If VNP is hard, then so are equations for it
Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, Anamay Tengse
aa r X i v : . [ c s . CC ] D ec If VNP is hard, then so are equations for it
Mrinal Kumar * C. Ramya † Ramprasad Saptharishi ‡ Anamay Tengse § Abstract
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, weshow that the class
VNP does not have efficiently computable equations. In other words, anynonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class
VNP requires algebraic circuits of super-polynomial size.In a recent work of Chatterjee and the authors [CKR + VP and VNP consisting of polynomials with bounded integer coefficients do have equationswith small algebraic circuits. Their work left open the possibility that these results could per-haps be extended to all of VP or VNP . The results in this paper show that assuming the hard-ness of Permanent, at least for
VNP , allowing polynomials with large coefficients does indeedincur a significant blow up in the circuit complexity of equations. * [email protected]. Department of Computer Science & Engineering, IIT Bombay, Mumbai, India. † [email protected]. School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai,India. Research supported by a fellowship of the DAE, Government of India. ‡ [email protected]. School of Technology and Computer Science, Tata Institute of Fundamental Research, Mum-bai, India. Research supported by Ramanujan Fellowship of DST, and by DAE, Government of India. § [email protected]. School of Technology and Computer Science, Tata Institute of Fundamental Research,Mumbai, India. Research supported by a fellowship of the DAE, Government of India. Introduction
In the context of proving lower bounds in complexity theory, many of the existing approaches forproving Boolean circuit lower bounds were unified by Razborov and Rudich under the
NaturalProofs framework [RR97] and they showed that, under standard cryptographic assumptions, anytechnique that fits into this framework cannot yield very strong lower bounds. In the last few yearsthere has been some work (e.g. [Gro15], [GKSS17, FSV18]) aimed at developing an analogue ofthe Natural Proofs framework for algebraic circuit lower bounds. A crucial notion in this contextis that of an equation for a class of polynomials which we now define.For a class C of polynomials, an equation for C is a family of nonzero polynomials such that itvanishes on the coefficient vector of polynomials in C . Informally, an algebraic natural proof for aclass C is a family of equations for C which can be computed by algebraic circuits of size and degreepolynomially bounded in their number of variables. Thus, a lower bound for C can be proved byexhibiting an explicit polynomial on which an equation for C does not vanish.Many of the known algebraic circuit lower bounds fit into this framework of algebraically nat-ural proofs as observed by several authors [AD08, Gro15, FSV18, GKSS17], thereby motivatingthe question of understanding whether techniques in this framework can yield strong algebraiccircuit lower bounds; in particular, whether such techniques are sufficient to separate VNP from VP . Thus, in this framework, the first step towards a lower bound for VP is to understand whether VP has a family of equations which itself is in VP , that is its degree and its algebraic circuit sizeare polynomially bounded in the number of the variables. The next step, of course, would be toshow the existence of a polynomial family in VNP which does not satisfy this family of equations.This work is motivated by the first step of this framework, that is the question of understand-ing whether natural and seemingly rich circuit classes like VP and VNP can have efficiently con-structible equations. We briefly discuss prior work on this problem, before describing our results.
In one of the first results on this problem, Forbes, Shpilka and Volk [FSV18] and Grochow, Kumar,Saks and Saraf [GKSS17] observe that the class VP does not have efficiently constructible equationsif we were to believe that there are hitting set generators for algebraic circuits with sufficiently suc-cinct descriptions. However, unlike the results of Razborov and Rudich [RR97], the plausibility ofthe pseudorandomness assumption in [FSV18, GKSS17] is not very well understood. The questionof understanding the complexity of equations for VP , or in general any natural class of algebraiccircuits, continues to remain open.In a recent work of Chatterjee and the authors [CKR + VP (in fact, even VNP ) consisting of polynomial families with bounded integercoefficients, then we indeed have efficiently computable equations. More formally, the main result Strictly speaking, these notions need us to work with families of polynomials, even though we sometimes drop theword family for ease of exposition.
2n [CKR +
20] was the following.
