Monotone Classes Beyond VNP

In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran and Perifel 2009, Malod 2011, Mahajan and Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotone VPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat's definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not. Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrube\v{s} and Yehudayoff (2021). In that context, we show that transparent polynomials of large sparsity are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.


Introduction
The aim of algebraic complexity is to classify polynomials in terms of how hard it is to compute them, and the most standard model for computing polynomials is that of an algebraic circuit.
An algebraic circuit is a rooted, directed acyclic graph where the leaves are labelled with variables or field constants and internal nodes are labelled with addition (+) or multiplication (×).Every node therefore naturally computes a polynomial and the polynomial computed by the root is said to be the polynomial computed by the circuit.A formal definition can be dependent on coefficients (for example the determinant (Det n ) is efficiently computable by algebraic circuits and this is expected to not be the case for Perm n , even though they have the same set of monomials), the τ -conjecture for Newton polygons is believed to be false.The approach suggested by Hrubeš & Yehudayoff [11] used shadows of Newton polytopes as a means to move from the multivariate setting to the bivariate setting, and use polynomials like determinant (Det n ) to refute the conjecture.The difficulty in this strategy however, is to find a polynomial in VP that exhibits high shadow complexity (maximum number of vertices in its projection), since even when a candidate polynomial is fixed, say Det n , it is not easy to design a suitable bivariate projection.
As a means to tackle this issue, Hrubeš & Yehudayoff introduced the notion of transparent polynomials -polynomials that can be projected to bivariates in such a way that all of their monomials become vertices of the resulting Newton polygon.Further, they also gave examples of polynomials with exponentially large sets of monomials that are provably transparent.Therefore, a proof of any one of these polynomials being in VP would directly refute the τ -conjecture for Newton polytopes.
Even though Hrubeš and Yehudayoff [11] were not able to actually use this approach to refute the conjecture, they used the notions of shadows & transparency to come up with yet another method for proving lower bounds against monotone algebraic circuits.They showed that the monotone circuit complexity of a polynomial is lower bounded by its shadow complexity when the polynomial is transparent.
As a corollary, they present an n-variate polynomial such that any monotone algebraic circuit computing it must have size Ω(2 n/3 ).

Our Contribution
Here we state our contributions informally; the formal statements can be found in Section 3. Throughout this work we assume that the underlying field is either the field of real numbers or the field of rational numbers.The goal of this work is two-fold.
The first goal is to understand how restrictive the notion of transparency is.Our search begins with an observation by Yehudayoff [26], that any lower bound against mVP depending solely on the support of the hard polynomial, automatically "lifts" to mVNP with the same parameters 1 .Since transparency is a property solely of the Newton polytope, and hence of the support of the polynomial, the above observation shows that any transparent polynomial that is non-sparse (has super-polynomially large support) is hard to compute even for mVNP.However, we suspect that transparency is an even stronger property.Therefore, a natural question for us is whether there are even larger classes of monotone polynomials that do not contain non-sparse, transparent polynomials.This brings us to the second goal of this work -studying monotone models of computation that can possibly compute polynomials outside even mVNP.Classes larger than VNP had not been defined in the monotone world prior to this work.We therefore turn to the literature in the non-monotone setting.Here, VPSPACE is a well studied class [19,14,18,17] 1 [26]: "If a monotone circuit-size lower bound for q(x) holds also for all polynomials that are equivalent to q(x) then the same lower bound also holds for every mVNP circuit computing q(x)."Here mVNP circuit denotes z∈{0,1} m C(x, z1, . . ., zm) where m = poly(n) and C(x, z) is a monotone algebraic circuit.

F S T T C S 2 0 2 3 11:4
Monotone Classes Beyond VNP that is believed to be strictly larger than VNP.Interestingly there are multiple definitions of VPSPACE, resulting from varied motivations, all of which are known to be essentially equivalent [18,17].We study the natural monotone analogues of these definitions and show that unlike the non-monotone setting, the powers of the different resulting models vary greatly.This allows us to then analyse if the technique of Hrubeš & Yehudayoff also works against monotone classes that are possibly larger than mVNP.
The following figure succinctly describes some of our main results.
mVP msuccABP mVNP mVP quant mVP sum,prod mVP proj [26] Theorem 3.12 Theorem 3.2 In Figure 1, the node labels refer to the following classes of polynomial families that have degree-poly(n) and poly(n)-complexity under the corresponding models.
msuccABP -monotone succinct ABPs (Definition 3.1), mVP quant -quantified monotone circuits (Definition 3.4), mVP sum,prod -monotone circuits with summation and production gates (Definition 3.8), mVP proj -monotone circuits with projection gates (Definition 3.11).The orange, rectangular nodes denote the classes in which sparsity of transparent polynomials in it is bounded by a constant factor of the size of the smallest M computing it, if M is the computational model corresponding to the class (Theorem 3.10).
An interesting point to note here is that there is an exponential separation between mVP quant and mVP proj , which means that at least one of the inclusions: mVP quant to mVP sum,prod , and mVP sum,prod to mVP proj is strict with an exponential separation.
Additionally, we show the following two statements about mVP quant .mVP quant = mVNP if and only if homogeneous components of polynomials in mVP quant are contained in mVP quant (Corollary 3.6).In particular, we show that homogeneous polynomials in mVP quant are also in mVNP (Theorem 3.5).mVP quant = mVP sum,prod if and only if quantified monotone circuits are closed under compositions (Observation 3.9).
Finally, we also show that the homogeneous components of polynomials in mVP proj are in mVP proj (Theorem 3.13).This property, along with the fact that Perm n ∈ mVP proj (Theorem 6.1), is the reason we propose "monotone VPSPACE" (mVPSPACE) to be defined as the class of polynomial families that are efficiently computable by monotone circuits with projection gates (without any restriction on degree).

