condition. To prove the second result, we exploit the combinatorial properties of expander graphs; this approach allows us to establish an exponential lower bound on the OBDD size of formulas satisfying strong syntactic restrictions.


Introduction
Motivation.A fundamental theoretical task in the study of Boolean functions is to relate the size of their encodings in different representation languages.In particular, the representation of circuits as binary decision diagrams (also known as branching programs) has been the subject of intense study in complexity theory (see, for instance [28,Chapter 14] and [19,Part V]).In this paper, we study the ordered binary decision diagram (OBDD) representations of Boolean functions given as propositional formulas in conjunctive normal form (CNF).In contrast to other variants of binary decision diagrams, equivalence of OBDDs can be decided in polynomial time, a crucial feature for basic applications in the areas of verification and synthesis [4].
Perhaps somewhat surprisingly, the question of which classes of CNFs can be represented as (or compiled into, in the jargon of knowledge representation) OBDDs of polynomial size is largely unexplored [28,Chapter 4].We approach this classification problem by considering structurally characterized CNF classes, that is, classes of CNF formulas defined in terms of properties of their incidence graphs (the incidence graph of a formula is the bipartite graph on clauses and variables where a variable is adjacent to the clauses it occurs in). Figure 1 depicts a hierarchy of well-studied bipartite graph classes as considered by Lozin and Rautenbach [22,Figure 2].This hierarchy is particularly well-suited for our classification project as it includes prominent cases such as beta acyclic CNFs [7] and bounded clique-width CNFs [26].When located within this hierarchy, the known bounds on the OBDD size of structural CNF classes leave a large gap (depicted on the left of Figure 1): • On the one hand, we have a polynomial upper bound on the OBDD size of bounded treewidth CNF classes proved recently by Razgon [25].The corresponding graph classes are located at the bottom of the hierarchy.
• On the other hand, there is an exponential lower bound for the OBDD size of general CNFs, proved two decades ago by Devadas [10].The corresponding graph class is not chordal bipartite, has unbounded degree and unbounded clique-width, and hence is located at the top of the hierarchy.Contribution.In this paper, we tighten this gap as illustrated on the right in Figure 1.More specifically, we prove new bounds for two structural classes of CNFs.
Our first result is a polynomial upper bound: Result 1. CNFs whose incidence graphs are variable convex have polynomial OBDD size (Theorem 2).
Convexity is a property of bipartite graphs that has been extensively studied in the area of combinatorial optimization [17,16,27], and that can be detected in linear time [5,21].To prove Result 1, we define a property of CNF classes, called the few subterms property, that naturally arises as a sufficient condition for polynomial size compilability when considering OBDD representations of CNF formulas (Theorem 1), and then prove that CNFs with variable convex incidence graphs have this property (Lemma 1).The few subterms property can also be invoked in proving the previously known result that classes of CNFs with incidence graphs of bounded treewidth have OBDD representations of polynomial size (Lemma 3).In fact, both the result on variable convex CNFs and the result on bounded treewidth CNFs can be improved to polynomial time compilation by appealing to a stronger version of the few subterms property (Theorem 2 and Theorem 3).
In an attempt to push the few subterms property further, we adopt the language of parameterized complexity to formally capture the idea that CNFs "close" to a class with few subterms have "small" OBDD representations.More precisely, defining the deletion distance of a CNF from a CNF class as the number of its variables or clauses that have to be deleted in order for the resulting formula to be in the class, we prove that CNFs have fixed-parameter tractable OBDD size parameterized by the deletion distance from a CNF class with few subterms (Theorem 4).This result can again be improved to fixed-parameter time compilation under additional assumptions (Theorem 5), yielding for instance fixed-parameter tractable time compilation of CNFs into OBDDs parameterized by the feedback vertex set size (Corollary 2).

Our second result is an exponential lower bound:
Result 2. There is a class of CNF formulas with incidence graphs of bounded degree such that every formula F in this class has OBDD size at least 2 Ω(size(F )) , where size(F ) denotes the number of variable occurrences in F (Theorem 7).
Observe that this bound is tight: every CNF on n variables has an OBDD of size O(2 n ).To establish the lower bound we use the powerful combinatorial machinery of expander graphs.Despite expander graphs appearing in many areas of mathematics and computer science [18,23], including circuit and proof complexity [19], their application in this setting is novel and allows us to improve the best known lower bound on the OBDD size of CNFs [10] in two ways.