Theorem 1.1 ([CKR + . For every constant c > , there is a polynomial family { P N , c } ∈ VP Q suchthat for all large n and N = ( n + n c n ) , the following are true. • For every family { f n } ∈ VNP Q , where f n is an n-variate polynomial of degree at most n c andcoefficients in {−
1, 0, 1 } , we haveP N , c ( coeff ( f n )) = There exists a family { h n } of n-variate polynomials and degree at most n c with coefficients in {−
1, 0, 1 } such thatP N , c ( coeff ( h n )) = Here, coeff ( f ) denotes the coefficient vector of a polynomial f . Many of the natural and well studied polynomial families like the Determinant, the Perma-nent, Iterated Matrix Multiplication, etc., have this property of bounded coefficients, and in factthe above result even holds when the coefficients are as large as poly ( N ) . Thus, Theorem 1.1 couldbe interpreted as some evidence that perhaps we could still hope to prove lower bounds for oneof these polynomial families via proofs which are algebraically natural. Extending Theorem 1.1to obtain efficiently constructible equations for all of VP (or even for slightly weaker models likeformulas or constant depth algebraic circuits) is an extremely interesting open question. In fact,even a conditional resolution of this problem in either direction, be it showing that the boundedcoefficients condition in Theorem 1.1 can be removed, or showing that there are no such equations,would be extremely interesting and would provide much needed insight into whether or not there is a natural-proofs-like barrier for algebraic circuit lower bounds. In this paper, we show that assuming the Permanent is hard, the constraint of bounded coefficientsin Theorem 1.1 is necessary for efficient equations for
VNP . More formally, we show the followingtheorem.
Theorem 1.2 (Conditional Hardness of Equations for
VNP ) . Let ε > be a constant. Suppose, for anm large enough, we have that Perm m requires circuits of size m ε .Then, for n = m ε /4 , any d ≤ n and N = ( n + dn ) , we have that every nonzero polynomial P ( x , . . . , x N ) that vanishes on all coefficient vectors of polynomials in VNP C ( n , d ) has size ( P ) = N ω ( ) . For a field F , VP F denotes the class VP where the coefficients of the polynomials are from the field F . Similarly, VNP F denotes the class VNP where the coefficients of the polynomials are from the field F . emark. Our proof of the above theorem easily extends to any field of characteristic zero. We shall justwork with complexes for better readability. ♦ Extending the result in Theorem 1.2 to hardness of equations for VP , even under the assump-tion that Permanent is sufficiently hard, is an extremely interesting open question. Such an exten-sion would answer the main question investigated in [FSV18, GKSS17] and show a natural-proofs-like barrier for a fairly general family of lower bound proof techniques in algebraic complexity.Our proof of Theorem 1.2 however crucially relies on some of the properties of VNP and does notappear to extend to VP .Although the proof of the above theorem is quite elementary, the main message (in our opin-ion) is that we do not have compelling evidence to rule out, or accept, the efficacy of algebraicnatural proofs towards proving strong lower bounds for rich classes of algebraic circuits. As was observed in [FSV18, GKSS17], a lower bound for equations for a class of polynomials isequivalent to showing the existence of succinctly describable hitting sets for this class. For ourproof we show that, assuming that the permanent is sufficiently hard, the coefficient vectors ofpolynomials in
VNP form a hitting set for the class VP . The connection between hardness and ran-domness in algebraic complexity is well known via a result of Kabanets and Impagliazzo [KI04],and we use this connection, along with some additional ideas for our proof. We briefly describe ahigh level sketch of our proof in a bit more detail now.Kabanets and Impagliazzo [KI04] showed that using any explicit polynomial family { f n } thatis sufficiently hard, one can construct a hitting set generator for VP , that is, we can construct apolynomial map Gen f : F k → F t that “fools” any small algebraic circuit C on t variables in thesense that C ( y , y , . . . , y t ) is nonzero if and only if the k -variate polynomial C ◦ Gen f is nonzero. Ina typical invocation of this result, the parameter k is much smaller than t (typically k = poly log t ).Thus, this gives a reduction from the question of polynomial identity testing for t -variate polyno-mials to polynomial identity testing for k -variate polynomials. Another related way of interpretingthis connection is that if { f n } is sufficiently hard then Gen f is a polynomial map whose image doesnot have an equation with small circuit size. Thus, assuming the hardness of the Permanent, thisimmediately gives us a polynomial map (with appropriate parameters) such that its image doesnot have an efficiently constructible equation.For the proof of Theorem 1.2, we show that the points in the image of the map Gen
Perm , canbe viewed as the coefficient vectors of polynomials in
VNP , or, equivalently in the terminology in[FSV18, GKSS17], that the Kabanets-Impagliazzo hitting set generator is
VNP -succinct. To this end,we work with a specific instantiation of the construction of the Kabanets-Impagliazzo generatorwhere the underlying construction of combinatorial designs is based on Reed-Solomon codes.Although this is perhaps the most well known construction of combinatorial designs, there are Or rather, the results of [CKR +
20] and the above theorem seem to provide some evidence for both sides!