Organization of the paper
We begin in Section 2 with formal definitions for all the models of computation that we will be using.Next, we define the monotone analogues of the various definitions of VPSPACE, and outline our results about them in Section 3. The proofs of our results are discussed in Section 4, Section 5 and Section 6.We conclude with Section 7, where we discuss some important open threads from our work.

Preliminaries
We shall use the following notation for the rest of the paper.We use the standard shorthand [n] = {1, 2, . . ., n}.
We use boldface letters like x, z, e to denote tuples/sets of variables or constants, individual members are expressed using indexed version of the usual symbols: e = (e 1 , e 2 , . . ., e n ), x = {x 1 , . . ., x n }.We also use |y| to denote the size/length of a vector y.
For vectors x and e of the same length n, we use the shorthand x e to denote the monomial For a polynomial f (x), we denote by deg(f ) the degree of f in x.For a polynomial f (x) and a monomial m = x e , we refer to the coefficient of m in f by coeff f (m).The support supp(f ) of a polynomial f is given by {m : coeff f (m) ̸ = 0}, and the sparsity of a polynomial is the size of its support, |supp(f )|.For any polynomial f (x) and any k ≤ deg(f ), we denote by hom k (f ) the k-th homogeneous degree component of f in terms of x.That is, if The permanent of an n × n symbolic matrix shall be denoted by Perm n and is defined as , where S n is the set of all permutations of [n].We use {f n } to denote families of polynomials indexed by N. All complexity classes are defined in terms of asymptotic properties of "polynomials" and are technically sets of such polynomial families.Sometimes however, this technicality is ignored for the sake of brevity, especially when the analogous statement about polynomial families is obvious.
▶ Definition 2.1 (Algebraic circuits).An algebraic circuit is a directed acyclic graph with leaves (nodes with in-degree zero) labelled by formal variables and constants from the field, and other nodes labelled by addition (+) and multiplication (×) have in-degree 2.
The leaves compute their labels, and every other node computes the operation it is labelled by, on the polynomials along its incoming edges.A node of out-degree zero is called the output of the circuit, and the circuit is said to compute the polynomial computed by the output gate.
In case there are multiple output gates, the circuit is said to be multi-output, and computes a set of polynomials.
The size of a circuit, C, denoted by size(C), is the number of nodes in the graph.An algebraic circuit over Q or R is said to be monotone, if all the constants appearing in it are non-negative.▶ Definition 2.2 (Algebraic Branching Programs (ABPs)).An algebraic branching program is specified by a layered graph where each edge is labelled by an affine linear form and the first and the last layer have one vertex each, called the "source" and the "sink" vertex respectively.The polynomial computed by an ABP is equal to the sum of the weights of all paths from the start vertex to the end vertex in the ABP, where the weight of a path is equal to the product of the labels of all the edges on it.
The width of a layer in an ABP is the number of vertices in it and the width of an ABP is the width of the layer that has the maximum number of vertices in it.
The size of an ABP is the number of vertices in it.

Basic monotone classes
An expression of the above form is alternatively called an exponential sum computing f n .

Various definitions of VPSPACE
Koiran & Perifel [14,15] were the first to define VPSPACE as the class of polynomials (of degree that is potentially exponential in the number of underlying variables) whose coefficients can be computed in PSPACE/ poly, and VPSPACE b to be the polynomials in VPSPACE that have degree bounded by a polynomial in the number of underlying variables.
Poizat defined algebraic circuits with projection gates and then defined VPSPACE to be the class of polynomial families that are efficiently computable by this model.Poizat showed2 that this definition is equivalent to that of Koiran & Perifel.