• First, the formulas used to prove the latter bound give rise to OBDDs of size 2 Ω(n) but "only" yield lower bounds of the form 2 Ω( √ size(F )) .
• Second, our lower bound is established for CNF formulas that satisfy strong syntactic restrictions: each clause has exactly two positive literals and each variable occurs at most 3 times; in particular, it holds for read 3 times monotone 2-CNF formulas.This nicely complements the known fact that read-once formulas have polynomial OBDD size [15]; to the best of our knowledge, it was not even known that 3 times formulas have super-polynomial OBDD size.
Organization.The paper is organized as follows.In Section 2, we introduce basic notation and terminology.In Section 3, we prove that the few subterms property implies polynomial OBDD size for CNF classes, and prove that variable-convex CNFs (and bounded treewidth CNFs) have the few subterms property (fixed-parameter tractable size and time compilability results based on the few subterms property are presented in Section 3.4).
In Section 4, we prove an exponential lower bound on the OBDD size of CNF formulas based on expander graphs.Finally, we present our conclusions in Section 5.

Preliminaries
Let X be a countable set of variables.A literal is a variable x or a negated variable ¬x.If x is a variable we let var(x) = var(¬x) = x.A clause is a finite set of literals.For a clause c we define var(c) = {var(l) | l ∈ c}.
If a clause contains a literal negated as well as unnegated it is tautological.A conjunctive normal form (CNF) is a finite set of non-tautological clauses.If F is a CNF formula we let var(F ) = c∈F var(c).The size of a clause c is |c|, and the size of a CNF F is size(F ) = c∈F |c|.An assignment is a mapping f : X → {0, 1}, where X ⊆ X; we identify f with the set {¬x | x ∈ X , f (x) = 0} ∪ {x | x ∈ X , f (x) = 1}.An assignment f satisfies a clause c if f ∩ c = ∅; for a CNF F , we let F [f ] denote the CNF containing the clauses in F not satisfied by f , restricted to variables in X \ var(f ), that is, A binary decision diagram (BDD) D on variables {x 1 , . . ., x n } is a labelled directed acyclic graph satisfying the following conditions: D has at at most two vertices without outgoing edges, called sinks of D. Sinks of D are labelled with 0 or 1; if there are exactly two sinks, one is labelled with 0 and the other is labelled with 1.Moreover, D has exactly one vertex without incoming edges, called the source of D. Each non-sink node of D is labelled by a variable x i , and has exactly two outgoing edges, one labelled 0 and the other labelled 1.Each node v of D represents a Boolean function An ordering σ of a set {x 1 , . . ., x n } is a total order on {x 1 , . . ., x n }.If σ is an ordering of {x 1 , . . ., x n } we let var(σ) = {x 1 , . . ., x n }.Let σ be the ordering of {1, . . ., n} given by x i1 < x i2 < • • • < x in .For every integer 0 < j ≤ n, the length j prefix of σ is the ordering of {x i1 , . . ., x ij } given by x i1 < • • • < x ij .A prefix of σ is a length j prefix of σ for some integer 0 < j ≤ n.For orderings σ = x i1 < • • • < x in of {x 1 , . . ., x n } and ρ = y i1 < • • • < y im of {y 1 , . . ., y m }, we let σρ denote the ordering of {x 1 , . . ., x n , y 1 , . . ., y m } given by Let D be a BDD on variables {x 1 , . . ., x n } and let σ = x i1 < • • • < x in be an ordering of {x 1 , . . ., x n }.The BDD D is a σ-ordered binary decision diagram (σ-OBDD) if x i < x j (with respect to σ) whenever D contains an edge from a node labelled with x i to a node labelled with x j .A BDD D on variables {x 1 , . . ., x n } is an ordered binary decision diagram (OBDD) if there is an ordering σ of {x 1 , . . ., x n } such that D is a σ-OBDD.For a Boolean function F = F (x 1 , . . ., x n ), the OBDD size of F is the size of the smallest OBDD on {x 1 , . . ., x n } computing F .
We say that a class F of CNFs has polynomial-time compilation into OBDDs if there is a polynomial-time algorithm that, given a CNF F ∈ F, returns an OBDD computing the same Boolean function as F .We say that a class F of CNFs has polynomial size compilation into OBDDs if there exists a polynomial p : N → N such that, for all CNFs F ∈ F, there exists an OBDD of size at most p(size(F )) that computes the same function as F .