VNP and not VP . Details of the proof.
Let us assume that for some constant ε > m ∈ N , Perm m requires circuits of size 2 m ε . Kabanets and Impagliazzo [KI04] showed that, for every combinato-rial design D (a collection of subsets of a universe with small pairwise intersection) of appropriateparameters, the map Gen
Perm ( z ) = ( Perm ( z S ) : S ∈ D ) where z S denotes the variables of in z restricted to the indices in S , is a hitting set generator forcircuits of size 2 o ( m ε ) . Our main goal is to construct a polynomial F ( y , z ) in VNP such that F ( y , z ) = ∑ S ∈D mon S ( y ) · Perm ( z S ) (1.3)By choosing parameters carefully, this would immediately imply that any equation on N -variables,for N = ( n + dd ) , that vanishes on the coefficient vector of polynomials in VNP ( n , d ) (which are n -variate polynomials in VNP of degree at most d ) requires size super-polynomial in N .To show that the polynomial F ( y , z ) in Equation 1.3 is in VNP , we use a specific combinatorialdesign. For the combinatorial design D obtained via Reed-Solomon codes, every set in the designcan be interpreted as a univariate polynomial g of appropriate degree over a finite field. Thedegree of g (say δ ) and size of the finite field (say p ) are related to the parameters of the design D .Now, F ( y , z ) = ∑ g ∈ F p [ v ] deg ( g ) ≤ δ δ ∏ i = y g i i ! · Perm ( z S ( g ) ) , (1.4)where ( g , . . . , g δ ) is the coefficient vector of the univariate polynomial g . Expressing F ( y , z ) inEquation 1.4 as a polynomial in VNP requires us to implement the product (cid:18) δ ∏ i = y g i i (cid:19) as a polyno-mial when given the binary representation of coefficients g , . . . , g δ via a binary vector t of appro-priate length (say r ). This is done via the polynomial Mon ( t , y ) in Section 3.1 in a straightforwardmanner. Furthermore, we want to algebraically implement the selection z S for a set S in the com-binatorial design when given the polynomial g corresponding to S . This is implemented via the To be more precise, we should work with this condition for “infinitely often” m ∈ N and obtain that VNP does nothave efficient equations infinitely often. We avoid this technicality for the sake of simplicity and the proof continues tohold for the more precise version with suitable additional care. ( t , z ) in Section 3.2. Finally, we have F ( y , z ) = ∑ t ∈{ } r Mon ( t , y ) · Perm ( RS-Design ( t , z )) which is clearly in VNP as Perm p is in VNP and polynomials Mon ( t , y ) and RS-Design ( t , z ) areefficiently computable. We refer the reader to Section 3 for complete details. Related results.
