Characterization and Lower Bounds for Branching Program Size using Projective Dimension

We study projective dimension, a graph parameter (denoted by pd$(G)$ for a graph $G$), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving lower bounds for pd$(G_f)$ for bipartite graphs $G_f$ associated with a Boolean function $f$ imply size lower bounds for branching programs computing $f$. Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy 2000), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between the projective dimension and size of the optimal branching program computing $f$ (denoted by bpsize$(f)$), is $2^{\Omega(n)}$. Motivated by the argument in (Pudl\'ak, R\"odl 1992), we define two variants of projective dimension - projective dimension with intersection dimension 1 (denoted by upd$(G)$) and bitwise decomposable projective dimension (denoted by bitpdim$(G)$). As our main result, we show that there is an explicit family of graphs on $N = 2^n$ vertices such that the projective dimension is $O(\sqrt{n})$, the projective dimension with intersection dimension $1$ is $\Omega(n)$ and the bitwise decomposable projective dimension is $\Omega(\frac{n^{1.5}}{\log n})$. We also show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between upd$(G_f)$ and bpsize$(f)$ is $2^{\Omega(n)}$. In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a constant $c>0$ and for any function $f$, $\textrm{bitpdim}(G_f)/6 \le \textrm{bpsize}(f) \le (\textrm{bitpdim}(G_f))^c$. We also study two other variants of projective dimension and show that they are exactly equal to well-studied graph parameters - bipartite clique cover number and bipartite partition number respectively.


Introduction
A central question in complexity theory -the P vs L problem -asks if a deterministic Turing machine that runs in polynomial time can accept any language that cannot be accepted by deterministic Turing machines with logarithmic space bound.A stronger version of the problem asks if P is separate from L/poly (deterministic logarithmic space given polynomial sized advice).The latter, recast in the language of circuit complexity theory, asks if there exists an explicit family of functions (f : {0, 1} n → {0, 1}) computable in polynomial time (in terms of n), such that any family of deterministic branching programs computing them has to be of size 2 Ω(n) .However, the best known non-trivial size lower bound against deterministic branching programs, due to Nechiporuk [14] in 1970s, is Ω( n 2 log 2 n ).Pudlák and Rödl [15] described a linear algebraic approach to show size lower bounds against deterministic branching programs.They introduced a linear algebraic parameter called projective dimension (denoted by pd F (f ), over a field F) defined on a natural graph associated with the Boolean function f .For a Boolean function f : {0, 1} 2n → {0, 1}, fix a partition of the input bits into two parts of size n each, and consider the bipartite graph G f (U, V, E) defined on vertex sets U = {0, 1} n and V = {0, 1} n , as (u, v) ∈ E if and only if f (uv) = 1.We call G f as the bipartite realization of f .For a bipartite graph G(U, V, E), the projective dimension of G over a field F, denoted by pd F (G), is defined as the smallest d for which there is a vector space W of dimension d (over F) and a function φ mapping vertices in U, V to linear subspaces of W such that for all (u, v) ∈ U × V , (u, v) ∈ E if and only if φ(u) ∩ φ(v) = {0}.We say that φ realizes the graph G.
Pudlák and Rödl [15] showed that if f can be computed by a deterministic branching program of size s, then pd F (f ) ≤ s over any field F. Thus, in order to establish size lower bounds against branching programs, it suffices to prove lower bounds for projective dimension of explicit family of Boolean functions.
Pudlák and Rödl in [15] showed that for most Boolean functions f : {0, In a subsequent work, the same authors [16] also established an upper bound pd R (f ) = O( 2 n n ) for all functions.More recently, Rónyai, Babai and Ganapathy [19] established the same lower bound over all fields.Over finite fields F, Pudlák and Rödl [15] also showed (by a counting argument) that there exists a Boolean function f : {0, 1} n × {0, 1} n → {0, 1} such that pd F (f ) is Ω( √ 2 n ).However, till date, obtaining an explicit family of Boolean functions (equivalently graphs) achieving such lower bounds remain elusive.The best lower bound for projective dimension for an explicit family of functions is for the inequality function (on 2n bits, the graph is the bipartite complement of the perfect matching) where a lower bound of n for an absolute constant > 0 is known [15] over R. For a survey on projective dimension and related linear algebraic techniques, refer [16,11].Thus, the best known size lower bound that was achieved using this framework is only Ω(n) which is not better than trivial lower bounds.Our Results : The starting point of our investigation is the observation that projective assignment appearing in the proof of [15] also has the property that the dimension of the intersection of two subspaces assigned to the vertices is exactly 1, whenever they intersect (See Proposition 2.2(2)).We denote, for a function f , the variant of projective dimension defined by this property as upd(f ) (See Section 4).From the above discussion, for any Boolean function f , pd(f ) ≤ upd(f ) ≤ bpsize(f ).A natural question is whether this restriction helps in proving better lower bounds for the branching programs.By observing properties about projective dimension and choosing a new candidate function 1 , we demonstrate that the above restriction can help by proving the following quadratic gap between the two measures.
Theorem 1.1.For any d ≥ 0, for the function SI d (on 2d 2 variables, See Definition 2.3), the projective dimension is exactly equal to d, while the projective dimension with intersection dimension 1 is Ω(d 2 ).However, this does not directly improve the known branching program size lower bound for SI d , since it leads to only a linear lower bound on upd(SI d ).We demonstrate the weakness of this measure by showing the existence of a function (although not explicit) for which there is an exponential gap between upd over any partition and the branching program size (Proposition 5.1).This motivates us to look for variants of projective dimension of graphs, which is closer to the optimal branching program size of the corresponding Boolean function.We observe more properties (see Proposition 2.2) about the subspace assignment from the proof of the upper bound from [15].We call the projective assignments with these properties bitwise decomposable projective assignment and denote the corresponding dimension 2 as bitpdim(f ) (See Definition 5.2).Thus, for any Boolean function f , pd(f ) ≤ bitpdim(f ).We also show that bitpdim(f ) ≤ 6 • bpsize(f ) (Lemma 5.3).To demonstrate the tightness of the definition, we first argue a converse with respect to this new parameter.