We freely use neigh(v, G) as a shorthand for neigh({v}, G), and we write neigh(W ) instead of neigh(W, G) if the graph G is clear from the context.
A graph G = (V, E) is bipartite if it its vertex set V can be partitioned into two blocks V and V such that, for every edge vw ∈ E, we either have v ∈ V and w ∈ V , or v ∈ V and w ∈ V .In this case we may write G = (V , V , E).The incidence graph of a CNF F , in symbols inc(F ), is the bipartite graph (var(F ), F, E) such that vc ∈ E if and only if v ∈ var(F ), c ∈ F , and v ∈ var(c); that is, the blocks are the variables and clauses of F , and a variable is adjacent to a clause if and only if the variable occurs in the clause.
A bipartite graph G = (V, W, E) is left convex if there exists an ordering σ of V such that the following holds: if wv and wv are edges of G and v < v < v (with respect to the ordering σ) then wv is an edge of G.The ordering σ is said to witness left convexity of G.
For an integer d, a CNF F has degree d if inc(F ) has degree at most d.A class F of CNFs has bounded degree if there exists an integer d such that every CNF in F has degree d.

Polynomial Time Compilability
In this section, we introduce the few subterms property, a sufficient condition for a class of CNFs to have polynomial size compilation into OBDDs (Section 3.1).We prove that the classes of variable convex CNFs and bounded treewidth CNFs have the few subterms property (Section 3.2 and Section 3.3).Finally, we establish fixed-parameter tractable size and time OBDD compilation results for CNFs, where the parameter is the distance to a few subterms CNF class (Section 3.4).

Few Subterms
In this section, we introduce a property of classes of CNFs called the few subterms property (Definition 1), and prove that classes of CNFs with the few subterms property admit polynomial time compilation into OBDDs (Corollary 1).
Definition 1 (Few Subterms).Let F be a CNF, let V ⊆ var(F ), and let f : Let σ be an ordering of var(F ).The subterm width of F with respect to σ, in symbols stw(F, σ), is equal to the subterm width of F is the minimum subterm width of F with respect to σ, where σ ranges over all orderings of var(F ).
Let F be a class of CNFs.A function b : N → N is called a subterm bound of F if for all F ∈ F, the subterm width of F is bounded above by b(size(F )).Let b : N → N be a subterm bound of F, let F ∈ F, and let σ be an ordering of var(F ).We call σ a witness of subterm bound b with respect to F if stw(F, σ) ≤ b(size(F )).
The class F has few subterms if it has a polynomial subterm bound p : N → N; if, in addition, for all F ∈ F, an ordering σ of var(F ) witnessing p with respect to F can be computed in polynomial time, F is said to have constructive few subterms.
The following statement describes how the few subterms property naturally presents itself as a sufficient condition for a polynomial size construction of OBDDs from CNFs.
Theorem 1.There exists an algorithm that, given a CNF F and an ordering σ of var(F ), returns a σ-OBDD for F of size at most |var(F )| stw(F, σ) in time polynomial in |var(F )| and stw(F, σ).
Proof of Theorem 1.Let F be a CNF and σ = x 1 • • • x n be an ordering of var(F ).The algorithm computes a σ-OBDD D for F as follows.
At step i = 1, create the source of D, labelled by F , at the level 0 of the diagram; if ∅ ∈ F (respectively, F = ∅), then identify the source with the 0-sink (respectively, 1-sink) of the diagram, otherwise make the source an x 1 -node.
At step i + 1 for i = 1, . . ., n − 1, let v 1 , . . ., v l be the x i -nodes at level i − 1 of the diagram, respectively labelled F 1 , . . ., F l .For j = 1, . . ., l and b = 0, 1, compute then identify v with the 0-sink of D, and if ∅ = F j [x i = b], then identify v with the 1-sink of D.
