Permanent Does Not Have Succinct PolynomialSize Arithmetic Circuits of Constant Depth

Maurice Jansen and Rahul Santhanam

School of Informatics, The University of Edinburghmaurice.julien.jansen@gmail.com, rsanthan@inf.ed.ac.uk

Abstract We show that over fields of characteristic zero there does notexist a polynomial p(n) and a constant-free succinct arithmetic circuitfamily Φn using division by constants1, where Φn has size at mostp(n) and depth O(1), such that Φn computes the n × n permanent.A circuit family Φn is succinct if there exists a nonuniform Booleancircuit family Cn with O(logn) many inputs and size no(1) such thatthat Cn can correctly answer direct connection language queries aboutΦn - succinctness is a relaxation of uniformity.To obtain this result we develop a novel technique that further strength-ens the connection between black-box derandomization of polynomialidentity testing and lower bounds for arithmetic circuits. From this weobtain the lower bound by explicitly constructing a hitting set againstarithmetic circuits in the polynomial hierarchy.

1 Introduction

Proving super-polynomial arithmetic circuit lower bounds for explicit polyno-mials is one of the hardest challenges in theoretical computer science. For un-restricted circuits the best-known lower bounds are Ω(n log r), for polynomialsin n variables and degree r, due to Baur and Strassen [BS82]. To make furtherprogress one popular approach has been to aim at proving lower bounds forsmall depth circuits first. This restriction is well-motivated by a recent result ofAgrawal and Vinay [AV08], who show that proving a 2Ω(n) size lower bound fora multilinear polynomial in n variables for depth four arithmetic circuits trans-lates into a 2Ω(n) lower bound for the unrestricted model. Over finite fields, Grig-oriev and Karpinski [GK98] show that depth three arithmetic formulas that aresums-of-products-of-sums require size 2Ω(n) to compute the determinant poly-nomial detn =

∑σ∈Sn sgn(σ)

∏nj=1 xiσ(j). Over fields of characteristic zero, e.g.

Q or R, super-polynomial lower bounds have been especially difficult to obtain.The best-known lower bound for depth three sums-of-products-of-sums formulasfor detn, and also for the permanent polynomial pern =

∑σ∈Sn

∏nj=1 xiσ(j) is

Ω(n4/ log n), due to Shpilka and Wigderson [SW01]. Over fields of characteristic

1In the constant-free model the only allowed constants in the circuit are −1, 0 and1, cf. [Bur09,KP07]. Additionally, we allow the circuit to use divisions of the form 1/g,for any previously computed nonzero constant g.

zero, no lower bounds for this depth three model are known beyond Ω(n2), forexplicit polynomials in n variables of degree at most poly(n). For higher depth,the best-known bounds are due to Raz [Raz10], who constructs a polynomialwith coefficients in 0, 1 of degree O(d) in n variables that requires depth dcircuits of size n1+Ω(1/d) over any field. Adding the restriction of multilinearity2

Raz and Yehudayoff [RY09] prove that detn and pern require size 2nΩ(1/d)

formultilinear circuits of product depth d.

We remark that for nonuniform constant depth arithmetic circuits using in-teger constants an exponential lower may be derived for Permanent from theRazborov-Smolensky [Raz87,Smo87] result that MOD3 requires exponential sizebounded depth Boolean circuits with parity gates. Namely, consider the followingargument3, which was communicated to us by Pavel Hrubes: By the Razborov-Smolensky result we have a Boolean function f with a poly size Boolean formulaF that requires exponential size bounded depth Boolean circuits with paritygates. Interpret F as an arithmetic circuit over GF (2) by replacing AND withmultiplication and parity with addition. Let f be the computed polynomial.Observe that f requires exponential size arithmetic circuits of constant depthover GF (2), as we can also perform the reverse replacement of gates to obtain aBoolean circuit for f . By Valiant’s [Val79a] completeness result for determinant,we can write f = det(A), where A is a poly-size matrix with variables, 0s and 1s,working over GF (2). However, over GF (2), Determinant equals Permanent andthus also Permanent requires exponential size arithmetic bounded depth circuitsover GF (2). This implies the exponential lower bound over Z.

For constant-free uniform4 arithmetic circuits it is known that deptho(log log n) circuits for pern must have super-polynomial size, as proved byKoiran and Perifel [KP09]. We interpret this result as a lower bound for cir-cuits that are succinct in some extreme sense. In order to make progress wewill prove lower bounds for a far less restrictive notion of succinctness. To makethis precise, we give some definitions. Consider a family Φn of constant-freearithmetic circuits, i.e. only using constants in −1, 0, 1. We say that Φn is(a(n), b(n))-succinct4, if there exists a family of nonuniform Boolean circuitsCn, where Cn has at most a(n) inputs and is of size at most b(n), such thatCn can answer direct connection language queries about Φn. This means thatgiven names of gates in Φn, the circuit Cn should be able to answer querieslike whether two gates are connected, what the type of a gate is: +, ×, /, or‘input’, and in the latter case whether the variable or constant label equals somegiven string. Regarding division gates, to express our lower bound result in itsstrongest form, we allow divisions of the form 1/g, where g is any nonzero con-stant previously computed by the circuit. We will call such circuits “constant-freearithmetic circuits using divisions by constants”.