Theorem 1.2.
There is an absolute constant c > 0 such that if bitpdim(f n ) ≤ d(n) for a function family {f n } n≥0 on 2n bits, then there is a deterministic branching program of size (d(n)) c computing it.
Thus, super-polynomial size lower bounds for branching programs imply super-polynomial lower bounds for bitpdim(f ).The function SI d (on 2d 2 input bits -See Definition 2.3) is a natural candidate for proving bitpdim lower bounds as the corresponding language is hard 3 for the complexity class C = L under logspace Turing reductions.
However, the best known lower bound for branching program size for an explicit family of functions is Ω n 2 log 2 n by Nechiporuk [14] which uses a counting argument on the number of subfunctions.By Theorem 1.2 , bitpdim(f ) (for the same explicit function) is at least Ω n 2/c log 2/c n .The constant c is more 4 than 3 and hence implies only weak lower bounds for bitpdim.Despite this weak connection, by combining the counting strategy with the linear algebraic structure of bitpdim, we show a super-linear lower bound for SI d matching the branching program size lower bound 5 .
Theorems 1.1 and 1.3 demonstrate gaps between the pd and the new measures considered.In particular, for log n .We remark that Theorem 1.3 implies a size lower bound of Ω( n 1. 5  log n ) for branching programs computing the function SI d (where n = d 2 ).However, note that this can also be derived from Nechiporuk's method.For the Element Distinctness function, the above linear algebraic adaptation of Nechiporuk's method for bitpdim gives Ω( n 2 log 2 n ) lower bounds (for bitpdim and hence for bpsize) which matches with the best lower bound that Nechiporuk's method can derive.This shows that our modification of approach in [15] can also achieve the best known lower bounds for branching program size.
Continuing the quest for better lower bounds for projective dimension, we study two further restrictions.In these variants of pd and upd, the subspaces assigned to the vertices must be spanned by standard basis vectors.We denote the corresponding dimensions as spd(f ) and uspd(f ) respectively.It is easy to see that for any 2n-bit function, both of these dimensions are upper bounded by 2 n .
We connect these variants to some of the well-studied graph parameters.The bipartite clique cover number (denoted by bc(G)) is the smallest collection of complete bipartite subgraphs of G such that every edge in G is present in some graph in the collection.If we insist that the bipartite graphs in the collection be edge-disjoint, the measure is called bipartite partition number denoted by bp(G).By definition, bc(G) ≤ bp(G).These graph parameters are closely connected to communication complexity as well.More precisely, log(bc(G f )) is exactly the non-deterministic communication complexity of the function f , and log(bp(G f )) is a lower bound on the deterministic communication complexity of f (See [8]).In this context, we show the following: Thus, if for a function family, the non-deterministic communication complexity is Ω(n), then we will have spd(f ) = 2 Ω(n) .Thus, both spd(DISJ) and uspd(DISJ) are 2 Ω(n) .

Preliminaries
In this section, we introduce the notations used in the paper.For definitions of basic complexity classes and computational models, we refer the reader to standard textbooks [8,20].
Unless otherwise stated, we work over the field F 2 .We remark that our arguments do generalize to any finite field.All subspaces that we talk about in this work are linear subspaces.Also 0 and {0} denote the zero vector, and zero-dimensional space respectively.For a subspace U ⊆ F n , we call the ambient dimension of U as n.We denote e i ∈ F n as the i th standard basis vector with i th entry being 1 and rest of the entires being zero.
For a graph G(U, V, E), recall the definition of projective dimension of G over a field F(pd F (G)), defined in the introduction.For a Boolean function f : {0, 1} 2n → {0, 1}, fix a partition of the input bits into two parts of size n each, and consider the bipartite graph G f defined on vertex sets U = {0, 1} n and V = {0, 1} n , as (u, v) ∈ E if and only if f (uv) = 1.A φ is said to realize the function f if it realizes G f .Unless otherwise mentioned, the partition is the one specified in the definition of the function.We denote by bpsize(f ) the number of vertices (including accept and reject nodes) in the optimal branching program computing f .Theorem 2.1 (Pudlák-Rödl Theorem [15]).For a Boolean function f computed by a deterministic branching program of size s and F being any field, pd F (G f ) ≤ s.
The proof of this result proceeds by producing a subspace assignment for vertices of G f from a branching program computing f .We reproduce the proof of the above theorem in our notation, in Appendix A and derive the following proposition from the same.Proposition 2.2.For a Boolean function f : {0, 1} n ×{0, 1} n → {0, 1} computed by a deterministic branching program of size s, there is a collection of subspaces of and D = {V b j } j∈[n],b∈{0,1} , where we associate the subspace U a i with a bit assignment x i = a and V b j with y j = b such that if we define the map φ assigning subspaces from 3. For any W ∈ C ∪ D, ∃S ⊆ S such that W = span {S }.
Proof.We reuse the notations introduced in proof of Theorem 2.1 which we have described in the Appendix A. If H x denotes the set of edges that are closed on an input a, then the subspace assignment φ(a) is span of vectors associated with edges of H x .Denote by H x i =a i , the subgraph consisting of edges labeled x i = a i .Hence H a can be written as span of vectors associated with H x i =a i .Hence φ(a) can be expressed as span n i=1 U i where U i = span (u,v)∈Hx i =a i (e u − e v ).A similar argument shows that φ(y) also has such a decomposition.We now argue the properties of φ.
Note that the first and third property directly follow from proof.To see second property, observe that the branching program is deterministic and hence there can be only one accepting path.Since we observed that the vectors in the accepting path contribute to the intersection space and since there is only one such path, dimension of the intersection spaces is bound to be 1.
We define the following family of functions and family of graphs based on subspaces of a vector space and their intersections.Definition 2.3 (SI d , P d ).Let F be a finite field.Denote by SI d , the Boolean function defined on F d×d ×F d×d → {0, 1} as for any A, B ∈ F d×d SI d (A, B) = 1 if and only if rowspan(A)∩rowspan(B) = {0}.Note that the row span is over the field F (which, in our case, is F 2 ).Denote by P d , the bipartite graph (U, V, E) where U and V are the set of all subspaces of F d .And for any (I, J) ∈ U × V , (I, J) ∈ E ⇐⇒ I ∩ J = {0} We collect the definitions of Boolean functions which we deal with in this work.For (x, y) ∈ {0, x i ∧ y i = 0 and is 0 otherwise.Note that all the functions discussed so far has branching programs of size O(n) computing them and hence have projective dimension O(n) by Theorem 2.1.
Let m ∈ N and n = 2m log m.The Boolean function, Element Distinctness, denoted ED n is defined on 2m blocks of 2 log m bits, x 1 , . . ., x m and y 1 , . . ., y m bits and it evaluates to 1 if and only if all the x i s and y i s take distinct values when interpreted as integers in [m 2 ].Let q be a power of prime congruent to 1 modulo 4. Identify elements in {0, 1} n with elements of F * q .For x, y ∈ F * q , the Paley function PAL q n (x, y) = 1 if x − y is a quadratic residue in F * q and 0 otherwise.We observe for any induced subgraph H of G, if G is realized in a space of dimension d, then H can also be realized in a space of dimension d.For any d ∈ N, P d appears as an induced subgraph of the bipartite realization of SI d .Hence, pd(SI d ) ≥ pd(P d ).
We need the following definition of Gaussian coefficients.For non-negative integers n, k and a prime power q, n k q is the expression, (q n −1)(q n −q)...(q n −q k−1 ) (q k −1)(q k −q)...
Linear Algebra : We recall some basic lemmas from linear algebra which we use later.Unless otherwise mentioned, all our algebraic formulations are over finite fields (F of size q).For vector spaces V 1 , V 2 with dimensions k 1 , k 2 respectively, the direct sum V 1 ⊕ V 2 is the vector space formed by the column space of the matrix We now state a useful property of direct sum.
Proposition 2.4.For an arbitrary field Let U, V be two vector spaces.Then the vector space formed by Span {uv | u ∈ U, v ∈ V } is called the tensor product of vector spaces U, V denoted as U ⊗ V .Here u, v are column vectors.A basic fact about tensor product that we need is the following.Let U be a vector space having basis u 1 , u 2 , . . .u k and V be a vector space having basis v 1 , v 2 , . . ., v over some field F then, vector space U ⊗ V has a basis B = {u i v j | i ∈ {1, 2, . . ., k}, j ∈ {1, 2, . . ., }} where u, v are column vectors.Hence, for any two vector spaces The proofs of the two Propositions 2.4 and 2.5 are fairly elementary and follows from basic linear algebra.For example Proposition 2.5 follows as an easy corollary from an exercise from [18, Chapter 14, exercise 12]6 .
Let V be a finite dimensional vector space.For any Then any vector w ∈ V can be uniquely expressed as w = Π A (w) + Π B (w).It is easy to see that, for any A, B ⊆ S F d , with A ∩ B = {0}, and any w ∈ F d , Π A+B (w) = Π A (w) + Π B (w).