At termination, the diagram obtained computes F and respects σ.We analyze the runtime.At step i + 1 (0 ≤ i < n), the nodes created at level i are labelled by CNFs of the form F [f ], where f ranges over all assignments of {x 1 , . . ., x i } not falsifying F ; that is, these nodes correspond exactly to the {x 1 , . . ., x i }subterms st(F, {x 1 , . . ., x i }) of F not containing the empty clause, whose number is bounded above by stw(F, σ).As level i is processed in time bounded above by its size times the size of level i −

Variable Convex
In this section, we prove that the class of variable convex CNFs has the constructive few subterms property (Lemma 1), and hence admits polynomial time compilation into OBDDs (Theorem 2); as a special case, CNFs whose incidence graphs are cographs admit polynomial time compilation into OBDDs (Example 1).
Lemma 1.The class F of variable convex CNFs has the constructive few subterms property.
We now introduce a partially ordered set P , representing the entailment relation among var(π)-active clauses restricted to literals on variables in var(π).Formally, we define P as follows: We now establish a correspondence between the var(π)-subterms of F and the elements in P , which allows to bound above the size of st(F, var(π)) by the size of P .
• There exists c ∈ ac(F, var(π)) such that [c] ≡ is maximal in P with the property that f does not satisfy c.
• Let t ∈ ac(F, var(π)) be such that [t] ≡ is maximal in P with the property that f does not satisfy t.Then, Proof of Claim.For the first part, let f : var(π) → {0, 1} be an assignment not satisfying ac(F, var(π)).By the first claim, there is a unique inclusion maximal clause c among the clauses in ac(F, var(π)) not satisfied by f .If [c] ≡ is maximal in P , then we are done.Otherwise, assume that [c] ≡ is not maximal in P , and assume for a contradiction that there exists holds that d contains at least one literal l, on a variable in var(π), such that l is not in c; a contradiction, since c is chosen inclusion maximal among the clauses in ac(F, var(π)) not satisfied by f .
For the second part, let t ∈ ac(F, var(π)) be such [t] ≡ is maximal in P with the property that f does not satisfy t.By definition, if c ∈ [s] ≡ and [s] ≡ ≤ [t] ≡ , then c entails t upon restriction to variables in var(π).Hence, if f does not satisfy t, it holds that f does not satisfy c.Hence, ac(F, var(π) The claim is settled.This shows that stw(F, σ) is linear in the size of F , where σ is an ordering witnessing left convexity of inc(F ).This proves that the class of variable convex CNFs has few subterms.Moreover, an ordering witnessing the left convexity of inc(F ) can be computed in polynomial (even linear) time [5,21], so the class of variable convex CNFs even has the constructive few subterms property.Example 1 (Bipartite Cographs).Let F be a CNF such that inc(F ) is a cograph.Note that inc(F ) is a complete bipartite graph.Indeed, cographs are characterized as graphs of clique-width at most 2 [9], and it is readily verified that if a bipartite graph has clique-width at most 2, then it is a complete bipartite graph.A complete bipartite graph is trivially left convex.Then Theorem 2 implies that CNFs whose incidence graphs are cographs have polynomial time compilation into OBDDs.

Bounded Treewidth
In this section, we prove that if a class of CNFs has bounded treewidth, then it has the constructive few subterms property (Lemma 3), and hence admits polynomial time compilation into OBDDs (Theorem 3).
A tree decomposition T of a graph G is a rooted tree whose elements, called bags, are subsets of the vertices of G satisfying the following: • for every vertex v of G, there is a bag containing v; • for every edge vw of G, there is a bag containing v and w; • for every three bags B, B , B A graph G has treewidth k if it has a tree decomposition T such that each bag contains at most k + 1 vertices; T is said to witness treewidth k for G.The notions of path decomposition and pathwidth are defined analogously using paths instead of trees.
Let F be a CNF.We say that inc(F ) = (var(F ), F, E) has treewidth (respectively, pathwidth) k if the graph (var(F ) ∪ F, E) has treewidth (respectively, pathwidth) k.We identify the pathwidth (respectively, treewidth) of a CNF with the pathwidth (respectively, treewidth) of its incidence graph.If inc(F ) has pathwidth k, then an ordering σ of var(F ) is called a forget ordering for F if, with respect to an arbitrary linearization of some path decomposition witnessing pathwidth k for inc(F ), if the first bag containing v is less than or equal to the first bag containing v , then σ(v) < σ(v ).
A proof of the following lemma already appears, in essence, in previous work by Ferrara, Pan, and Vardi [13, Theorem 2.1] and Razgon [25,Lemma 5].
Proof.Let F be a CNF such that inc(F ) has pathwidth k − 1, let σ be a forget ordering for F , and let π be any prefix of σ.