▶ Definition 2.6 (Algebraic circuits with projection gates [19]
).An algebraic circuit with projection gates is an algebraic circuit (Definition 2.1) in which the internal nodes can also be projection gates (Definition 2.5), in addition to + or ×.
The size of an algebraic circuit with projection gates is the number of nodes in the underlying graph.
Adding to Poizat's work, Malod [18] characterized VPSPACE using exponentially large algebraic branching programs (ABPs) that are succinct.Malod's work defines the complexity of an ABP as the size of the smallest algebraic circuit that encodes its graph -outputs the corresponding edge label when given the two endpoints as input.An n-variate ABP is then said to be succinct, if its complexity is poly(n).
▶ Definition 2.7 (Succinct ABPs [18]).A succinct ABP over the n variables x = {x 1 , . . ., x n } is a triple (B, s, t) with |s| = |t| = r, where s is the label of the source vertex, and t is the label of the sink(target) vertex.B(u, v, x) is an algebraic circuit that describes a directed acyclic graph G B on the vertex set {0, 1} r in the following way.For any two vertices a, b ∈ {0, 1} r , the output is the label of the edge from a to b in the ABP.The polynomial computed by the ABP is the sum of polynomials computed along all s to t paths in G B ; where each path computes the product of the labels of the constituent edges.
The size of the circuit B is said to be the complexity of the succinct ABP.The number of vertices 2 r is the size of the succinct ABP, and the length of the longest s to t path is called the length of the succinct ABP.
In the same work [18], Malod alternatively characterized VPSPACE using an interesting algebraic model that resembles (totally) quantified boolean formulas that are known to characterize PSPACE.This model, which we refer to as "quantified algebraic circuits", is defined using special types of projection gates called summation and production gates.
▶ Definition 2.8 (Summation and Production gates [18]).Summation and production gates are unary gates that are labelled by a variable z, and are denoted by sum z and prod z respectively.A summation gate returns the sum of the (z = 0) and (z = 1) evaluations of its input, and a production gate returns the product of those evaluations.That is, We sometimes use sum {z1,...,z k } to refer to the nested expression sum z1 • • • sum z k (similarly for prod); it can be checked that the order does not matter here.
A quantified algebraic circuit has the form where each Q i is a summation or a production, and C(x, z) is a usual algebraic circuit.
▶ Definition 2.9 (Quantified Algebraic Circuits [18]).A quantified algebraic circuit is an algebraic circuit that has the form, and C is an algebraic circuit.The size of such a quantified algebraic circuit is m + size(C).
Finally, Mahajan & Rao [17] defined algebraic analogues of small space computation (e.g.L, NL) using the notion of width of an algebraic circuit.They use their definitions to import some relationships known in the boolean world to the algebraic world (e.g, they show VL ⊆ VP).They further show that their definition of uniform polynomially-bounded-space computation coincides with that of uniform-VPSPACE as defined by Koiran & Perifel [14].
We now narrow our focus to the definitions due to Poizat [19] and Malod [18].We choose these definitions because they are algebraic in nature, and have fairly natural monotone analogues.We elaborate a bit more about this decision in Appendix B.
▶ Remark.It should be noted that all the above-mentioned definitions of VPSPACE allow for the polynomial families to have large degree -as high as exp(poly(n)).The main focus of our work, however, is to compare the monotone analogues of these models with mVP and mVNP.Since the latter classes only contain low-degree polynomials, we will only work with polynomials of degree poly(n), or VPSPACE b as defined in [14], unless mentioned otherwise.

Monotone analogues of VPSPACE, and our contributions
We now define monotone analogues for the various definitions of VPSPACE outlined in the previous section, and compare the powers of the resulting monotone models/classes.

Monotone succinct ABPs
We first consider the natural monotone analogue of the definition due to Malod [18] which uses succinct algebraic branching programs (Definition 2.7).Malod showed that every family {f n } in VPSPACE can be computed by 2 poly(n) sized ABPs that have complexity poly(n).Recall that the complexity of a succinct ABP is the size of the smallest algebraic circuit that encodes its graph.