The concept of algebraically natural proofs was first studied in the works ofForbes, Shpilka and Volk [FSV18] and Grochow, Kumar, Saks and Saraf [GKSS17] who showedthat constructing efficient equations for a class directly contradicts a corresponding succinct deran-domization of the polynomial identity testing problem. In fact, Forbes, Shpilka and Volk [FSV18]unconditionally ruled out equations for depth-three multilinear formulas computable by certainstructured classes of algebraic circuits using this connection. However, this does not imply any-thing about complexity of equations for general classes of algebraic circuits such as VP and VNP .In the context of proving algebraic circuit lower bounds, Efremenko, Garg, Oliveira and Wigder-son [EGOW18] and Garg, Makam, Oliveira and Wigderson [GMOW19] explore limitations ofproving algebraic circuit lower bounds via rank based methods. However, these results are notdirectly concerned with the complexity of equations for circuit classes.Recently, Bläser, Ikenmeyer, Jindal and Lysikov [BIJL18] studied the complexity of equationsin a slightly different context. They studied a problem called “matrix completion rank”, a mea-sure for tensors that is NP -hard to compute. Assuming coNP * ∃ BPP , they construct an explicittensor of large (border) completion rank such that any efficient equation for the class of tensors ofsmall completion rank must necessarily also vanish on this tensor of large completion rank. Thatis, efficient equations cannot certify that this specific tensor has large (border) completion rank.Subsequently, this result was generalized to min-rank or slice-rank [BIL + every equation for this variety requires large complexity.In the context of equations for varieties in algebraic complexity, Kumar and Volk [KV20]proved polynomial degree bounds on the equations of the Zariski closure of the set of non-rigidmatrices as well as small linear circuits over all large enough fields. • We use [ n ] to denote the set {
1, . . . , n } and J n K to denote the set {
0, 1, . . . , n } . We also use N ≥ to denote the set of non-negative integers.6 We use boldface letters such as x , y to denote tuples, typically of variables. When necessary,we adorn them with a subscript such as y [ n ] to denote the length of the tuple.• We also use x e to denote the monomial ∏ x e i i . We write x ≤ d for the set of all monomials ofdegree at most d in x , and F [ x ] ≤ d for the set of polynomials in x over the field F of degree atmost d .• As usual, we identify the elements of F p with {
0, 1, . . . , p − } and think of J n K as a subset of F p in the natural way for any n < p . Circuit classesDefinition 2.1 (Algebraic circuits) . An algebraic circuit is specified by a directed acyclic graph, withleaves (indegree zero; also called inputs ) labelled by field constants or variables, and internal nodes labelledby + or × . The nodes with outdegree zero are called the outputs of the circuit. Computation proceeds inthe natural way, where inductively each + gate computes the sum of its children and each × gate computesthe product of its children.The size of the circuit is defined as the number of nodes in the underlying graph. ♦ Definition 2.2 ( VP and VNP ) . A family of polynomials { f n } , where f n is n-variate, is said to be in VP if deg ( f n ) and the algebraic circuit complexity of f n is bounded by a polynomial function of n. That is, thereis a constant c ≥ such that for all large enough n we have deg ( f n ) , size ( f n ) ≤ n c .A family of polynomials { f n } is said to be in VNP if there is a family n g n ( x [ n ] , y [ m ] ) o ∈ VP such thatm is bounded by a polynomial function of n andf n ( x ) = ∑ y ∈{ } m g n ( x , y ) . ♦ For some n , d ∈ N , let C n , d be a class of n -variate polynomials of total degree at most d . That is, C n , d ⊆ F [ x ] ≤ d . Similarly, we will use VP ( n , d ) and VNP ( n , d ) to denote the intersection of VP and VNP respectively, with F [ x [ n ] ] ≤ d . Equations and succinct hitting setsDefinition 2.3 (Equations for a class) . For N = ( n + dn ) , a nonzero polynomial P N ( Z ) is called an equa-tion for C n , d if for all f ( x ) ∈ C n , d , we have that P N ( coeff ( f )) = , where coeff ( f ) is the coefficient vectorof f . ♦ Alternatively, we also say that a polynomial P ( Z ) vanishes on the coefficient vectors of polyno-mials in class C if P N ( coeff ( f )) = f ∈ C . Definition 2.4 (Hitting Set Generator (HSG)) . A polynomial map G : F ℓ → F n given by G ( z , . . . , z ℓ ) =( g ( z ) , . . . , g n ( z )) is said to be a hitting set generator (HSG) for a class C ⊆ F [ x ] of polynomials if forall nonzero P ∈ C , P ◦ G = P ( g , . . . , g n ) . ♦
7e review the definition of succinct hitting sets introduced [GKSS17, FSV18].