Let v be the last variable in var(π) relative to the ordering σ, and let B be the first bag (in the total order of P ) that contains v.

Claim. V ⊆ B.
Proof of Claim.Let c be a var(π)-active clause occurring only in bags strictly larger than B in P .Let v ∈ var(c) ∩ var(π).By the choice of v and the properties of the forget ordering σ, it holds that the first bag containing v is less than or equal to B. Since B is the first bag that contains v, it holds that v ∈ B by the properties of P (the edge cv is witnessed in a bag strictly larger than B in P ).
Proof of Claim.Define an equivalence relation on var(π)-assignments as follows: For all f, f : Proof.Let c−1 be the treewidth bound of F and let F ∈ F, so that the treewidth of inc(F ) is at most c−1.We can compute a width c − 1 tree decomposition of inc(F ) in linear time O(size(F )) [3].From this decomposition, we can compute a path decomposition of inc(F ) of width at most (c−1) Corollary 24] and a corresponding forget ordering of var(F ) in polynomial time.By Lemma 2, the subterm width of F with respect to σ is at most ).Thus F has a polynomial subterm bound, and a witnessing ordering σ can be computed for each F ∈ F in polynomial time.We conclude that F has the constructive few subterms property.Theorem 3. Let F be a class of CNFs of bounded treewidth.Then, F has polynomial time compilation into OBDDs.
Proof.Immediate from Lemma 3 and Corollary 1.

Almost Few Subterms
In this section, we use the language of parameterized complexity to formalize the observation that CNF classes "close" to CNF classes with few subterms have "small" OBDD representations [12,14].
Let F be a CNF and D a set of variables and clauses of F .Let E be the formula obtained by deleting D from F , that is, we call D the deletion set of F with respect to E.
The following lemma shows that adding a few variables and clauses does not increase the subterm width of a formula too much.
The number of subterms C[f ] for f ∈ {0, 1} X is bounded from above by the number of subsets of C, so st(C, X) Splitting the assignments f into two parts, we can write this as Let f ∈ {0, 1} V ∩X be an assignment.The formula E is obtained from and the right hand side of this inequality corresponds to |st(E, X \ V )|.Combining this with (2), we obtain Inserting into (1), we get where k = k + k , and the lemma is proved.
In this section, the standard of efficiency we appeal to comes from the framework of parameterized complexity [12,14].The parameter we consider is defined as follows.Let F be a class of CNF formulas.We say that F is closed under variable and clause deletion if E ∈ F whenever E is obtained by deleting variables or clauses from F ∈ F. Let F be a CNF class closed under variable and clause deletion.The F-deletion distance of F is the minimum size of a deletion set of F from any E ∈ F. An F-deletion set of F is a deletion set of F with respect to some E ∈ F.
Let F be a class of CNF formulas with few subterms closed under variable and clause deletion.We say that CNFs have fixed-parameter tractable OBDD size, parameterized by F-deletion distance, if there is a computable function f : N → N, a polynomial p : N → N, and an algorithm that, given a CNF F having F-deletion distance k, computes an OBDD equivalent to F in time f (k) p(size(F )).
Theorem 4. Let F be a class of CNF formulas with few subterms closed under variable and clause deletion.CNFs have fixed-parameter tractable OBDD size parameterized by F-deletion distance.
The assumption that F is closed under variable and clause deletion is technically necessary to have, for every CNF, a finite deletion distance from F; it is a mild assumption though, as it is readily verified that if F has few subterms with polynomial subterm bound p : N → N, then also the closure of F under variable and clause deletion has few subterms with the same polynomial subterm bound.
Proof.Let F be a class of CNF formulas with few subterms closed under variable and clause deletion.Since F has few subterms, it has a polynomial subterm bound p : N → N. Let k be the F-deletion distance of F .Let E ∈ F be a formula such that the deletion distance of F from E is k, and let D the deletion set of F with respect to E. Let π be an ordering of var(E) witnessing p for E, and let σ be an ordering of var(F ) ∩ D. By Lemma 4, the subterm width of F with respect to ρ = σπ is at most 2 k p(size(E)), so by Theorem 1 there is a ρ-OBDD for F of size at most 2 k p(size(E)) |var(F )|.