F S T T C S 2 0 2 3 11:8 Monotone Classes Beyond VNP
We therefore define monotone succinct ABPs as ABPs that can be succinctly described by monotone algebraic circuits of size poly(n).However, this restriction forces that if the monomial x e appears in any edge-label (a, b), then it also appears in the label of ( 1, 1).Therefore, self-loops are inevitably present in succinct ABPs in the monotone setting.To handle this, we additionally allow the length of the ABP, say ℓ, to be predefined3 so that now the polynomial computed by the ABP can be defined to be the sum of polynomials computed by all st paths of length at most ℓ.
is the label of the edge from a to b in the ABP.The polynomial computed by the ABP is the sum of polynomials computed along all s to t paths in G B of length at most ℓ; where each path computes the product of the labels of the constituent edges.
The size of the circuit B is said to be the complexity of the monotone succinct ABP.The number of vertices 2 r is the size of the succinct ABP.
Note that since B is a monotone algebraic circuit, all the edge-labels in the ABP are monotone polynomials over x.It is also not hard to see that any polynomial f ∈ mVP is computable by this model.If C is the monotone circuit computing f , then the monotone succinct ABP computing f is (C ′ , 0, 1, 1) where We show that the computational power of monotone succinct ABPs when computing polynomials of bounded degree does not even go beyond mVNP.
▶ Theorem 3.2.If a polynomial family {f n } of degree poly(n) is computable by monotone succinct ABPs of complexity poly(n), then {f n } ∈ mVNP.
In contrast, Malod [18] showed that every family in VPSPACE admits succinct ABPs of polynomial complexity, and we expect VPSPACE b to be a much bigger class than VNP.The proof of Theorem 3.2 is quite straightforward and relies on the following claim.▷ Claim 3.3.For any f n , let A = (B, s, t, ℓ) be the monotone succinct ABP computing it, This bound allows us to write the sum of all s to t paths in the ABP as an exponential sum of an mVP expression, finishing the proof.A complete proof can be found in Subsection C.4.
It is not clear to us if the converse of Theorem 3.2 is true.Any obvious attack seems to fail due to the restriction that the circuit encoding the ABP needs to be monotone.

Quantified monotone circuits
As mentioned earlier, Malod [18] had also characterized the class VPSPACE using the notion of quantified algebraic circuits (Definition 2.9).We now consider its natural monotone analogue, which we call quantified monotone circuits.

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▶ Definition 3.4 (Quantified Monotone Algebraic Circuits).A quantified monotone algebraic circuit has the form and C is a monotone algebraic circuit.The size of the quantified monotone algebraic circuit above is m + size(C).
We denote by mVP quant the class of all n-variate polynomial families of degree poly(n) that are computable by quantified monotone algebraic circuits of size poly(n).
Clearly mVNP ⊆ mVP quant .It is therefore interesting to check if the inclusion is tight.We show that mVNP ̸ = mVP quant if and only if there is a family {f n } ∈ mVP quant such that the k-th homogeneous component of f n is not in mVP quant for some n and k ≤ deg(f ).
In particular we show the following statement.
▶ Theorem 3.5.Let f be computable by a quantified monotone circuit of size s.If f is homogeneous, then it is expressible as an exponential sum of size at most O(s Since mVNP is closed under addition, we get the following as a corollary.
▶ Corollary 3.6.The class mVP quant is closed under taking homogeneous components, if and only if, mVP quant = mVNP.That is, A proof of Theorem 3.5 and Corollary 3.6 can be found in Section 4.
Even though we believe mVNP ⊊ mVP quant , we feel this might be tricky to prove.The following theorem sheds some light on why that may be the case.
▶ Theorem 3.7.Suppose f (x) is an n-variate, degree-d polynomial computed by a quantified monotone circuit of size s, which uses ℓ summation gates.Then for a set of variables w of size at most d • ℓ, there is a monotone circuit h(x, w) of size at most d • s, and a monotone polynomial A(w) such that, where A(w) potentially has circuit size and degree that is exponential in n and ℓ.
Although the obvious size and degree bounds on A(w) above are exponential, it has a somewhat succinct quantified expression that can be inferred from the proof (see Subsection C.5).We now discuss how Theorem 3.7 helps us understand a possible difficulty in separating mVP quant from mVNP.

1.
If the polynomial A(w) from Theorem 3.7 were to have degree and size that is polynomial in n, then mVP quant would collapse to mVNP.Further, since A is free of x, its exponential degree and size can be leveraged only for designing coefficients of f .Moreover, the monotone nature of A and h ensures that A(1) is the largest value, and contributes equally to all monomials in the support of f , since supp(f ) = supp(h(x, w = 1)).2. Another consequence that is quite interesting is the following.Suppose there is a different monotone polynomial B(w) of small degree and size that agrees with A(w) on all {0, 1}inputs, then That is, we can replace A by B in our expression and then f clearly has an efficient "mVNP-expression".Thus, any separation between mVNP and quantified monotone VP will provide a polynomial A(w) which is hard to compute for mVNP, even as a function over the boolean hypercube; a result that perhaps stands on its own.

F S T T C S 2 0 2 3 11:10 Monotone Classes Beyond VNP
A proof sketch of Theorem 3.7 can be found in Section 4 and a complete proof can be found in Subsection C.5.