Properties of Projective Dimension
In this section, we observe properties about projective dimension as a measure of graphs and Boolean functions.We start by proving closure properties of projective dimension under Boolean operations ∧ and ∨.The proof is based on direct sum and tensor product of vector spaces.Lemma 3.1.Let F be an arbitrary field.For any two functions Proof.In this proof, for a Boolean f with bipartite representation G f (U, V, E) we define the map φ to be from {0, 1} n × {0, 1} where φ(u, 0) denotes the subspace assigned to u ∈ U and φ(v, 1) denotes the subspace assigned to v ∈ V of G f .Let f 1 and f 2 be of projective dimensions k 1 and k 2 realized by maps • From φ 1 and φ 2 we construct a subspace assignment φ : {0, 1} n × {0, 1} → F k 1 +k 2 which realize f 1 ∨ f 2 thus proving the theorem.The subspace assignment is : By Proposition 2.4, it must be the case that (φ • From φ 1 and φ 2 we construct a subspace assignment φ : {0, 1} n × {0, 1} → F k 1 k 2 , realizing f 1 ∧ f 2 thus proving the theorem.Consider the following projective dimension assignment φ: ).The proof is similar to the previous case and applying Proposition 2.5, completes the proof.
The ∨ part of the above lemma was also observed (without proof) in [16].A natural question is whether we can improve any of the above bounds.In that context, we make the following remarks: (1) the construction for ∨ is tight up to constant factors, (2) we cannot expect a general relation connecting pd R (f ) and pd R (¬f ).
• We prove that the construction for ∨ is tight up to constant factors.Assume that n is a multiple of 4. Consider the functions ) and g(x n 4 +1 , . . ., x n 2 , x 3n 4 +1 , . . ., x n ) each of which performs inequality check on the first n 4 and the second n 4 variables.It is easy to see that f ∨ g is the inequality function on n 2 variables x 1 , . . ., x n 2 and the next n 2 variables x n 2 +1 , . . ., x n .By the fact that they are computed by n size branching programs and using Theorem 2.1 (Pudlák-Rödl theorem) we get that pd(f ) ≤ n and pd(g) ≤ n.Hence by Lemma 3.1, pd(f ∨ g) ≤ pd(f ) + pd(g) ≤ 2n.Lower bound on projective dimension of inequality function comes from [15,Theorem 4], giving pd(f ∨ g) ≥ .n 2 for an absolute constant .This shows that pd(f ∨ g) = Θ(n).
• A natural idea to improve the upper bound of pd(f 1 ∧ f 2 ) is to prove upper bounds for pd(¬f ) in terms of pd(f ).However, we remark that over R, it is known [15] Hence we cannot expect a general relation connecting pd R (f ) and pd R (¬f ).
We now observe a characterization of bipartite graphs having projective dimension at most d over F.
Proposition 3.2.For any subspace assignment φ realizing G f , no two vertices from the same partition whose neighborhoods are different can get the same subspace assignment.
Proof.Suppose there exists x, x ∈ S from the same partition, i.e., either X or Y ,such that φ(x) = φ(x ).Since N (x) = N (x ), without loss of generality, there exists z ∈ N (x) \ N (x ).Now since φ(x) = φ(x ), x will be made adjacent to z by the assignment and hence φ is no longer a realization of G f since z should not have been adjacent to x .Proof.Suppose G appears as an induced subgraph of P d .To argue, pd(G) ≤ d, simply consider the assignment where the subspaces corresponding to the vertices in P d are assigned to the vertices of G.
On the other hand, suppose pd(G) ≤ d.Let U 1 , . . ., U N and V 1 , . . ., V N be subspaces assigned to the vertices.Since the neighborhoods of the associated vertices are different, by Proposition 3.2, no two subspaces assigned to these vertices can be the same.Hence corresponding to each vertex in G, there is a unique vertex in P d which corresponds to the assignment.Now the subgraph induced by the vertices corresponding to these subspaces in P d must be isomorphic to G as the subspace assignment map for G preserves the edge non-edge relations in G.
It follows that pd(P d ) ≤ d.Observe that, in any projective assignment, the vertices with different neighborhoods should be assigned different subspaces.For pd(P d ), all vertices on either partitions have distinct neighborhoods.The number of subspaces of a vector space of dimension d−1 is strictly smaller than the number of vertices in P d .Thus, we conclude the following theorem.For an N vertex graph G, the number of vertices of distinct neighborhood can at most be N .Thus, the observation that we used to show the lower bound for the graph pd(P d ) cannot be used to obtain more than a √ log N lower bound for pd(G).Also, for many functions, the number of vertices of distinct neighborhood can be smaller.
We observe that by incurring an additive factor of 2 log N , any graph G on N vertices can be transformed into a graph G on 2N vertices such that all the neighborhoods of vertices in one partition are all distinct.Let f : {0, 1} 2n → {0, 1} be such that the neighborhoods of G f are not necessarily distinct.We consider a new function f whose bipartite realization will have two copies of G f namely G 1 (A 1 , B 1 , E 1 ) and G 2 (A 2 , B 2 , E 2 ) where A 1 , A 2 , B 1 , B 2 are disjoint and a matching connecting vertices in A 1 to B 2 and A 2 to B 1 respectively.Since the matching edges associated with every vertex is unique, the neighborhoods of all vertices are bound to be distinct.Applying Lemma 3.1 and observing that matching (i.e, equality function) has projective dimension at most n, pd(f ) ≤ 2pd(f ) + 2n.This shows that to show super-linear lower bounds on projective dimension for f where the neighborhoods may not be distinct, it suffices to show a super-linear lower bound for f .