Analogously, we say that CNFs have fixed-parameter tractable time computable OBDDs (respectively, F-deletion sets), parameterized by F-deletion distance, if an OBDD (respectively, a F-deletion set) for a given CNF F of F-deletion distance k is computable in time bounded above by f (k) p(size(F )).
Theorem 5. Let F be a class of CNFs closed under variable and clause deletion satisfying the following: • F has the constructive few subterms property.
• CNFs have fixed-parameter tractable time computable F-deletion sets, parameterized by F-deletion distance.
CNFs have fixed-parameter tractable time computable OBDDs parameterized by F-deletion distance.
Proof.Given an input formula F , the algorithm first computes a smallest F-deletion set D of F .Let E be the formula obtained from F by deleting D.

Polynomial Size Incompilability
In this section, we introduce the subfunction width of a graph CNF, to which the OBDD size of the graph CNF is exponentially related (Section 4.1), and prove that expander graphs yield classes of graph CNFs of bounded degree with linear subfunction width, thus obtaining an exponential lower bound on the OBDD size for graph CNFs in such classes (Section 4.2).

Many Subfunctions
In this section, we introduce the subfunction width of a graph CNF (Definition 2), and prove that the OBDD size of a graph CNF is bounded below by an exponential function of its subfunction width (Theorem 6).
A graph CNF is a CNF F such that F = {{u, v} | uv ∈ E} for some graph G = (V, E) without isolated vertices.
Definition 2 (Subfunction Width).Let F be a graph CNF.Let σ be an ordering of var(F ) and let π be a prefix of σ.We say that a subset {c 1 , . . ., c e } of clauses in F is subfunction productive relative to σ and π if there exist {a 1 , . . ., a e } ⊆ var(π) and {u 1 , . . ., u e } ⊆ var(F ) \ var(π) such that for all i, j ∈ {1, . . ., e}, i = j, and all c ∈ F , • c i = {a i , u i }; • c = {a i , a j } and c = {a i , u j }.
The subfunction width of F , in symbols sfw(F ), is defined by where σ ranges over all orderings of var(F ) and π ranges over all prefixes of σ.
Intuitively, in the graph G underlying the graph CNF F in Definition 2, there is a matching of the form a i u i with a i ∈ var(π) and u i ∈ var(F ) \ var(π), i ∈ {1, . . ., e}; such a matching is "almost" induced, in that G can contain edges of the form u i u j , but no edges of the form a i a j or a i u j , i, j ∈ {1, . . ., e}, i = j.Theorem 6.Let F be a graph CNF.The OBDD size of F is at least 2 sfw(F ) .
Proof.Let F be a graph CNF.Let D be any OBDD computing F , let σ be the ordering of var(F ) respected by D, and let π be a prefix of σ such that {c 1 , . . ., c e } ⊆ F is subfunction productive relative to σ and π and e ≥ sfw(F ).Let {a 1 , . . ., a e } ⊆ var(π) and {u 1 , . . ., u e } ⊆ var(F ) \ var(π) be as in Definition 2, so that in particular c i = {a i , u i }, i ∈ {1, . . ., e}.Let in words, L is the set containing, for each assignment of {a 1 , . . ., a e }, its extension to var(π) that sends all variables in var(π) \ {a 1 , . . ., a e } to 1.
• Settle a j = w j , u j = w j , and c j = {a j , u j }.
Each iteration deletes at most 2d vertices in var(π) (the neighbors of w j in var(π), at most d vertices, and the neighbors of w j in var(π), at most d vertices, including w j ), and at most d vertices in neigh(var(π)) (the neighbors of w j in neigh(var(π)), including w j ).Since |var(π)| = n/2 and |neigh(var(π))| ≥ a n/2 by ( 5), the number of steps is We now check Definition 2. By construction, c j = {a j , u j } for all j ∈ {1, . . ., e}.Moreover, let j, j ∈ {1, . . ., e}, j = j , and let c ∈ F .Say without loss of generality that j < j .Assume that c = {a j , a j }.Then there exist w j = a j ∈ var(π) at step j, and w j = a j ∈ var(π) at step j , such that w j w j ∈ F .Then, w j ∈ neigh(w j ), so that it is deleted at step j; but w j exists at step j > j, a contradiction.Finally assume that c = {a j , u j } or c = {a j , u j }.If c = {a j , u j }, then there exist w j = a j ∈ var(π) at step j, and w j = u j ∈ neigh(var(π)) at step j , such that w j w j ∈ F .Then, w j ∈ neigh(w j ) is deleted at step j, but it exists at step j > j, a contradiction.If c = {a j , u j }, then there exist w j = u j ∈ neigh(var(π)) at step j, and w j = a j ∈ var(π) at step j , such that w j w j ∈ F .Then, w j ∈ neigh(w j ) ∩ var(π) is deleted at step j, but it exists at step j > j, a contradiction.