Monotone circuits with summation and production gates
Note that it is unclear if quantified monotone circuits are closed under compositions.We therefore also consider a model that generalizes quantified monotone circuits and is trivially closed under compositions.Here summation and production gates are allowed to appear anywhere in the circuit.
▶ Definition 3.8 (Algebraic circuits with summation and production gates).An algebraic circuit with summation and production gates is an algebraic circuit (Definition 2.1) in which the internal nodes can also be summation or production gates (Definition 2.8), in addition to + or ×.A subset of the variables used by the circuit are marked as auxiliary.These variables do not appear in the output polynomial(s) of the circuit, and the labels for all the summation and production gates are required to be auxiliary variables.
The size of an algebraic circuit with summation and production gates is the number of nodes in the graph.
An algebraic circuit with summation, production gates is said to be monotone, if all the constants appearing in it are non-negative.
We denote by mVP sum,prod the class of all n-variate polynomial families of degree poly(n) that are computable by monotone algebraic circuits with summation and production gates of size poly(n).
Note that even in the non-monotone setting this model is clearly as powerful as quantified circuits, but can be simulated by circuits with projection gates.Again, Malod [18] showed that quantified circuits and circuits with projection gates are equivalent in power.So the class of polynomials efficiently computable by this model is also VPSPACE.
In the monotone setting, however, it is not clear if the power of quantified monotone circuits is the same as that of this model.In particular, we observe the following.Here, we mean "closure under compositions" in a strong sense: if C 1 and C 2 are quantified monotone circuits of size s 1 and s 2 respectively, then the polynomial computed by their composition to have a quantified monotone circuit of size at most s 1 + s 2 .
▶ Observation 3.9 (Informal).Quantified monotone circuits are closed under compositions, if and only if, mVP quant = mVP sum,prod .Theorem 5.1 gives a formal statement and its proof can be found in Subsection C.3.We, however, show that even this seemingly stronger model does not help in computing transparent polynomials.

▶ Theorem 3.10. Any monotone algebraic circuit with summation and production gates that computes a transparent polynomial f , has size at least |supp(f )| /4.
This shows that transparent polynomials with large support are hard even for this model.A proof sketch can be found in Section 5.
Recall that one way to refute the τ -conjecture for Newton polygons is to show a transparent polynomial in (non-monotone) VP.Theorem 3.10 shows that any transparent polynomial from VP that refutes the conjecture would also witness a separation between VP and a class P. Chatterjee, K. Gajjar, and A. Tengse 11:11 potentially much bigger than mVNP4 .Even though stark separations between monotone and non-monotone models are not unheard of [10,4], such a result would also be quite interesting and would further highlight the power of subtractions.

Monotone circuits with projection gates
Finally, adapting the definition of VPSPACE due to Poizat (Definition 2.6) [19], we define monotone circuits with projection gates.
▶ Definition 3.11 (Monotone algebraic circuits with projection gates).A monotone algebraic circuit with projection gates is an algebraic circuit with projections (Definition 2.6) in which only non-negative constants from the field are allowed to appear as labels of leaves.
As in Definition 3.8, only the auxiliary variables can be used as labels for the projection gates.The size of a monotone algebraic circuit with projection gates is the number of nodes in the underlying graph.
We denote by mVP proj the class of all n-variate polynomials of degree poly(n) that are computable by size-poly(n) monotone algebraic circuits with projection gates.This model is clearly at least as powerful as monotone circuits with summation and production gates, since sum z = fix (z=0) + fix (z=1) and prod z = fix (z=0) × fix (z=1) .It would therefore be interesting to show a separation between the power of the two models.
Even though we are unable to do that, we show that monotone circuits with projection gates are indeed more powerful than quantified monotone circuits, with a 2 Ω( √ m) separation.
▶ Theorem 3.12.The polynomial family {Perm n } can be computed by monotone circuits with projection gates of size O(n 3 ), but quantified monotone circuits computing it must have size 2 Ω(n) .
Finally we show that mVP proj is closed under taking homogeneous components.
▶ Theorem 3.13.Suppose f is computed by a size s monotone circuit with projections.Then for any k ≤ deg(f ), hom k (f ) has a monotone circuit with projections of size O(k 2 • s).
Proof sketches of Theorem 3.12 and Theorem 3.13 can be found in Section 6.

Defining Monotone VPSPACE (mVPSPACE)
We propose the following definition for mVPSPACE.