2This means that at any gate the computed polynomial must be multilinear.3We missed this point previously. This has prompted us to strengthen the model

to include divisions by constants in order to properly demonstrate the power of ourtechniques.

4See Section 3 for a formal definition.

Succinctness interpolates between uniformity and non-uniformity. It canbe observed that DLOGTIME-uniform circuits of polynomial size are(O(log n), O(log n log log n))-succinct. On the other hand, if a sequence Φnof circuits is (a(n), b(n))-succinct, the circuit Φn can be constructed in time2O(a(n))b(n)O(1), given O(b(n) log(b(n))) bits of advice.

In what follows we take a(n) = O(log n), which limits the size of the arith-metic circuit to be nO(1). By convention, whenever a(n) = O(log n), we will dropit from the notation, and just write “b(n)-succinct”. Within this setting there isa spectrum of scenarios to study. At one extreme, poly(n)-succinctness forms norestriction at all. At the other end, for polylog(n)-succinctness, a lower boundcan be derived with an advice elimination argument using the known uniformTC0 lower bound for permanent by Allender [All99]. Our main result is to pushthe envelope towards the high end by proving the following theorem:

Theorem 1. Let d ≥ 0 be an integer. Over fields of characteristic zero, theredoes not exist a polynomial p(n) such that pern has no(1)-succinct size p(n)depth d constant-free arithmetic circuits using divisions by constants.

Thus either the permanent does not have polynomial size constant depthconstant-free arithmetic circuits using divisions by constant, or in (the unlikely)case that it does, the direct connection language L for such circuits is hard in thesense that for deciding inputs of size k = O(log n), for some ε > 0 one requiresnonuniform Boolean circuits of size nε = 2Ω(k), i.e. exponential in the inputsize5.

Why are we interested in lower bounds against succinct circuits? If we’reinterested in actually computing the permanent, the mere fact of there beingpolynomial-size circuits for it doesn’t seem to be sufficient, for there is the ques-tion of how we can find these circuits. Succinctness is a measure of how easy thecircuits themselves are.

Also, ultimately, much of the motivation for arithmetic complexity comesfrom classical problems like NP vs P and PSPACE vs P, which have resistedsolution, or even progress, for many decades. Often the nonuniform versions ofthese questions are studied because we have few ways of taking advantage ofuniformity apart from diagonalization. One aspect of our proof is that we do notmake use of hierarchy theorems, although it is conceivable a more traditional di-agonalization argument6 can also yield results similar to ours. However, we thinkour approach has more potential as it comes to breaking new ground, depend-ing on whether one can manage to use more sophisticated algebraic techniquesfor constructing hitting sets. As this paper shows, already a naive counting ar-gument for constructing a hitting set in the polynomial hierarchy manages toestablish a strong lower bound theorem. We next describe the actual techniquesused in the proof.

5It is possible to give a uniform upper of ENPRPfor L.

6One of the referees of this paper suggested such an argument.

1.1 Techniques

At a high level, we exploit the well-known connection between lower boundsand derandomization of arithmetic circuit identity testing (ACIT). In the ACITproblem one is given an arithmetic circuit, and the question is to decide whetherthe polynomial computed by the circuit is identically zero or not. Using theSchwartz-Zippel-deMillo-Lipton Lemma [DL78,Sch80,Zip79], Ibarra and Moran[IM83] show this problem is in coRP. It is known that non-trivial deterministicalgorithms would lead to circuit lower bounds against Boolean or arithmeticcircuits [HS80,KI04,Agr05]. This is often interpreted as evidence for the difficultyof derandomizing ACIT. However Agrawal turns this interpretation on its headand advocates derandomization of ACIT as an approach toward proving circuitlower bounds [Agr05]. Our result is a step in this direction.

There are two major parts to our argument. The first is to show that thelower bound of Theorem 1 follows from a black-box derandomization hypothesis(Working Hypothesis 1) for ACIT. Our Working Hypothesis 1 poses the existenceof integer sequences definable by no(1)-succinct TC0 circuits that form a hittingset against small constant-free arithmetic circuits. We prove the implicationby combining ideas of Agrawal [Agr05], Burgisser [Bur09] and Koiran [Koi10],and making critical use of our succinctness assumption to indirectly performcomputations which we cannot afford to do directly.