The claim implies that sfw(F ) ≥ min{1,a} 8d • n.Theorem 7.There exist a class F of CNF formulas and a constant c > 0 such that, for every F ∈ F, the OBDD size of F is at least 2 c•size(F ) .In fact, F is a class of read 3 times, monotone, 2-CNF formulas.
Proof.Let G = {G i | i ∈ N} be a family of graphs as in (6), so that for all i ∈ N the graph G i = (V i , E i ) is a (n i , d, a)-expander (n i ≥ 2, d = 3, a > 0) and n i → ∞ as i → ∞.Note that, using the expansion property, it is readily verified that each graph in G is connected; in particular, it does not have isolated vertices.Therefore F = {E i : i ∈ N} is a class of graph CNFs; indeed, it is a class of read 3 times, monotone, 2-CNF formulas.
Let F ∈ F. By Lemma 5, we have that It follows from Theorem 6 that the OBDD size of F is at least 2 c•|var(F )| where c = min{1, a}/16d 2 , and we are done.

Conclusion
We have proved new lower and upper bound results on the OBDD size of structurally characterized CNF classes, pushing the frontier significantly beyond the current knowledge, as depicted in Figure 1.We conclude mentioning that tightening the gap left by this work in the considered hierarchy of structural CNF classes seems to require new ideas.
As far as upper bounds are concerned, the few subterms property is a natural source of polynomial upper bounds; for instance, the width measure recently introduced by Oztok and Darwiche in the compilation of CNFs into DNNFs (a more general formalism than OBDDs), once instantiated to OBDDs, is closely related to our subterm width measure [24].However, the frontier charted in this work seems to push the few subterms property to its limits, in the sense that natural variable orderings do not yield the few subterms property for classes lying immediately beyond the frontier, namely (clause) convex CNFs and bounded clique-width CNF classes.
As for lower bounds, the technique based on expander graphs essentially requires bounded degree, but the candidate classes for improving lower bounds in our hierarchy, bounded clique-width CNFs and beta acyclic CNFs, have unbounded degree.In fact, in both cases, imposing a degree bound leads to classes of bounded treewidth [20] and thus polynomial bounds on the size of OBDD representations.

Figure 1 :
Figure 1: The diagram depicts a hierarchy of classes of bipartite graphs under the inclusion relation (thin edges).B, H, D k , C, Cv, and Cc denote, respectively, bipartite graphs, chordal bipartite graphs (corresponding to beta acyclic CNFs), bipartite graphs of degree at most k (k ≥ 3), convex graphs, left (variable) convex graphs, and right (clause) convex graphs.The class Cv ∩ Cc of biconvex graphs and the class D k of bipartite graphs of degree at most k have unbounded clique-width.The class H ∩ D k of chordal bipartite graph of degree at most k has bounded treewidth.The green and red curved lines enclose, respectively, classes of incidence graphs whose CNFs have polynomial time OBDD compilation, and classes of incidence graphs whose CNFs have exponential size OBDD representations; the right hand picture shows the compilability frontier, updated in light of Results 1 and 2.
the following way.Let (b 1 , . . ., b n ) ∈ {0, 1} n and let w be a node labelled with x i .We say that (b 1 , . . ., b n ) activates an outgoing edge of w labelled with b ∈ {0, 1} if b i = b.Since (b 1 , . . ., b n ) activates exactly one outgoing edge of each non-sink node, there is a unique sink that can be reached from v along edges activated by (b 1 , . . ., b n ).We let F v (b 1 , . . ., b n ) = b, where b ∈ {0, 1} is the label of this sink.The function computed by D is F s , where s denotes the (unique) source node of D. The size of a BDD is the number of its nodes.