Projective Dimension with Intersection Dimension 1
Motivated by the proof of Theorem 2.1 (presented in Appendix A) we make the following definition.U, V, E) is said to have projective dimension with intersection dimension 1 (denoted by upd(f )) d over field F, if d is the smallest possible dimension for which there exists a vector space K of dimension d over F with a map φ assigning subspaces of K to U ∪ V such that By the properties observed in Proposition 2.2, Theorem 4.2.For a Boolean function f computed by a deterministic branching program of size s, upd F (f ) ≤ s for any field F. Thus, it suffices to prove lower bounds for upd(f ) in order to obtain branching program size lower bounds.We now proceed to show lower bounds on upd.
Our approaches use the fact that the adjacency matrix of P d has high rank.
Lemma 4.3.Let M be the bipartite adjacency matrix of Proof.For 0 ≤ i ≤ k ≤ d, and subspace and 0 otherwise.This matrix has dimension d i q × d k q .Consider the submatrix M i of M with rows and columns indexed by subspaces of dimension exactly i. Observe that W ii = J − M i where J is an all ones matrix of appropriate order.These matrices are well-studied (see [7]).Closed form expressions for eigenvalues are computed in [5,12] and the eigenvalues are known to be non-zero.Hence for 0 ≤ i ≤ d/2 the matrix W ii has rank We now present two approaches for showing lower bounds on upd(f ) -one using intersection families of vector spaces and the other using rectangle arguments on M f .Lower Bound for upd(P d ) using intersecting families of vector spaces : To prove a lower bound on upd(P d ) we define a matrix N from a projective assignment with intersection dimension 1 for P d , such that it is equal to (q − 1)M .Let D = upd(P d ).We first show that rank (N ) is at most 1 + D 1 q .Then by Lemma 4.3 we get that rank (N ) is at least q d 2 4 .Let G = {G 1 , . . ., G m }, H = {H 1 , . . ., H m } be the subspace assignment with intersection dimension 1 realizing P d with dimension D. Lemma 4.4.For any polynomial p in q x of degree s, with matrix N of order |G| × |H| defined as Proof.This proof is inspired by the proof in [6] of a similar claim where a non-bipartite version of this lemma is proved.To begin with, note that p is a degree s polynomial in q x , and hence can be written as a linear combination of polynomials p i = x i q , 0 ≤ i ≤ s.Let the linear combination be given by p(x) = s i=0 α i p i (x).For 0 ≤ i ≤ s define a matrix N i with rows and columns indexed respectively by G, H defined as To bound the rank of N i 's we introduce the following families of inclusion matrices.For any j ∈ [D], consider two matrices Γ j corresponding to G and ∆ j corresponding to H defined as Γ j (G, I) = 1 if dim(I) = j, G ∈ G, I ⊆ s G and 0 otherwise.∆ j (H, I) = 1 if dim(I) = j, H ∈ H, I ⊆ s H and 0 otherwise.Note that rank of the these matrices are upper bounded by the number of columns which is D j q .We claim that for any i ∈ {0, 1, . . ., s}, rank (N i ) ≤ D i q .This completes the proof since N = i∈[s] α i N i .
To prove the claim, let F i denote the set of all i dimensional subspace of F D q .We show that We apply Lemma 4.4 on N defined via p(x) = q x − 1 with s = 1, to get . By definition, rank (N ) = rank (M ).This gives that D = Ω(d 2 ) and proves Theorem 1.1.Lower Bound for upd(P d ) from Rectangle Arguments : We now give an alternate proof of for Theorem 1.1 using combinatorial rectangle arguments.
where F is a finite field of size q.
Proof.Let φ be a subspace assignment realizing f of dimension d with intersection dimension 1.Let . Consider all 1 dimensional subspaces which appear as intersection space for some input (x, y).Fix a basis vector for each space and let T denote the collection of basis vectors of all the intersection spaces.Note that for any (x, y) ∈ f −1 (1), there is a unique v ∈ F d q (up to scalar multiples) such that (x, y) ∈ S(v) for otherwise intersection dimension exceeds 1.Then The fact that the number of 1 dimensional spaces in F d can be at most q d −1 q−1 completes the proof.Note that the rank of M f can be over any field (we choose R).
We get an immediate corollary.Any function f , such that the adjacency matrix of M f of the bipartite graph G f is of full rank 2 n over some field must have upd(f ) = Ω(n).There are several Boolean functions with this property, well-studied in the context of communication complexity (see textbook [10]).Hence, we have for f ∈ {IP n , EQ n , INEQ n , DISJ n , PAL q n }, upd F (f ) is Ω(n) for any finite field F.
For arguing about PAL q n , it can be observed that the graph is strongly regular (as q ≡ 1 mod 4) and hence the adjacency matrix has full rank over R [4].Except for PAL q n , all the above functions have O(n) sized deterministic branching programs computing them and hence the Pudlák-Rödl theorem (Theorem 2.1) gives that upd for these functions (except PAL q n ) are O(n) and hence the above lower bound is indeed tight.From Lemma 4.3, it follows that the function SI d also has rank 2 Ω(d 2 ) .To see this, it suffices to observe that P d appears as an induced subgraph in the bipartite realization of SI d .Thus, upd(SI d ) is Ω(d 2 ).We proved in Theorem 3.4 that pd(SI d ) = d.This establishes a quadratic gap between the two parameters.This completes the proof of Theorem 1.1.
Let D(f ) denote the deterministic communication complexity of the Boolean function f .We observe that the rectangle argument used in the proof of Lemma 4.5 is similar to the matrix rank based lower bound arguments for communication complexity.This yields the Proposition 4.6.If upd(f ) ≤ D, the assignment also gives a partitioning of the 1s in M f into at most q D −1 q−1 1-rectangles.However, it is unclear whether this immediately gives a similar partition of 0s into 0-rectangles as well.Notice that if D(f ) ≤ d, there are at most 2 d monochromatic rectangles (counting both 0-rectangles and 1-rectangles) that cover the entire matrix.However, our proof does not exploit this difference.Proposition 4.6.For a Boolean function f : {0, 1} n × {0, 1} n → {0, 1} and a finite field We give a proof of the first inequality.Any deterministic communication protocol computing , there is exactly one i ∈ [k] where f i (x, y) = 1.Hence for each j ∈ [k], j = i, the intersection vector corresponding to the edge (x, y) in the assignment of f j is trivial.Hence the assignment obtained by applying Lemma 3.1, to f 1 , ∨f 2 ∨ . . .f k will have the property that for any (x, y) with f (x, y) = 1, the intersection dimension is 1.Hence upd F (f ) ≤ k ≤ 2 D(f ) .To prove the second inequality, consider the protocol where Alice sends the subspace associated with her input as a pd F (f ) × pd F (f ) matrix.Bob then checks if this subspace intersects with his own subspace and sends 1 if it does so and sends 0 otherwise.An immediate consequence of Proposition 4.6 is that all symmetric functions f on 2n bits have have projective dimension O(n).Note that the first inequality is tight, up to constant factors in the exponent.To see this, consider the function f : {0, 1} n × {0, 1} n → {0, 1} whose pd F (f ) = Ω(2 n/2 ) [15, Proposition 1] and note that D(f ) for any f is at most n.Tightness of second inequality is witnessed by SI d since by Lemma 4.3, D(SI d ) = Ω(d 2 ) while pd(SI d ) = d.