▶ Definition 3.14 (Monotone VPSPACE). A family of polynomials {f n } is said to be in mVPSPACE if for all large n, f n is computable by a monotone algebraic circuit with projection gates (Definition 3.11) of size poly(n).
Further if {f n } has degree poly(n), then it is said to be in mVPSPACE b .
That is, we define mVPSPACE b := mVP proj and define mVPSPACE along the same lines, but without the restriction on the degree being bounded (since VPSPACE does not impose any restrictions on degree).Some of our reasons for this choice are as follows.
Firstly, being a complexity class, mVPSPACE b should be closed under (monotone) affine projections, i.e. setting a few variables to monotone affine polynomials.All of mVP quant , mVP sum,prod and mVP proj have this property.
Further, as mVP and mVNP are closed under taking homogeneous components, it is desirable for a more powerful class to also have this property.Even if mVP quant satisfies this, it would not lead to a larger class (Corollary 3.6).Also, it is not clear mVP sum,prod is closed under homogenization, while mVP proj is (Theorem 3.13).
Finally, we believe that having Perm n ∈ mVP proj is an interesting property that further strengthens the case for mVP proj being the definition for mVPSPACE b .

4
Quantified monotone circuits Computing homogeneous polynomials ▶ Theorem 3.5.Let f be computable by a quantified monotone circuit of size s.If f is homogeneous, then it is expressible as an exponential sum of size at most O(s • deg(f )).
Proof.Let d = deg(f ), and let C be a quantified monotone circuit computing f , that uses exactly k production gates.We can then assume that, without loss of generality, by using some empty y j s whenever necessary.Note that the y j s are sets of variables, whereas each of the z j s are single variables.We now prove the statement in two steps.First, we use the homogeneity of f , and the monotonicity of the quantified circuit, to show that k ≤ log(d)., y, z).Here w i denotes all the auxiliary variables that are alive after "i rounds" of quantifiers.Further, let h i (x, w i ) = prod zi g i (z i , x, w i ).Now, f (x) = sum y0 h 1 (x, y 0 ), and it is homogeneous.Therefore, since h 1 is monotone, it is also homogeneous in x with degree exactly d.But deg x (h 1 ) = deg x (prod z1 g 1 ) = deg x (g 1 (z 1 = 0)) + deg x (g 1 (z 1 = 1)).If we write g 1 (z 1 , x, w 1 ) = g 1,0 (x, w 1 ) + z • g 1,1 (z 1 , x, w 1 ), then we have that g 1 (z 1 = 0) = g 1,0 (x, w 1 ) and g 1 (z 1 = 1) = g 1,0 (x, w 1 ) + g 1,1 (z 1 = 1, x, w 1 ).Since h 1 is homogeneous in x and g 1 is monotone in all the variables, this must mean that deg x (g Also, g 1 is homogeneous in x, and thus we can repeat the same argument for h 2 , g 2 , and so on. As a result, we see that deg(f ) = 2 k • deg x (g), and hence k ≤ log d. ◁ We can now make 2 k ≤ d many copies of the "inner circuit" g(x, y, z), one for each fixing of the z variables.We then obtain the final exponential sum computing f by using the following "product rule" for summations repeatedly.
Note that in the above case the two summations are over disjoint sets of variables.This can easily be ensured in our case, by treating the y variables in each of the 2 k ≤ d copies as mutually disjoint.It is easy to see that the exponential sum has size O(size(C), d).◀ ▶ Remark 4.2.The first step in the above proof extends more or less as it is, to an arbitrary circuit with summation and production gates.Thus, any circuit with arbitrary summations and productions that computes a homogeneous polynomial can be assumed to not contain any production gates, with a polynomial blow-up in size.

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However, this does not directly give an efficient exponential sum, because of the second step in the above argument.It crucially uses the fact that for any summation gate g, the number of production gates on a path from g to the root was O(log d).This ensures that no summation gate (or its auxiliary variable) has to be replicated more than poly(d) times, which is not necessarily true if we start with an arbitrary circuit with summation gates.

Large exponential sums for arbitrary polynomials
▶ Theorem 3.7.Suppose f (x) is an n-variate, degree-d polynomial computed by a quantified monotone circuit of size s, which uses ℓ summation gates.Then for a set of variables w of size at most d • ℓ, there is a monotone circuit h(x, w) of size at most d • s, and a monotone polynomial A(w) such that, where A(w) potentially has circuit size and degree that is exponential in n and ℓ.
We shall need the following simple observation, which follows from the "product-rule" for summations stated earlier.