We do not know how to prove Working Hypothesis 1. Instead what we willestablish for the second part of our argument, is a weaker derandomization ofACIT using integer sequences so-called weakly-definable in the polynomial hi-erarchy. However, this statement appears not to be strong enough to get thelower bound of Theorem 1 directly. To resolve this, we argue by contradictionas follows. Assuming that the conclusion of Theorem 1 fails, then due to Toda’sTheorem [Tod91] and an improvement by Zanko [Zan91], cf. [All99], of Valiant’s[Val79b] completeness result for pern, this induces a collapse of the polynomialhierarchy, which makes the integer sequence we explicitly construct good enoughto satisfy the requirements of Working Hypothesis 1. This way we obtain a lowerbound for pern after all, but this contradicts the assumption that the conclusionof Theorem 1 does not hold.

1.2 Overview

The rest of this paper is organized as follows. In Section 2 we import somerequired definitions and results from the literature. In Section 3 we define ourcomputational model of succinct circuits, and we prove some preliminary resultspertaining to this model. In Section 4 we state our derandomization hypothesis(Working Hypothesis 1), and we show this statement implies a lower bound forsuccinct circuits computing permanent. Section 5 contains a simple constructionof a hitting set against arithmetic circuits in the polynomial hierarchy. Puttingeverything together, we prove Theorem 1 in Section 6. In Section 7 we switchgears to give the advice elimination argument alluded to before. Finally, wediscuss some open problems in Section 8.

2 Preliminaries

We use standard base 2 representation of integers by binary strings, with onebit representing the sign. Per abuse of notation, type casting between integersand binary strings is omitted throughout the paper.

An arithmetic circuit Φ over a set of variables X = x1, x2, . . . , xn and afield F is given by a directed acyclic graph such that nodes with in-degree = 0are labeled with elements of X ∪ F. Nodes with higher in-degree are labeled by+ or ×. To each vertex in Φ we can associate an element of the polynomial ringF[X] in the obvious way. A polynomial f ∈ F[X] is said to be computed by Φ, ifthere exists a node in Φ where the associated polynomial equals f . The size s(Φ)of the circuit is defined to be the number of edges in Φ. We will also write |Φ|for s(Φ). The depth d(Φ) of the circuit is defined to be the length of the longestdirected path in Φ. For a polynomial f ∈ F[X]. its arithmetic circuit complexitys(f) is defined by s(f) = mins(Φ) : Φ computes f. If the underlying graph ofΦ is a tree, then Φ is called a formula. Arithmetic formula size of f is denotedby e(f).

For constant-free arithmetic circuits the only field constants that are allowedfor labels are ∈ −1, 0, 1, cf. [Bur09,KP07]. Constant-free arithmetic circuitscompute polynomials with integer coefficients. To express our main lower boundresult in its strongest form, we extend the constant-free model to include so-called divisions by constants by allowing gates of the form f/g, where the inputsf and g are computed previously in the circuit, and such that g is a nonzeroconstant. Constant-free arithmetic circuits using division by constants definedas such compute polynomials in Q[X].

For definitions of standard complexity classes like P, PH, PP, etc., we referthe reader to Scott Aaronson’s Complexity Zoo7. Next we list several of theimportant ones we use for the reader’s convenience.

#P is the class of all function f : 0, 1∗ → 0, 1∗ such that there existsa language A ∈ P and a polynomial p(n) such that f(x) = |w ∈ 0, 1p(|x|) :(x,w) ∈ A|. GapP is the class of all function f − g, where f, g ∈ #P. We willmake combined use of Valiant’s [Val79b] result that computing pern(M) for Mwith entries in 0, 1 over Z is complete for #P, and Toda’s [Tod91] Theorem,which states that PH ⊆ P#P[1].

We define the majority operator C. acting on a complexity class as follows.Given a complexity class C, C.C is the class of all languages L for which there ex-ists L′ ∈ C and a polynomial p(n) such that x ∈ L⇔ |w ∈ 0, 1p(|x|) : (x,w) ∈L′| > 2p(|x|)−1. The counting hierarchy, introduced by Wagner [Wag86], is de-fined to be

⋃i≥0 CiP, where C0P = P, and for all i ≥ 1, CiP = C.Ci−1P.

The first level C1P of this hierarchy corresponds to the standard complexityclass PP. We will use Toran’s [Tor91] characterization of the counting hierarchywhich states that Ci+1P = PPCiP, for all i ≥ 0.

Next we define nonuniform versions of complexity classes. An advice functionis a function of type h : N→ 0, 1∗. For a complexity class C, we define C/poly

7http://http://qwiki.stanford.edu/index.php/Complexity Zoo

to be the class of languages for which there exists L′ ∈ C, and advice function hwith |h(n)| = nO(1), such that x ∈ L⇔ (x, h(|x|)) ∈ L′.

We use the Boolean circuit complexity classes AC0, TC0 and NC1. AC0

is the class of all Boolean functions computable by polynomial size constantdepth circuits with unbounded fan-in gates in ∨,∧,¬. TC0 is the class of allBoolean functions computable by polynomial size constant depth unboundedfan-in threshold circuits. A threshold circuit is a Boolean circuit in which allgates either compute the negation, or the majority function of their inputs.NC1 is the class of all Boolean functions that can be decided by polynomialsize O(log n) depth circuits of bounded fan-in. For these classes we have thatAC0 ⊆ TC0 ⊆ NC1.