Bitwise Decomposable Projective Dimension
The restriction of intersection dimension being 1, although potentially useful for lower bounds for branching program size, does not capture the branching program size exactly.We start the section by demonstrating a function where the gap is exponential.We show the existence of a Boolean function f such that the size of the optimal branching program computing it is very high but has a very small projective assignment with intersection dimension 1 for any balanced partition of the input.
Proposition 5.1.(Implicit in Remark 1.30 of [8]) There exist a function f : {0, 1} n × {0, 1} n that requires size Ω( 2 n n ) for any branching program computing f but the upd(f ) ≤ O(n) for any balanced partitioning of the input into two parts.
Proof.Consider the function EQ n .The graph G EQ n (U, V, E) with U = V = N is a perfect matching where N = {0, 1} n .Relabel the vertices in U of this graph to produce a family of G of N !different labeled graphs.Let F be the set of Boolean functions whose corresponding graph is in G (or equivalently F of N !different functions).Let t be the smallest number such that any function in F can be computed by a branching program of size at most t.The number of branching programs of size ≤ t (bounded by O(t t ) [8]) forms an upper bound on |F|.Thus, 2 O(t log t) ≥ N !, and hence t is Ω 2 n n .Hence there must exist a function f ∈ F such that upd(f ) = upd(EQ n ) ≤ n but bpsize(f ) is Ω 2 n n for this partition.We now argue upper bound for upd(f ) for any balanced partition.Consider the function f π obtained by a permutation π ∈ S N on the U part of EQ n graph.Consider a partition Π of EQ n , G Π fπ be the corresponding bipartite graphs EQ Π n and f Π π be the corresponding functions) with respect to the partition Π, of EQ n and f π respectively.
We claim that upd(G ).Thus for any input (u , v ) of f Π π there is unique input (x , y ) of EQ Π n obtained via the above procedure.Thus, from the upd assignment for EQ Π n we can get a upd assignment for f Π π .Observing that Theorem 4.2 holds for any partition Π of the input, we get a upd assignment for EQ Π n .
The above proposition can be shown by adapting the counting argument presented in Remark 1.30 of [8].