▶ Observation 4.3 (Product of exponential sums).
prod z sum y g(x, y, z) = sum y0,y1 (g(x, y 0 , 0) • g(x, y 1 , 1)) Let us see a toy case of trivially moving from a quantified expression to an exponential sum, using Observation 4.3.f (x) = sum y1 prod z1 sum y2 prod z2,z3 sum y3 g(x, y 1 , y 2 , y 3 , z 1 , z 2 , z 3 ) = sum y1 prod z1 sum y2 prod z2 sum y3,0,y3,1 g(x, y 1 , y 2,a1 , y 3,(a1a2a3) , a 1 , a 2 , a 3 )   In the last line, each * runs over {0, 1}, so there are 1 + 2 + 8 = 11 auxiliary variables in total.Note that y 3 has 8 copies, which is due to the 3 production gates "above" the summation gate labelled by it.Similarly, y 2 has just 2 copies, while y 1 has just one.Also, if instead of single auxiliary variables y 2 and y 3 we had sets of auxiliary variables y 2 and y 3 , nothing much would change.That is, we would have had 8 copies of the set y 3 and 2 copies of y 2 , irrespective of their sizes.
What this shows in general, is that we can trivially move from a quantified expression to an expression which has the form where Y = ∪ a {y a }, r is the number of production gates in the quantified expression, |Y| is potentially exponential (since the number of copies of some auxiliary variable might be exponential) but g a (x, y a ) = g(x, y = y a , z = a) for a poly-sized circuit g(x, y, z).

F S T T C S 2 0 2 3 11:14 Monotone Classes Beyond VNP
The key observation that allows us to prove Theorem 3.7 is that if f has degree d, then the number of copies of each auxiliary variable needed in the outer summation gate is at most d.This is because, due to monotonicity, deg x (g a (x, y a )) ̸ = 0 for only d many a ∈ {0, 1} r .A complete proof of Theorem 3.7 can be found in Subsection C.5.

Monotone circuits with summation and production gates
In this section, we give the proof overview of Theorem 3.10.
▶ Theorem 3.10.Any monotone algebraic circuit with summation and production gates that computes a transparent polynomial f , has size at least |supp(f This result is an extension of the ideas in the work of Hrubeš & Yehudayoff [11].Their argument shows that any bivariate monotone circuit of size s that computes a polynomial with convexly independent support outputs a polynomial with support at most 4s.They achieve this by keeping track of the largest polygon (in terms of the number of vertices) that one can build using the polynomials computed at all the gates in the circuit.They then inductively show that no gate (leaf, addition, multiplication) can increase the number of vertices by 4. We are able to show the same bound for production and summation gates, by working with a monotone bivariate circuit over y 1 , y 2 that is allowed some auxiliary variables z for summations and productions.
An important component of the proof in [11] is that if the sum or product of two monotone polynomials is convexly independent, then so are each of the two inputs.However, allowing for summations and productions means that some monomials that are computed internally could get "zeroed out".In fact, summation and production gates do not quite "preserve convex dependencies".For example, the convexly dependent support y 1 y 2 , y 1 y 2 z, y 1 y 2 z 2 when passed through sum z produces just {y 1 y 2 }, which is convexly independent.
In order to prove Theorem 3.10, one can get around this by working directly with the support projected down to the "true" variables, which we call y-support in our arguments.It turns out that summations and productions indeed preserve convex dependencies that are in the y support of the input polynomial.Since the proof follows exactly along the same lines as the one in [11], we omit the proof here.A formal proof can be found in the full version [2].

Quantified monotone circuits and compositions
▶ Observation 3.9 (Informal).Quantified monotone circuits are closed under compositions, if and only if, mVP quant = mVP sum,prod .
Even though this statement appears to be straightforward, formally stating it requires a bit more care.In particular, we require quantified circuits to be closed under compositions in a strong sense, similar to usual algebraic circuits.Doing that yields the following theorem, which we prove in Subsection C.3.▶ Theorem 5.1.Suppose that for any multi-output quantified monotone circuit C of size s with r inputs, and any multi-output quantified monotone circuit C ′ of size s ′ with r outputs, we have that the polynomial computed by C • C ′ has a quantified monotone circuit of size at most (s + s ′ ).
Then, any multi-output, monotone circuit with summation and production gates of size s can be simulated by a multi-output quantified monotone circuit of size at most s, and hence mVP quant = mVP sum,prod .
The converse is also true.Monotone circuits with projection gates Exponential separation from quantified circuits ▶ Theorem 3.12.The polynomial family {Perm n } can be computed by monotone circuits with projection gates of size O(n 3 ), but quantified monotone circuits computing it must have size 2 Ω(n) .
We begin by proving that Perm n ∈ mVP proj .
▶ Theorem 6.1.There is a monotone circuit with projection gates of size O(n 3 ) that computes Perm n .
Proof.We first define a polynomial P 0 such that all its monomials contain exactly one x-variable from each row.
Note that P 0 has n 2 auxiliary variables y, one attached to each "true" variable x i,j .We now want to use these to progressively prune the monomials that pick up multiple variables from the jth column by projecting the n variables y 1,j , . . ., y n,j .Let e 1 , . . ., e n ∈ {0, 1} n be such that e i (k) = 1 ⇔ i = k, and define for each j ∈ [n], The following claim is now easy to verify.
▷ Claim 6.2.For all j ∈ [n], P j contains all the monomials from P j−1 that are supported on exactly one x-variable from the jth column.
As a result, the monomials in P n are exactly those of the monomials in Perm n .Additionally, for each j, the auxiliary variables in P j are only from the columns j + 1, . . ., n; thus P n = Perm n .The size of our circuit is O(n 3 ), since size(P 0 ) = O(n 2 ) and size(P j ) = size(P j−1 ) + O(n 2 ).This proves Theorem 6.1.◀ ▶ Remark 6.3.Our upper bound above also implies that any polynomial (family) that can be expressed as the permanent of a monotone matrix of size poly(n) (called monotone pprojection of Perm n ) can also be computed by efficient monotone circuits with projection gates.Although Perm n is complete for non-monotone VNP, it is not the case that all monotone polynomials in VNP are monotone p-projections of Perm n , as shown by Grochow [8].
The proof of Theorem 3.12 now follows from the following simple extension of an observation due to Yehudayoff [26] and the classical lower bound of Jerrum & Snir [12] against monotone algebraic circuits for Perm n .A complete proof can be found in Subsection C.1.
▶ Lemma 6.4.Let f (x) be a monotone polynomial whose support cannot be written as a non-trivial product of two sets, and for some monotone polynomial g(x, z), suppose we have  We show this using the classical argument of "gate replication" and the complete proof can be found in Subsection C.2.