For a Boolean circuit family Cn, if there are no requirements on con-structability, we call the family nonuniform. For the uniform versions of Booleancomplexity classes we will always be using the notion of DLOGTIME-uniformity.We postpone defining this until Section 3.

We will use the notion of definability of Ref. [KP07]. To distinguish thisfrom definability as in Ref. [Bur09], we use the term weakly-definable. An in-teger sequence of bit size q(n) is given by a function a(n, k1, k2, . . . , kt), forsome fixed number t, such that there exist polynomials p(n) and q(n) so thata(n, k1, k2, . . . , kt) ∈ Z is defined for all n ≥ 0, and all 0 ≤ k1, k2, . . . , kt < 2p(n),and where the bit size of a(n, k1, k2, . . . , kt) is bounded by q(n). We will oftenwrite an(k1, k2, . . . , kt) instead of a(n, k1, k2, . . . , kt). We define the language

uBit(a) = (1n, k1, k2, . . . , kt, j, b) : the jth bit of a(n, k1, k2, . . . , kt) equals b.

Here k1, k2, . . . , kt and j are encoded in binary. For a sequence a(n, k1, k2, . . . , kt)and a complexity class C, if uBit(a) ∈ C, then we say that the sequencea(n, k1, k2, . . . , kt) is weakly-definable in C.

We make use of the well-known Schwartz-Zippel-deMillo-Lipton Lemma.

Lemma 1 ([DL78,Sch80,Zip79]). Let A be an arbitrary nonempty subsetof the field F. Then for any nonzero polynomial f ∈ F[X] of degree d,Pr[f(a1, a2, . . . , an) = 0] ≤ d

|A| , where the ais are picked independently and uni-formly at random from A.

Finally, we require the following completeness result for permanent:

Proposition 1 (Proposition 2.17 in [Bur00]). Suppose char(F) 6= 2. As-sume the polynomial g ∈ F[x1, . . . , xn, y1, . . . , ym] can be computed by a formulaof size < e that uses constants ∈ A ⊂ F. Put f(X) =

∑a∈0,1m g(X, a).

Then there exists a digraph G of size |G| ≤ 6e with edge weights in A ∪−1,− 1

2 , 0,12 , 1 ∪ x1, . . . , xn such that f = per(G). In other words, f is a

projection of per6e.

3 Representing Arithmetic Circuits by Boolean Circuits

For the set x1, x2, . . . , xn∪−1, 0, 1∪+,×, /, we assume that we have fixedsome naming scheme that assigns to each element an O(log n) bit binary string,

which is called a type. We assume that all gates in a circuit have been labeledby unique binary strings that also specify the type.

Definition 1. A representation of a constant-free arithmetic circuit Φ usingdivisions by constants is given by a Boolean circuit Cn that accepts precisely alltuples (t, a, b, q) such that

1. In case q = 1 (connection query), a and b are numbers of gates in Φ, b is achild of a, and a has type t.

2. In case q = 0 (type query only), a is a number of a gate in Φ, and a is oftype t.

For a constant-free arithmetic circuit using divisions by constants over nvariables that is of size poly(n), gate names can be encoded using O(log n) bits.

Definition 2. Let a(n), b(n) be two functions N→ N. For a family of arithmeticcircuits Φn, we say it is (a(n), b(n))-succinct, if there exists a non-uniformfamily of Boolean ∨,∧,¬-circuits Cn, such that Cn represents Φn, where forall large enough n, Cn has ≤ a(n) inputs and is of size ≤ b(n). As a matterof convention, if a(n) = O(log n), we drop it from the notation, and just writeb(n)-succinct.

We want to study the notion of no(1)-succinctness. We will fix some arbitrarilyslow growing function γ(n) and consider n1/γ(n)-succinctness instead. A typicalexample to think of would be γ(n) = log∗ n.

Notation 1 For the rest of the paper we let γ(n) : N → N be an unboundedmonotone function, such that ∀n, γ(n) < log log n.

Similarly to the above, we define the notion of (a(n), b(n))-succinct Booleancircuits. In this case type names are assumed to form a naming scheme for theelements of x1, x2, . . . , xn ∪ 0, 1 ∪ ∨,∧,¬,MAJ.

Regarding DLOGTIME-uniformity we refer the reader to Ref. [BIS90] foran extensive treatment. In our set-up this can be defined as follows. A poly sizeBoolean circuit family Cn is DLOGTIME-uniform, if given (n, t, a, b, q) with nin binary, we can answer the queries of Definition 1 in time O(log n) on a Turingmachine. Using standard conversions from Turing machines to Boolean circuits,observe that if a Boolean circuit family Cn is DLOGTIME-uniform, then it isO(log n log log n)-succinct, but that a converse of this does not generally hold.