A Characterization for Branching Program Size
Motivated by strong properties observed in Proposition 2.2, we make the following definition.Definition 5.2 (Bitwise Decomposable Projective Dimension).Let f be a Boolean function on 2n bits and G f be its bipartite realization.The bipartite graph G f (X, Y, E) is said to have bit projective dimension, bitpdim(G) ≤ d, if there exists a collection of subspaces of where a projective assignment φ is obtained by associating subspace U a i with a bit assignment x i = a and V b j with y j = b satisfying the conditions listed below.

for any
. Same property must hold for subspaces in D.
We show that the new parameter bitwise decomposable projective dimension (bitpdim) tightly characterizes the branching program size, up to constants in the exponent.Proof.The subspace assignment obtained by applying (Theorem A.1) on an arbitrary branching program need not satisfy Property 3 because there can be a vertex z that has two edges incident on it reading different variables from the same partition.To avoid this, we subdivide every edge.We show that this transformation is sufficient to get a bitpdim assignment.We now give a full proof.Let B be a deterministic branching program computing f .Denote the first n variables of f as x x i x j Now label every vertex of B with standard basis vectors as it is done in Pudlák-Rödl Theorem (Theorem A.1).Let φ be projective assignment obtained from B via Pudlák-Rödl theorem.We claim that φ satisfies all the requirements of bitpdim(f ).
1. Since φ is obtained via Pudlák-Rödl it captures adjacencies of G f .Hence property 1 holds.
Property 2 is satisfied by Pudlák-Rödl assignment.(See appendix A) 2. The standard basis vector e u corresponding to vertex u appears only in edges incident on u in Pudlák-Rödl assignment.For any edge (u, v) querying a variable x i = b the other edges incident to v must query variables from y.All the edges incident on u, except (u, v) must also query variables from y. Otherwise, there is an edge (w, u) which queries a variable x j and our transformation would have subdivided the edge.Hence e u , e v belongs only to . This implies Property 3.
We show that given a bitpdim assignment for a function f , we can construct a branching program computing f .Theorem 5.4 (Theorem 1.2 restated).For a Boolean function f : {0, 1} n × {0, 1} n → {0, 1} with bitpdim(f ) ≤ d, there exists a deterministic branching program computing f of size d c for some absolute constant c.
Proof.Consider the subspace associated with the variables C, D of the bitpdim assignment as the advice string.These can be specified by a list of n basis matrices each of size d 2 .Note that for any any f which has a polynomial sized branching program, d = bitpdim(f ) is at most poly(n), and hence the advice string is poly(n) sized and depends only on n.
We construct a deterministic branching program computing f as follows.On input x, y, from the basis matrices in C, D, construct an undirected graph7 G * with all standard basis vectors in C, D as vertices and add an edge between two vertices u, v if where C 1 consists of edges from basis matrices in C and C 2 contain edges from basis matrices in D. Note that if one of C 1 or C 2 is empty then there is a cycle consisting only of vectors from C which implies a linear dependence among vectors in C.But this contradicts Property 3 of bitpdim assignment.Hence both C 1 and C 2 are non-empty.
Then, it must be that (u,v)∈C 1 e u − e v + (w,z)∈C 2 e w − e z = 0. Hence (u,v)∈C 1 e u − e v = − (w,z)∈C 2 e w − e z .Hence we get a vector in the intersection which gives f (x, y) = 1.Note that if f (x, y) = 1, then clearly there is a non-zero intersection vector.If we express this vector in terms of basis, we get a cycle in G * .
Hence, to check if f evaluates to 1, it suffices check if there is a cycle in G * which is solvable in L using Reingold's algorithm [17].The log-space algorithm can also be converted to an equivalent branching program of size n c for a constant c.
We can improve the constant c to 3 + .We achieve this using the well known random walk based RL algorithm for reachability [1], amplifying the error and suitably fixing the random bits to achieve a non-uniform branching program of size d 3+ .
The RL algorithm requires to store log d bits to remember the current vertex while doing the random walk and another log d bits to store the next vertex in the walk.It performs a walk of length 4d 3 and answers correctly with probability of 1/2 [13].Amplifying the error does not incur any extra space as the algorithm has a one-sided error and it never errs when it accepts.This gives a probabilistic Turing machine using 2 log d + 1 work space.By amplifying the success probability, we can obtain a choice of a random bits which works for all inputs of a fixed length.The conversion of this machine to a branching program will incur storing of the head index position of the work tape and input tape position which incur an additional log log d + log d space.Hence overall space is 3 log d + log log d = (3 + ) log d for small fixed > 0, thus proving that c ≤ 3 + .
Assuming C = L ⊆ L/poly, the function SI d (a language which is hard for C = L under Turing reductions) cannot be computed by deterministic branching programs of polynomial size.Proof.We start with the following claim.Claim 5.6 (Corollary 2.3 of [2]).Fix an n ∈ N.There exists a logspace computable function g : F n×n → F n×n such that for any matrix M over F n×n , det(M ) = 0 =⇒ rank(g(M )) = n and det(M The reduction is as follows.Given an M ∈ F d×d , apply g (defined in Claim 5.6) on M to get N , and define for 1 where M i 1 is the matrix consisting of i th row of N repeated n times and M i 2 as same as N with i th row replaced by all 0 vectors.For each 1 ≤ i ≤ d, we make oracle query to L checking if H i ∈ L and if all answers are no, reject otherwise accept.
We now argue the correctness of the reduction.Suppose det(M ) is 0, then N = g(M ) (by Claim 5.6) must have full rank.Hence for all 1 ≤ i ≤ d, rowspan(M i 1 ) and rowspan(M i 2 ) does not intersect.If det(M ) = 0, then N = g(M ) (by Claim 5.6) must have a linearly dependent column and hence there is some i for which rowspan(M i 1 ) and rowspan(M i 2 ) is non-zero.Also the overall reduction runs in logspace as g is logspace computable.
The upper bound follows by observing that given two d×d matrices M 1 and M 2 , their individual ranks r 1 and r 2 can be computed in L C=L [2].Consider the matrix M of size d × 2d by adjoining M 1 and M 2 .It follows that the rowspace(M 1 ) ∩ rowspace(M 2 ) = φ if and only if rank (M ) < r 1 + r 2 .The latter condition can also be tested using a query to C = L oracle.