1, and |var(F )| levels are processed, the diagram D has size at most |var(F )| • stw(F, σ) and is constructed in time bounded above by a polynomial in |var(F )| and stw(F, σ).Corollary 1.Let F be a class of CNFs with constructive few subterms.Then F has has polynomial time compilation into OBDDs.Proof of Corollary 1.Let F be a class of CNFs with constructive few subterms, and let p : N → N be a polynomial subterm bound of F. The algorithm, given a CNF F , computes in polynomial time an ordering of var(F ) witnessing p with respect to F , and invokes the algorithm in Theorem 1, which runs in time polynomial in |var(F )| and stw(F, σ).Since stw(F, σ) ≤ p(size(F )) the overall runtime is polynomial in size(F ).

Theorem 2 .
The class of variable convex CNF formulas has polynomial time compilation into OBDDs.Proof.Immediate from Corollary 1 and Lemma 1.
A clause c ∈ F is called var(π)-active in F if var(c) ∩ var(π) = ∅ and var(c) ∩ (var(F ) \ var(π)) = ∅.Let ac(F, var(π)) denote the CNF containing the var(π)-active clauses of F .Let C = ac(F, var(π)) ∩ B, C = {c ∈ ac(F, var(π)) | c ∈ B only if B > B in P }; in words, C contains var(π)-active clauses in the bag B, and C contains var(π)-active clauses occurring only in bags strictly larger than B in the total order of P .Clearly, C ∩ C = ∅.Claim.ac(F, var(π)) = C ∪ C .Proof of Claim.First observe that a var(π)-active clause c cannot occur only in bags strictly smaller than B in the total order of P .For otherwise, since var(c) ∩ (var(F ) \ var(π)) = ∅, let v ∈ var(c) ∩ (var(F ) \ var(π)); if B is the first bag that contains v , then B ≤ B (by the choice of v), hence v is not contained in any bag strictly smaller than B, and the edge cv is not witnessed in P , a contradiction.Thus var(π)-active clauses either occur in B (including the case where they occur in B and in bags smaller or larger than B in P ), or occur only in bags strictly larger than B in P .Thus, ac(F, var(π)) ⊆ C ∪ C ; the other inclusion holds by definition.The claim and the fact that C ∩ C = ∅ imply that |st(ac(F, var(π)))| ≤ |st(C , var(π))| • |st(C , var(π))|; thus, suffices to bound above the size of the two sets on the right so that the product of the individual bounds is at most 2 k .Let k = |C |.Obviously, Claim.|st(C , var(π))| ≤ 2 k .Let V = c∈C var(c) ∩ var(π) and let k = |V |.
[8] algorithm computes a variable ordering π of E witnessing a polynomial subterm bound p : N → N of F. Since F has the constructive few subterms property, this can be done in polynomial time.Next, the algorithm chooses an arbitrary ordering σ of var(F ) ∩ D. By Lemma 4 we have stw(F, σπ) ≤ 2 |D| stw(E, π) ≤ 2 k p(size(E)), where k is the F-deletion distance of F .Invoking the algorithm of Theorem 1, our algorithm computes and returns an OBDD for F in time polynomial in 2 k p(size(E)) |var(F )|.Since size(E) ≤ size(F ) there is a polynomial q : N → N such that last expression is bounded by 2 k q(size(F )).Corollary 2 (Feedback Vertex Set).Let F be the class of formulas whose incidence graphs are forests.CNFs have fixed-parameter tractable time computable OBDDs parameterized by F-deletion distance.Proof.Given a graph G = (V, E), a set D ⊆ V is called a feedback vertex set of G if the graph G \ D is a forest; here, G \ D is the graph (V \ D, E ) such that vw ∈ E if and only if vw ∈ E and v, w ∈ V \ D. For any CNF F , a subset D of its variables and clauses is a feedback vertex set of the incidence graph inc(F ) if and only if it is a F-deletion set, so a smallest feedback vertex set of inc(F ) is a smallest F-deletion set.There is fixed-parameter tractable algorithm that, given a graph G and a parameter k, computes a feedback vertex set D of G such that |D| ≤ k or reports that no such set exists[8].It follows that there is a fixed-parameter tractable algorithm, parameterized by the F-deletion distance, for computing a smallest F-deletion set of an input CNF.Moreover, the incidence graphs of formulas in F have treewidth 1, so F has the constructive few subterms property by Lemma 3. Clearly, F is closed under variable and clause deletion.Hence, applying Theorem 5, we conclude that CNFs have fixed-parameter tractable time computable OBDDs parameterized by F-deletion distance.