We use the following results. For iterated integer multiplication oneis given n integers A1, A2, . . . , An of n bits each, and the problem is to com-pute the bits of A1A2 . . . An. For the problem division, one is given two n bitnumbers A and B and the required output is bA/Bc. Hesse, Allender and Bar-rington [HAB01] show that both these computational tasks can be performedwith DLOGTIME-uniform TC0 circuits. For the problem of iterated integeraddition it is well-known it is in DLOGTIME-uniform TC0, cf. [Vol99].

Next we prove several technical lemmas that deduce consequences from theassumption that pern has n1/γ(n)-succinct arithmetic circuits. First however,

we prove an easy proposition which shows that we can assume that for a constant-free arithmetic circuit using divisions by constants only the output gate computesa division.

Proposition 2. Let Φ be a (t, t′)-succinct constant-free arithmetic circuit usingdivision by constants of size s and depth d computing f ∈ Q[X]. Then there ex-ists an equivalent (O(t), O(t′))-succinct constant-free arithmetic circuit Φ′ usingdivisions by constants of size O(s2) and depth 2d that has a single division-by-constant gate p/c at the output such that f = p/c for some c ∈ Z\0. Further-more, for the gate u computing c we have that the subcircuit rooted at u does notcontain variables.

Proof. For a gate u in an arithmetic circuit let us denote the polynomial itcomputes by [u]. The arithmetic circuit Φ′ is obtained by allocating for eachgate u in Φ a triple of gates (u1, u2, u3) in Φ′ with the following properties:

– [u1] ∈ Z[X] and [u2] ∈ Z\0.– [u] = [u1]/[u2].– [u3] = [u1](0).– The subcircuits rooted at u2 and u3 do not contain variables.

The construction goes as follows. The above properties are easily establishedusing structural induction, which we do not write in full, but we will makesome remarks when appropriate. For an input gate labeled with xi we create atriple of gates computing (xi, 1, 0). Similarly, for an input gate labeled with c ∈−1, 0, 1 the triple is (c, 1, c). For a ×-gate u in Φ with inputs g1, g2, . . . , gk, say(g1

1 , g12 , g

13), . . . , (gk1 , g

k2 , g

k3 ) are corresponding triples already created. We then

allocate a triple of gates in Φ′ computing [g11 ][g2

1 ] . . . [gk1 ], [g12 ][g2

2 ] . . . [gk2 ], and[g1

3 ][g23 ] . . . [gk3 ]. Similary, if u is an addition gate we first allocate a layer of k-ary

product gates computing [gj1] ·∏i∈[k],i6=j [g

i2] for each j ∈ [k]. Then we allocate

a gate computing the sum of these. For the second component of the triplewe compute [g1

2 ][g22 ] . . . [gk2 ]. For the third component of the triple we compute∑

j∈[k][gj3] ·∏i∈[k],i6=j [g

i2]

For a division gate with inputs g and h, say corresponding triples are(g1, g2, g3) and (h1, h2, h3). Then we allocate gates computing the triple([g1][h2], [g2][h3], [g3][h2]). Note that as division is allowed only if [h] is a constant,and we are given by induction hypothesis that h2 ∈ Z\0 and [h] = [h1]/[h2],we get that [h1] ∈ Z. Therefore [g2][h1] = [g2][h3]. Similarly, we have that[g1][h2](0) = [g1](0)][h2] = [g3][h2].

Note that for the second an third component of each triple inputs are nevertaken from a first component. This means that subcircuits rooted at such nodesdo not contain variables.

Finally, say for the output gate u we have created the triple (u1, u2, u3). Thenwe add a single division gate computing [u1]/[u2].

The construction we have described yields a circuit of size O(s2) and depth2d. If we have a Boolean circuit of size t′ with t inputs representing Φ we caneasily obtain a representation for Φ′ using O(t) input bits and of size O(t′). ut

Lemma 2. Assume that for some constant c0 > 0, pern can be computed byn1/γ(n)-succinct size nc0 depth d constant-free arithmetic circuits using divisionby constants, Then for some constants d′, d′′, we can compute pern(M) over Z,where entries of M are in 0, 1 by O(n1/γ(n))-succinct TC0 circuits of depthd′ · d+ d′′.

Proof. First we apply Proposition 2 to obtain two constant-free arithmetic cir-cuits computing f ∈ Z[X] and c ∈ Z\0 such that pern = f/c. The circuitcomputing c does not contain variables. We can bound the size and depth ofthese circuits by s := O(n2c0) and 2d, respectively. They can be representedby Boolean circuits of size O(n1/γ(n)). By a straightforward induction we canbound the magnitude of values at gates in these circuits for 0, 1 inputs by 2s

2d.