Definition 2.5 (Succinct Hitting Sets for a class of polynomials [GKSS17, FSV18]) . For N = ( n + dn ) ,we say that a class of N-variate polynomials D N has C n , d -succinct hitting sets if for all nonzero P ( Z ) ∈D N , there exists some f ∈ C n , d such that P N ( coeff ( f )) = . ♦ Hardness to randomness connection
For our proofs, we will need the following notion of combinatorial designs, which is a collectionof subsets of a universe with small pairwise intersection.
Definition 2.6 (Combinatorial designs) . A family of sets { S , . . . , S N } ⊆ [ ℓ ] is said to be an ( ℓ , m , n ) -design if • | S i | = m for each i ∈ [ n ] • | S i ∩ S j | < n for any i = j. ♦ Kabanets and Impagliazzo [KI04] obtain hitting set generators from polynomials that are hardto compute for algebraic circuits. The following lemma is crucial to the proof of our main theorem.
Lemma 2.7 (HSG from Hardness [KI04]) . Let { S , . . . , S N } be an ( ℓ , m , n ) -design and f ( x m ) be anm-variate, individual degree d polynomial that requires circuits of size s. Then for fresh variables y ℓ , thepolynomial map KI-gen ( N , ℓ , m , n ) ( f ) : F ℓ → F n given by ( f ( y S ) , . . . , f ( y S N )) (2.8) is a hitting set generator for all circuits of size at most (cid:16) s N ( d + ) n (cid:17) . Notation
1. For a vector t = ( t , . . . , t r ) , we will use the short-hand t ( a ) i , j to denote the variable t ( i · a + j + ) .This would be convenient when we consider the coordinates of t as blocks of length a .2. For integers a , p , we shall use Mod ( a , p ) to denote the unique integer a p ∈ [ p − ] suchthat a p = a mod p .As mentioned in the overview, the strategy is to convert the hitting set generator given in (2.8)into a succinct hitting set generator. Therefore, we would like to associate the coordinates of (2.8)into coefficients of a suitable polynomial. That is, we would like to build a polynomial in VNP ofthe form g ( y , . . . , y ℓ , z , . . . , z t ) = ∑ m ∈ y ≤ d m · f ( z S m ) m ∈ y ≤ d suitably indexing into the sets of the combinatorial design. Theabove expression already resembles a VNP -definition and with a little care this can be made ef-fective. We will first show that the different components of the above expression can be madesuccinct using the following constructions.
For n , r ∈ N , let a = ⌊ r / n ⌋ , and define Mon r , n ( t , y ) as follows.Mon r , n ( t , . . . , t r , y , . . . , y n ) = n − ∏ i = a − ∏ j = (cid:16) t ( a ) i , j y j i + + ( − t ( a ) i , j ) (cid:17) The following observation is now immediate from the definition above.
Observation 3.1.
For any ( e , . . . , e n ) ∈ J d K n , we have Mon r , n ( Bin ( e ) , . . . , Bin ( e n ) , y , . . . , y n ) = y e · · · y e n n , where Bin ( e ) is the tuple corresponding to the binary representation of e, and r = n · ⌈ log d ⌉ . Furthermore,the polynomial Mon r , n is computable by an algebraic circuit of size poly ( n , r ) . Next, we need to effectively compute the hard polynomial f on sets of variables in a combinato-rial design, indexed by the respective monomials. We will need to simulate some computationsmodulo a fixed prime p . The following claim will be helpful for that purpose. Claim 3.2.
For any i , b , p ∈ N ≥ with i ≤ p, there exists a unique univariate polynomial Q i , b , p ( v ) ∈ Q [ v ] of degree at most b such thatQ i , b , p ( a ) = if ≤ a < b and a ≡ i ( mod p ) ,0 if ≤ a < b and a i ( mod p ) . Proof.