Conclusion
Our work is an attempt at understanding the hardness of transparent polynomials for monotone algebraic models.We observe that the lower bound of Hrubeš & Yehudayoff [11] extends beyond monotone VNP, and therefore turn to exploring the class VPSPACE from the non-monotone world.This exploration reveals that the natural monotone analogues of the multiple equivalent definitions of VPSPACE have contrasting powers.Additionally, transparent polynomials turn out to be as hard for some of these analogues as they are for usual monotone circuits.The following are some interesting open threads from our work.
We suspect that transparency is a highly restrictive property, especially for monotone computation.Therefore, we conjecture that if f is a transparent polynomial being computed by a size-s monotone circuit with projection gates, then |supp(f )| ≤ 2 polylog(s) .It would be interesting (at least to us) to see a proof or a refutation of this conjecture.An immediate hurdle in extending the techniques in [11] (as in Theorem 3.10) to mVPSPACE, is that unlike summations and productions, 0-projections do not preserve convex dependencies, even if we restrict to the "true" variables.
Along similar lines, a possibly simpler goal is to show a non-monotone circuit upper bound for a transparent polynomial.Since transparency only restricts the support of the polynomial, one is free to choose any real coefficients to ensure that it is in VP.In particular, this brings powerful non-monotone tricks like interpolation into play.Among other things, such a result would refute the notoriously open τ -conjecture for Newton polygons.
Another question we would like to highlight is separating mVNP and quantified monotone circuits.As mentioned in the discussion following Theorem 3.7, such a separation would yield a (high degree) polynomial that is hard for mVNP even as a function over the boolean hypercube.Such a polynomial might be of interest, perhaps, even in the non-monotone setting.

Figure 1
Figure 1 Nodes represent classes of polynomial families; A B ≡ A ⊆ B and A −→ B ≡ A ⊊ B. Transparent polynomials are hard for all models corresponding to orange, rectangular nodes.

F▶ Theorem 3 . 13 .
Suppose f is computed by a size s monotone circuit with projections.Then for any k ≤ deg(f ), hom k (f ) has a monotone circuit with projections of size O(k 2 • s).

T C S 2 0 2 3 11:6 Monotone Classes Beyond VNP ▶ Definition 2.4 (
Monotone VNP (mVNP)).A family {f n } of monotone polynomials is said to be in mVNP, if there exists a constant c ∈ N, and an m-variate family {g m } ∈ mVP with m, size(g m ) ≤ n c , such that for all large enough n, f n satisfies the following.
They showed that if VP ̸ = VPSPACE b then either VP ̸ = VNP or P/ poly ̸ = PSPACE/ poly.Later, Poizat [19] gave an alternate definition that does not rely on any boolean machinery, but instead uses a new type of gate called a projection gate.
▶ Definition 2.5 (Projection gates [19]).A projection gate is a unary gate that is labelled by a variable z and a constant b ∈ {0, 1}, denoted by fix (z=b) .It returns the partial evaluation of its input polynomial, at Definition 3.1 (Monotone Succinct ABPs).A monotone succinct ABP over the n variables x = {x 1 , . . ., x n } is a four tuple (B, s, t, ℓ) with |s| = |t| = r, where ℓ is the length of the ABP.s is the label of the source vertex, and t is the label of the sink (target) vertex.
B(u, v, x) is a monotone algebraic circuit that describes a directed graph G B on the vertex set {0, 1} r in the following way.For any two vertices a, b ∈ {0, 1} r , the output