The value of pern(M) is less than 2n logn. Let m = max(2n2, 2n

c0(2d+1)). In order

to compute the permanent we evaluate both depth 2d arithmetic circuit, wherearithmetic is done modulo m. To do this we use that addition and multiplica-tion can be done in DLOGTIME uniform-TC0 [HAB01]. Computing remaindersmod m is easily done by discarding bits of too large significance. Finally usingthe fact that integer division can be done in DLOGTIME-uniform TC0 [HAB01],we add circuitry for computing the division f(M)/c.

As remarked on before, DLOGTIME uniformity implies O(log n log log n)-succinctness. Regarding succinctness of the circuits obtained this way, we repre-sent the resulting circuits by modifying the Boolean circuit family representingthe original arithmetic circuit for pern with additional circuitry of at mostO(log n log log n) many gates.

ut

Using an improvement of Valiant’s completeness result by Zanko [Zan91], cf.[All99], who shows that 0, 1-pern over Z is complete for #P under DLOGTIMEuniform-AC0 reductions, one can now easily verify the following statement:

Lemma 3. Assume that for some constant c0 > 0, pern can be computed byn1/γ(n)-succinct size nc0 depth d constant-free arithmetic circuits using divisionby constants. Then for any F ∈ GapP there exists constants d′, d′′ and c′ ≥ 1such that F can be computed by nc

′/γ(n)-succinct depth d′ · d+ d′′ TC0 circuits.

The following corollary is now easy, and is left as an exercise:

Corollary 1. If for some constant c0 > 0, pern can be computed by n1/γ(n)-succinct size nc0 depth d constant-free arithmetic circuits using division by con-stants, then CH/poly ⊆ nonuniform-TC0.

Finally, we need the following lemma.

Lemma 4. Assume for some constant c0 > 0, pern can be computed byn1/γ(n)-succinct size nc0 depth d constant-free arithmetic circuits using divisionby constants. Let F : 0, 1∗ → 0, 1∗ be a GapP function, and let c′ and d′, d′′

be the constants provided by Lemma 3 for this F . Let (An) be an integer se-quence of bit size at most `(n) that is weakly-definable in CH/poly. If it holds

that `(n)c′/γ(`(n)) = nO(1) and log `(n) ≤ nO(1), then bn := F (An) is an integer

sequence of bit size poly(`(n)) weakly-definable in CH/poly.

Proof. We use the technique of ‘scaling up to the counting hierarchy’ as in[Bur09]. First we use Lemma 3 to get a poly(`(n)) size depth d′ ·d+d′′ circuit Cnwith majority and negation gates for computing F on inputs of size `(n). Further-more, there is a Boolean circuit Bn which can answer connectivity queries of thiscircuit. This circuit has O(log `(n)) inputs and is of size `(n)c

′/γ(`(n)) = nO(1).For 0 ≤ i ≤ d + d′, let Li be the language of tuples (G, 1n, b) such that thecondition “G is the name of a gate on level i in Cn. It outputs b when Cn isgiven input An” holds. We will prove by induction on i that Li ∈ CH/poly.

For the base case i = 0 this is clear, since uBbit(An) ∈ CH/poly. Now assumeLi ∈ CH/poly. By Toran’s [Tor91] charaterization of the counting hierarchy itsuffices to show that we can decide Li+1 in PPLi/poly. This is done as follows.We assume the gate G is of majority type. Negation gates are handled similarly.Let N be an nondeterministic Turing machine that on input (G, 1n, b) nonde-terministically guesses the O(c log `(n)) = poly(n) size name of a gate H, usesthe advice Bn to check that H → G is a wire in Cn. If this is not true, non-deterministically flip a bit b′ and accept if b′ = 1, reject if b′ = 0. Otherwise,

query (H, 1n, b)?∈ Li. Accept if the answer to this query is yes, reject otherwise.

Observe that N accepts on the majority of its nondeterministic guesses iff themajority of the inputs to G are outputting b in Cn(An). ut

4 Conditional Lower Bound for Permanent

In this section we will pose a derandomization hypothesis for the polynomialidentity testing problem, and show that if the hypothesis can be met, then alower bound for the permanent follows.

Given an integer sequence an(i, j) with 0 ≤ i < n, 0 ≤ j < p(n) for somefunction p(n), we think of this as encoding a collection Hn of subsets Hn ⊆ Znwith |Hn| = p(n), where Hn = (an(0, j), an(1, j), . . . , an(n − 1, j)) : 0 ≤ j <p(n).

Working Hypothesis 1 There exists an integer sequence an(i, j) of polyno-mial bit size with 0 ≤ i < n, 0 ≤ j < p(n), with p polynomially bounded, suchthat uBit(an(i, j)) can be decided by n1/γ(n)-succinct TC0 circuits, and for whichthe following holds:

– For any constant-free arithmetic circuit Φ of size n over m ≤ n variables, ifΦ(x1, x2, . . . , xm) computes a nonzero polynomial, then there exist 0 ≤ j <p(n) such that Φ(an(0, j), an(1, j), . . . , an(m− 1, j)) 6= 0.