We can define a unique univariate polynomial Q i , b , p ( v ) satisfying the conditions of the claimvia interpolation to make a unique univariate polynomial take a value of 0 or 1 according to theconditions of the claim. Since, there are b conditions, there always exists such a polynomial ofdegree at most b .For any n , b , p ∈ N ≥ with n ≥ p , defineSel n , b , p ( u , . . . , u n , v ) , n ∑ i = u i · Q i , b , p ( v ) . 9 bservation 3.3. For any n , b , p ∈ N ≥ with n ≥ p, for any ≤ a < b, we have that Sel n , b , p ( u , . . . , u n , a ) = u Mod ( a , p ) = u a mod p The degree of
Sel n , b , p is at most ( b + ) and can be computed by an algebraic circuit of size poly ( b ) .Proof. From the definition of the univariate polynomial Q i , b , p ( v ) of degree b in Claim 3.2, Q i , b , p ( a ) outputs 1 if and only if i = a mod p . Hence, Sel n , b , p ( u , . . . , u n , a ) is u a mod p and is of degree atmost ( b + ) .And finally, we choose a specific combinatorial design to instantiate Lemma 2.7 with. For any prime p and any choice of a ≤ p , the following is an explicit construction of a ( p , p , a ) -combinatorial design of size p a , defined as follows:With the universe U = F p × F p , for every univariate polynomial g ( t ) ∈ F p [ t ] of degreeless than a , we add the set S g = (cid:8) ( i , g ( i )) : i ∈ F p (cid:9) to the collection.Since any two distinct univariate polynomials of degree less than a can agree on at most a points,it follows that the above is indeed a ( p , p , a ) -design.The advantage of this specific construction is that it can be made succinct as follows. For r = a · ⌊ log p ⌋ , let t , . . . , t r be variables taking values in {
0, 1 } . The values assigned to t -variablescan be interpreted as a univariate over F p of degree < a by considering t ∈ {
0, 1 } r as a matrixwith a rows and ⌊ log p ⌋ columns each . The binary vector in each row represents an element in F p . We illustrate this with an example. t = −→ ∼ = g ( v ) For p = a = g ( v ) = + v + v + v + v ∈ F [ v ] , t is a 5 × g ( v ) .Let z denote the p variables n z , . . . , z p o , put in into a p × p matrix. Let S be a set in theReed-Solomon based ( p , p , a ) -combinatorial design. We want to implement the selection z S alge-braically. In the following, we design a vector of polynomials that outputs the vector of variables Working with ⌊ log p ⌋ bits (as opposed to ⌈ log p ⌉ ) makes the proofs much simpler, and does not affect the size ofthe design by much. z ( p ) g ( ) mod p , . . . , z ( p ) p − g ( p − ) mod p (cid:17) . Note that as mentioned above the polynomial g can be speci-fied via variables t , . . . , t r . That is,RS-Design p , a ( t , . . . , t r , z , . . . , z p ) ∈ ( F [ t , z ]) p , for r = a · ⌊ log p ⌋ ,RS-Design p , a ( t , . . . , t r , z , . . . , z p ) i + = Sel p , p , p (cid:16) z ( p ) i ,0 , . . . , z ( p ) i , p − , R i , a , p ( t ) (cid:17) , for each i ∈ F p ,where R i , a , p ( t ) = a − ∑ j = " ℓ p − ∑ k = t ( ℓ p ) j , k · k ! · Mod ( i j , p ) ,with ℓ p = ⌊ log p ⌋ . Observation 3.4.