Lower Bounds for Bitwise Decomposable Projective dimension
From the results of the previous section, it follows that size lower bounds for branching programs do imply lower bounds for bitwise decomposable projective dimension as well.As mentioned in the introduction, the lower bounds that Theorem 1.2 can give for bitwise decomposable projective dimension are only known to be sub-linear.
To prove super-linear lower bounds for bitwise decomposable projective dimension, we show that Nechiporuk's method [14] can be adapted to our linear algebraic framework (thus proving Theorem 1.3 from the introduction).The overall idea is the following: given a function f and a bitpdim assignment φ, consider the restriction of f denoted f ρ where ρ fixes all variables except the ones in T i to 0 or 1 where T i is some subset of variables in the left partition.For different restrictions ρ, we are guaranteed to get at least c i (f ) different functions.We show that for each restriction ρ, we can obtain an assignment from φ realizing f ρ .Hence the number of different bitpdim assignments for ρ restricted to T i is at least the number of sub functions of f which is at least c i (f ).Let d i be the ambient dimension of the assignment when restricted to T i .By using the structure of bitpdim assignment, we count the number of assignments possible and use this relation to get a lower bound on d i .Now repeating the argument with disjoint T i , and by observing that the subspaces associated with T i s are disjoint, we get a lower bound on d as d = i d i .
Theorem 5.7.For a Boolean function f : {0, 1} n ×{0, 1} n → {0, 1} on 2n variables, let T 1 , . . ., T m are partition of variables to m blocks of size r i on the first n variables.Let c i (f ) be the number of distinct sub functions of f when restricted to T i , then bitpdim(f Proof.Let (x, y) denote the 2n input variables of f and ρ : {x 1 , . . ., x n , y 1 , . . ., y n } → {0, 1, * } be a map that leaves only variables in T i unfixed.Let φ be a bitpdim assignment realizing f and let Let S be a subspace of F d 2 .Define ρ S to be SI d (A, B) where B is a matrix whose rowspace is S.And A is the matrix whose all but ith row is 0's and ith row consists of variables (x i 1 , . . ., x in ).Thus for any v ∈ {0, 1} d , rowspace of A(x) is span {v}.
We claim that for any S, S ⊆ S log n where n = 2d 2 is the number of input bits of SI d .

Standard Variants of Projective Dimension
In this section, we study two stringent variants of projective dimension for which exponential lower bounds and exact characterizations can be derived.Although these measure do not correspond to restrictions on branching programs, they illuminate essential nature of the general measure.We define the measures and show their characterizations in terms of well-studied graph theoretic parameters.Definition 6.1 (Standard Projective Dimension).A Boolean function f : {0, 1} n × {0, 1} n → {0, 1} with the corresponding bipartite graph G(U, V, E) is said to have standard projective dimension (denoted by spd(f )) d over field F, if d is the smallest possible dimension for which there exists a vector space K of dimension d over F with a map φ assigning subspaces of K to U ∪ V such that • u ∈ U ∪ V , φ(u) is spanned by a subset of standard basis vectors in K.
In addition to the above constraints, if the assignment satisfies the property that for all (u, v) ∈ U × V , dim (φ(u) ∩ φ(v)) ≤ 1, we say that the standard projective dimension is with intersection dimension 1, denoted by uspd(f ).We make some easy observations about the definition itself.
For N × N bipartite graph G with m edges, consider the assignment of standard basis vectors to each of the edges and for any u ∈ U ∪ V , φ(u) is the span of the basis vectors assigned to the edges incident on u.Moreover, the intersection dimension in this case is 1.Hence for any G, spd(G) ≤ uspd(G) ≤ m.
Even though pd(G) ≤ spd(G), there are graphs for which the gap is exponential.For example, consider the bipartite realization G of EQ n with N = 2 n .We know pd(G) = Θ(log N ) but spd(G) ≥ N since each of the vertices associated with the matched edges cannot share any basis vector with vertices in other matched edges.Hence dimension must be at least N .We show that standard projective dimension of bipartite G is same as that of biclique cover number.Definition 6.2 (Biclique cover number).For a graph G, a collection of complete bipartite graphs defined on V (G) is said to cover G if every edge in G is present in some complete bipartite graph of the collection.The size of the smallest collection of bipartite graph which covers G is its biclique cover number (denoted by bc(G)).If in addition, we insist that bicliques must be edge-disjoint, the parameter is known as biclique partition number denoted by bp(G).
Let φ be the intersection dimension one standard assignment of ambient dimension d of f .For every e i ∈ F d , define the set is a bipartite partition of G f .Every C i thus defined is a biclique, because if φ(x, y) = e i then that implies e i ∈ φ(x) and e i ∈ φ(y).Note that for every (x, y) ∈ G f , there exists a unique i ∈ [d] such that φ(x, y) = e i .Hence any (x, y) ∈ G f belongs to exactly one of the sets C i thus implying that C i 's are edge disjoint biclique covers.Note that any (x, y) ∈ G f do not belong to any of C i 's as φ(x, y) = {0}. (uspd where d = bp(G f ) be a biclique partition cover.We give a standard assignment φ for G f defined as follows.For any x, φ(x) = span {e i | ∃y, (x, y) ∈ C i }.By definition φ is a standard assignment.We just need to prove that (x, y) ∈ G f if and only φ(x, y) = {0} and dimφ(x, y) = 1.To prove this we would once again employ the rectangle property of bicliques, that is if (x, y ) ∈ C i and (x , y) ∈ C i then so is (x, y).First we will argue that if there an intersection then it is dimension 1. Recall that intersection of two standard subspaces is a standard subspace.Suppose there is exists (x, y) with dimφ(x, y) > 1.Let e j , e k be any two standard intersection vectors in φ(x, y).By construction and rectangle property of bicliques, we get that (x, y) ∈ C j and (x, y) ∈ C k contradicting the disjoint cover property.Hence for any (x, y), dimφ(x, y) ≤ 1.If (x, y) ∈ G f , then there does not exist an i, (x, y) ∈ C i .But if φ(x, y) = e i for some i ∈ [d], then that implies by rectangle property of bicliques that (x, y) ∈ C i , a contradiction.