The following is our randomness-to-hardness theorem:

Theorem 2. If Working Hypothesis 1 is true, then there does not exist a polyno-mial p(n) and integer constant d such that pern has size p(n) depth d constant-free arithmetic circuits using division by constants, where in addition these cir-cuits are n1/γ(n)-succinct.

Proof. We will argue by contradiction, hence we start by assuming that for someconstant c0 > 0, pern can be computed by a family Φn of size nc0 depthd constant-free arithmetic circuits using division by constants. Furthermore, weassume that for all large enough n, these circuits are represented by a family ofBoolean circuits Cn with O(log n) inputs and size n1/γ(n). Hence by Corol-lary 1, we have the collapse CH/poly ⊆ nonuniform-TC0. We proceed as in Ref.[Agr05] by using the black-box derandomization assumption to construct a hardpolynomial by solving a system of linear equations.

Let an(i, j) be the integer sequence given by Working Hypothesis 1. LetHn be given as mentioned in the remark before Working Hypothesis 1. Letk be such that we can take p(n) ≤ nk in Working Hypothesis 1, and let c besuch that the bit size of any integer an(i, j) is bounded by nc. Choose positiveε < min(k−1, c−1). Let N = bnεγ(n)c, and let m = dγ(n) log ne. Then Nk < 2m.Let

fm =2m−1∑e=0

cm(e)xe11 xe22 . . . xemm , (1)

where ej denotes the jth bit of e. In other words, we sum over all strings in0, 1m in the above. We want to take cm(e) to be a nonzero integer solution tothe system (S) given by

f(b) = 0, for all b ∈ HN .

These are at most Nk linear equations in 2m variables. Since 2m > Nk, wecan get a nonzero solution. The system (S) is slightly too large to manipulatedirectly. Coding (S) as an integer sequence weakly-definable in CH/poly willallow us to indirectly let a solution finding procedure act on (S), due to Lemma 4.For this purpose we think of (S) as presented by a 2m × 2m matrix MS (with2m − Nk zero rows). We code MS as an integer sequence by letting An be theinteger represented by the binary string 12m01r0list(MS), where r := N cm isan upper bound on the maximum bit length of entries of M , and list(MS) isthe concatenation of length r binary representations of the entries of MS (sayleft-to-right, top-to-bottom). Define `(n) to be the bit length of An.

4.1 Checking that (An) is Weakly-Definable in CH/poly

The goal of this subsection is to prove the following lemma:

Lemma 5. Given that γ(n) is an unbounded monotone nondecreasing functionsuch that for all n, γ(n) < log log n, we have that (An) is an integer sequenceof bit size `(n), where for all but finitely many n, `(n) ≤ n4γ(n). Furthermore,(An) is weakly-definable in CH/poly.

Proof. For the bit size of An we can give an upper bound of 2m + 2 + N cm +22mN cm = O(nγ(n)(2+εc)dγ(n) log ne). This is at most n4γ(n), provided n is largeenough. Next we show that (An) is weakly-definable in CH/poly.

On input (1n, i, b), we assume that 1m = 1dγ(n) logne and N = bnεγ(n)c (inbinary) are given as advice (for technical convenience). Similarly, we assume thatthe bitlength r = N cm (in binary) in the coding of MS is given as advice. Fromthe binary index i, we can then easily compute e ∈ 0, 1m, j with 0 ≤ j < p(n),and k ≤ r such that (An)i equals the kth bit of dN (j, e), where dN (j, e) :=∏m−1p=0 aN (p, j)ep−1 .

Claim 1 There exist a TC0-circuit family Dn such that

– |Dn| = nO(γ(n)).– The family Dn is (O(γ(n) log n), 3nε)-succinct.– Dn has input gates so it can be given e ∈ 0, 1m, 0 ≤ j < p(n) in binary,

and k ≤ r in binary.– Dn(e, j, k) outputs the kth bit of dN (j, e).

Proof. Recall that uBit(an) = (1n, i, j, k, b) : the kth bit of an(i, j) equals b.Working Hypothesis 1 gives that we have a (O(log n), n1/γ(n))-succinct familyEn of TC0 circuits, such that En(1n, i, j, k, b) = 1 iff the kth bit of an(i, j)equals b. By adding O(log n) circuitry to the nonuniform circuits representingEn, we can hardcode the first n inputs to the circuit En to be 1, so let usassume wlog. En takes input tuple (i, j, k, b) instead.