For any prime p, a ≤ p, and t ∈ {
0, 1 } r for r = a · ⌊ log p ⌋ , we have RS-Design p , a ( t , z ) = (cid:16) z i , g ( i ) : i ∈ F p (cid:17) , where g ( v ) ∈ F p [ v ] is the univariate whose coefficient vector is represented by the bit-vector t . Furthermore,the polynomial RS-Design p , a is computable by an algebraic circuit of size poly ( p ) .Proof. Fix some t ∈ {
0, 1 } r . From the definition of R i , a , p ( t ) , it is clear that R i , a , p ( t ) returns aninteger α such that g ( i ) = α mod p where t encodes the coefficients of the polynomial g ( t ) inbinary. Furthermore, since Mod ( i j , p ) is the unique integer c ∈ [ p − ] with c = i j mod p , it alsofollows that R i , a , p ( t ) is an integer in the range [ p ] . Hence,Sel p , p , p (cid:16) z ( p ) i ,0 , . . . , z ( p ) i , p − , R i , a , p ( t ) (cid:17) = z i , g ( i ) as claimed. VNP -Succinct-KI generator
We are now ready to show the
VNP -succinctness of the Kabanets-Impagliazzo hitting set generatorwhen using a hard polynomial from
VNP and a Reed-Solomon based combinatorial design.For a prime p and for the largest number m such that m ≤ p , we will use Perm [ p ] ∈ F [ y [ p ] ] todenote Perm m applied to the first m variables of y .We now define the polynomial F n , a , p ( y [ n ] , z [ p ] ) as follows. F n , a , p ( y , . . . , y n , z , . . . , z p ) = ∑ t ∈{ } r Mon r , n ( t , y ) · Perm [ p ] ( RS-Design p , a ( t , z )) (3.5)where r = a · ⌊ log p ⌋ It is evident from the above definition that the polynomial F n , a , p ( y , z ) is in VNP for any p that ispoly ( n ) , when seen as a polynomial in y -variables with coefficients from C [ z ] .11rom the construction, we have that F n , a , p ( y , . . . , y n , z , . . . z p ) = ∑ e y e · Perm [ p ] ( z S e ) ,where { S e } is an appropriate ordering of the Reed-Solomon based ( p , p , a ) -combinatorial designof size p a , described in Section 3.3. We are now ready to show that if the Permanent polynomial is exponentially hard, then any poly-nomial P that vanishes on the coefficient vectors of all polynomials in the class VNP requiressuper-polynomial size to compute it.
Theorem 1.2 (Conditional Hardness of Equations for
VNP ) . Let ε > be a constant. Suppose, for anm large enough, we have that Perm m requires circuits of size m ε .Then, for n = m ε /4 , any d ≤ n and N = ( n + dn ) , we have that every nonzero polynomial P ( x , . . . , x N ) that vanishes on all coefficient vectors of polynomials in VNP C ( n , d ) has size ( P ) = N ω ( ) .Proof. Let p be the smallest prime larger than m ; we know that p ≤ m . We will again usePerm [ p ] ∈ F [ y [ p ] ] to denote Perm m acting on the first m variables of y . Therefore, if Perm m re-quires size 2 m ε then so does Perm [ p ] .Consider the polynomial F n , n , p ( y [ n ] , z [ p ] ) ∈ VNP defined in (3.5), which we interpret as a polyno-mial in y with coefficients in C [ z ] . The individual degree in y is at least d , and at most p .Let F ≤ dn , n , p ( y [ n ] , z [ p ] ) denote the polynomial obtained from F n , n , p by discarding all terms whose to-tal degree in y exceeds d . By standard homogenisation arguments, it follows that F ≤ dn , n , p ∈ VNP aswell. Therefore, F ≤ dn , n , p ( y , z ) = ∑ deg ( y e ) ≤ d y e · Perm [ p ] ( z S e ) ,where S e , for various e , is an appropriate indexing into a ( p , p , n ) -combinatorial design of size N . Since the individual degree in y of F n , n , p was at least d , every coefficient of F ≤ dn , n , p is Perm [ p ] ( z S ) for some S in the combinatorial design. In other words, the coefficient vector of F ≤ dn , n , p is preciselyKI-gen N , p , p , n ( Perm [ p ] ) .Suppose P ( x , . . . , x N ) is a nonzero equation for VNP ( n , d ) , then in particular it should bezero on the coefficient vector of F ≤ dn , n , p ( y , a ) ∈ VNP for any a ∈ C p . By the Polynomial IdentityLemma [Ore22, DL78, Zip79, Sch80], this implies that P must be zero on the coefficient vector of F ≤ dn , n , p ( y , z ) ∈ ( C [ z ])[ y ] , where coefficients are formal polynomials in C [ z ] . Since the coefficientvector of F ≤ dn , n , p ( y , z ) is just KI-gen N , p , p , n ( Perm [ p ] ) , the contrapositive of Lemma 2.7 gives thatsize ( P ) > size ( Perm [ p ] ) N · n > size ( Perm m ) N · n ⇒ size ( P ) > m ε N · n Since N = ( n + dn ) ≤ n ≤ o ( m ε ) , it follows that size ( P ) = N ω ( ) . Concluding that
VNP has no efficient equations