Discussion & Conclusion
In this paper we studied variants of projective dimension of graphs with improved connection to branching programs.We showed lower bounds for these measures indicating the weakness and of each of the variants.A pictorial representation of all parameters is shown in Fig. 2.
An immediate question that arises from our work is whether Ω(d 2 ) lower bound on upd(P d ) is tight.In this direction, since we have established a gap between upd(P d ) and pd(P d ), it is natural to study how pd and upd behave under composition of functions, in order to amplify this gap.
In another direction, we believe that the Ω(d 2 ) lower bound on upd(P d ) is not tight.It is natural to study composition of functions to improve this gap.The subspace counting based lower bounds for bitpdim that we proved are tight for functions like ED n .However, observe that under standard complexity theoretic assumptions the bitpdim assignment for P d is not tight.Hence it might be possible to use the specific linear algebraic properties of P d to improve the bitpdim lower bound we obtained for P d .when there is a cycle.It is to avoid this that we add a vertex labeled with variable from the other partition.
To show that φ is a valid subspace assignment, it remains to show that reverse implication of statement 1 holds.Suppose for (u, v) ∈ E(G f ), φ(u), φ(v) are linearly dependent.Hence there exists a non trivial combination giving a zero sum.Let S be the non-empty set of edges such that λ e = 0 and V (S) be its set of vertices.Now for any vertex u ∈ V (S) there must be at least two edges containing u because with just a single edge u , which being a basis vector and summing up to zero, must have a zero coefficient which contradicts that fact that e ∈ S.This shows that every vertex in S has a degree ≥ 2 (in the undirected sense).Hence it must have an undirected cycle.Fig. 3 shows the transformations done to the branching program as per the proof of Pudlák-Rödl Theorem and subspace assignment obtained for 00 and 01.The intersection vector for 00 and 01 is highlighted in blue on the left partition and in red on the right partition.Notice that this intersection vector corresponds to two halves of a cycle starting from start vertex of the modified BP.The subspace assignment for each of the vertices is listed in the table below.).For integers, k ≥ 0, n ≥ k.
Proof.Note that since n ≥ k, q n ≥ q k , we have q n −t q k −t ≥ q n q k for any 0 ≤ t < q k .Hence the lower bound follows.
Remark B.2.This shows that the total number of subspaces of an n dimensional space is upper bounded by 2c q n/2 i=0 q in ≤ 2c 2 q (n 2 +n)/2 .
C Proof of Proposition 2.5 Proof.For the reverse direction, suppose there is a non zero vector w 1 in U 1 ∩ V 1 and a non zero vector For the forward direction, let w be a non zero vector in (U be the set of basis vectors for F k 1 and {ẽ j } j∈[k 2 ] be the set of basis vectors for F k 2 .Hence for some λ ij , µ ij ∈ F, w can be written as, w = i,j λ ij e T i ẽj = i,j µ ij e T i ẽj .Hence, i,j (λ ij − µ ij )e T i ẽj = 0.By linear independence of tensor basis, Since w is non-zero, there exists i 1 , j 1 with (i 1 , j 1 ) ∈ [k 1 ] × [k 2 ] such that λ i 1 j 1 = 0. Applying equation 3, we get µ i 1 j 1 = 0. Hence for (i 1 , j 1 ), λ i 1 j 1 , µ i 1 j 1 are both non-zero.
Hence it must be that (U 1 ⊗ U 2 ) and (V 1 ⊗ V 2 ) has the vector e T i 1 ẽj 1 .So e i 1 must be present in U 1 and V 1 and e j 1 must be present in U 2 and V 2 (if not, e T i 1 ẽj 1 would not have appeared in the intersection).Hence U 1 ∩ V 1 = {0} and U 2 ∩ V 2 = {0}.

Lemma 3 . 3 (
Characterization).Let G be a bipartite graph with no two vertices having same neighborhood, pd(G) ≤ d if and only if G is an induced subgraph of P d .

Figure 1 :
Figure 1: Edge modification and the rest as y.We first apply Pudlák-Rödl transformation on B to obtain a branching program B computing f .We note that |V (B )| = |V (B)|.Obtain B from B by subdividing every edge (u, v) checking a variable x i = b from partition x to get three edges (u, V uv ) checking x i = b and add two edges between (V uv , v) one which checks y 1 = 0 and another which checks y 1 = 1 (see Figure 1).Clearly the transformation does not change the function computed by the branching program.Since we are taking every edge of the branching program B and introducing two more edges, the total number of edges in B is 3|E(B )|.Since B is a deterministic branching program, every vertex v ∈ B has out degree at most 2 and at least 1 for every node except sink node.Hence |E(B )| ≤ 2(|V (B )|).Along with |E(B )| = 3|E(B )|, we get |E(B )| ≤ 6(|V (B )|) = 6(|V (B)|).Now label every vertex of B with standard basis vectors as it is done in Pudlák-Rödl Theorem (Theorem A.1).Let φ be projective assignment obtained from B via Pudlák-Rödl theorem.We claim that φ satisfies all the requirements of bitpdim(f ).

Proposition 5 . 5 .
The function family {SI d } d≥0 is hard for C = L via logspace Turing reductions.Moreover, the negation of {SI d } d≥0 is in L C=L (and hence in NC 2 ).

d 2 where
S = S , (SI d ) ρ S ≡ (SI d ) ρ S .By definition (SI d ) ρ S ≡ SI d (A, B) and (SI d ) ρ S ≡ SI d (A, B ) where B and B are matrices whose rowspaces are S and S respectively.Since S = S there is at least one vector v ∈ F d 2 such that it belongs to only one of S, S .Without loss of generality let that subspace be S. Then SI d (A(v), B) = 1 as v ∈ S where as SI d (A(v), B ) = 0 as v ∈ S .Hence the number of different restrictions is at least number of different subspaces of F d 2 which is 2 Ω(d 2 ) .Hence the proof.This completes the proof of Theorem 1.3 from the introduction.This implies that for SI d , the branching program size lower bound is Ω d 2 log d × d = Ω d 3 log d = Ω n 1.5

Figure 2 :
Figure 2: Parameters considered in this work and their relations