Note that for a family { P N } to be a family ofequations for a class C , we want that for all large enough n , the corresponding polynomial P N shouldvanish on the coefficient vectors of all n -variate polynomials in C . This condition is particularlyimportant if we want to use equations for C to prove lower bounds against it, since a family ofpolynomials { f n } is said to be computable in size s ( n ) if size ( f n ) ≤ s ( n ) for all large enough n .Theorem 1.2 shows that, for m large enough, if there is a constant ε > ( Perm m ) ≥ m ε , then for n = m ε /4 and any d ≤ n , the coefficient vectors of polynomials in VNP ( n , d ) form ahitting set for all N -variate polynomials (where N = ( n + dd ) ) of degree poly ( N ) that are computableby circuits of size poly ( N ) . Now suppose the Permanent family is 2 m ε -hard for a constant ε > m is 2 m ε -hard for infinitely many m ∈ N . Then using Theorem 1.2, we canconclude that for any family { P N } ∈ VP , we must have for infinitely many n that P N ( coeff ( f n )) = f n ∈ VNP , which then shows that { P N } is not a family of equations for VNP . In the context of proving circuit lower bounds, and in relation to the notion of algebraically nat-ural proofs , an interesting question that emerges from the recent work of Chatterjee and the au-thors [CKR +
20] (stated in Theorem 1.1) is whether the condition of “small coefficients” is neces-sary for efficiently constructible equations to exist, especially for the class VP . While this questionremains open for VP , our result shows that this additional restriction on the coefficients is essen-tially vital for the existence of efficiently constructible equations for the class VNP , and thereforeprovides strong evidence against the existence of efficient equations for
VNP .In light of Theorem 1.1 and Theorem 1.2 for
VNP , one could make a case that equations for VP might also incur a super-polynomial blow up, without the restriction on coefficients. On theother hand, it could also be argued that an analogue of Theorem 1.2 may not be true for VP , sinceour proof crucially uses the fact that VNP is “closed under exponential sums”. In fact, our proofessentially algebraises the intuition that coefficient vectors of polynomials in
VNP “look random”to a polynomial in VP , provided that VNP was exponentially more powerful than VP .Thus, along with the previously known results on efficient equations for polynomials in VP with bounded coefficients, our result highlights that the existence of such equations for VP ingeneral continues to remain an intriguing mystery.13 pen Problems We now conclude with some possible directions for extending our results.• Perhaps the most interesting question here is to prove an analogue of Theorem 1.2 for equa-tions for VP . This would provide concrete evidence for the possibility that we cannot hopeto prove very strong lower bounds for algebraic circuits using proofs which proceed viaefficiently constructible equations, from a fairly standard complexity theoretic assumption.• At the moment, we cannot rule out the possibility of there being efficient equations for VP in general; it may be possible that the bounded coefficients condition in Theorem 1.1 can beremoved. In particular, the question of proving upper bounds on the complexity of equa-tions for VP is also extremely interesting, even if one proves such upper bounds under somereasonable complexity theoretic assumptions. A first step perhaps would be to prove upperbounds on the complexity of potentially simpler models, like formulas, algebraic branchingprograms or constant depth circuits. From the works of Forbes, Shpilka and Volk [FSV18],we know that such equations for structured subclasses of VP (like depth-3 multilinear cir-cuits) cannot be too simple (such as sparse polynomials, depth-3 powering circuits, etc.). Canwe prove a non-trivial upper bound for equations for these structured classes within VP ?• Another question of interest would be to understand if the hardness assumption in Theo-rem 1.2 can be weakened further. For instance, is it true that VNP does not have efficientlyconstructible equations if VP = VNP , or if Perm n requires circuits of size n poly log ( n ) ? Thecurrent proof seems to need an exponential lower bound for the Permanent. Acknowledgements
We thank an anonymous reviewer of FOCS 2020, and Joshua Grochow, whose questions pointedus in the direction of this result. We also thank Prerona Chatterjee and Ben Lee Volk for helpfuldiscussions at various stages of this work.
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