Let us now describe the circuit Dn. We take mr copies of EN by rang-ing over all values for 0 ≤ i ≤ m − 1 and 1 ≤ k ≤ r, and hardcodingthese values into the corresponding copy. One single copy of EN is of sizeNO(1) = nO(γ(n)), and it is represented by Boolean circuits with O(γ(n) log n)many inputs of size N1/γ(N) ≤ nεγ(n)/γ(N) ≤ nε. We have that mr = N cm2 =nO(γ(n))(dγ(n) log ne)2 = nO(γ(n)). Now the important observation is that we cando this mr-fold duplication by adding O(logmr) = O(γ(n) log n) bits to gatenames, and adding poly(logmr) = polylog(n) to the size of the Boolean circuitthat represents the individual copy. This way we construct TC0 circuits that are(O(γ(n) log n), 2nε))-succinct that computes m many r-bit sequences encodingaN (0, j), aN (1, j), . . . , aN (m− 1, j). Within the succinctness constraints, we ob-tain m many r-bit strings giving aN (0, j)e0 , aN (1, j)e1 , . . . , aN (m− 1, j)em−1 , byjust masking using the bits e0, e1, . . . , em−1. We add below this the DLOGTIME-uniform TC0 circuits that compute iterated multiplication from Ref. [HAB01].Note the input size in this case is nO(γ(n)). This means we have a Boolean circuitof size O(γ(n) log n(log γ(n) + log log n)) with O(γ(n) log n) inputs representingthe circuit for iterated multiplication. Merging this representation with the rep-resentation of the circuit computing aN (0, j), aN (1, j), . . . , aN (m−1, j) can easilybe done to yield a (O(γ(n) log n), 3nε)-succinct TC0 circuit family. This ends thedescription of the circuit Dn.

To summarize, we have given TC0 circuits computing dN (j, e) of size atmost nO(γ(n)) that are represented by Boolean circuits with O(γ(n) log n) manyinputs, and having size 3nε, for all large enough n. ut

We now use the above claim to finish the proof of Lemma 5. Similar to theproof of Lemma 4, we use the circuit family Dn to scale up to the counting

hierarchy. The Boolean circuits of size 3nε with O(γ(n) log n) many inputs thatrepresent this circuit family are given as advice. Let d′ be a bound on the depthof the TC0 family Dn.

For 0 ≤ i ≤ d′, let Li be the language of tuples (G, 1n, (e, j, k), b) such thatthe condition “G is the name of a gate on level i in Dn. It outputs b when Dn

is given input (e, j, k) with e ∈ 0, 1m, 0 ≤ j < p(n) in binary, and k ≤ r inbinary, where m = dγ(n) log ne, N = bnεγ(n)c, r = N cm” holds.

We will prove by induction on i that Li ∈ CH/poly. Consider the in-put (G, 1n, (e, j, k), b). Note that it is of length n + O(γ(n) log n) = n +O(log n log log n), where n bits of this input are used for expressing n in unary.Note that for technical convenience we can again assume N , r (in binary) and1m are given as advice. Names of gates in Dn have length O(γ(n) log n) =O(log n log log n). We have a size nε Boolean circuits Bn with O(γ(n) log n) =O(log n log log n) that can answer connection queries about Dn. This will be partof the advice as well.

For the base case i = 0, it is easy to check which variable the gate G islabeled with, since for a gate labeled with a variable x`, ` (in binary of lengthO(log n log log n)) is part of the gate name. Then one just need to fetch the `thbit of (e, j, k). Note that (e, j, k) has length O(γ(n) log n), so we can just scanthe input string and keep a counter. Gates labeled by Boolean constants aredealt with even more easily as these constants appear in the gate name itself.To check whether G is on level 0 we can either assume wlog. this informationcan be obtained from the gate name (or if we want we can add another level oforacle calling to the below argument by making existential queries to Bn of theform “Does there exist H such that G is a child of H?”), so we will ignore thisdetail.

Now assume Li ∈ CH/poly. By Toran’s [Tor91] charaterization of the count-ing hierarchy it suffices to show that we can decide Li+1 in PPLi/poly. This isdone as follows. We assume the gate G is of majority type. Negation gates arehandled by a straightforward extension of the below argument. Let N be thefollowing nondeterministic Turing machine. On input (G, 1n, (e, j, k), b) nonde-terministically guesses the O(γ(n) log n) = O(log n log log n) size name of a gateH, uses the advice Bn to check that H → G is a wire in Cn. If this is not true,nondeterministically flip a bit b′ and accept if b′ = 1, reject if b′ = 0. Other-

wise, query (H, 1n, (e, j, k), b)?∈ Li. Accept if the answer to this query is yes,

reject otherwise. Observe that N accepts on the majority of its nondeterministicguesses iff the majority of the inputs to G are outputting b in Dn(e, j, k). Themachine N runs in nondeterministic time O(n). This puts Li+1 in PPLi/poly.

The conclusion is that the language consisting of all tuples (1n, j, e, k, b) sub-ject to the condition

– The kth bit of dN (j, e) = b, where N = bnεγ(n)c, and– with k ≤ r, 0 ≤ j < p(n), and e ∈ 0, 1m,

is in CH/poly. Hence, by the remarks before Claim 1, we get that (An) is weakly-definable in CH/poly. ut

4.2 Coefficients of fm Are Weakly-Definable in CH/poly

Next we want to apply a solution finding procedure to (S). For this purpose,we let F : 0, 1∗ → 0, 1∗ be the following poly-time computable mapping:On input x of length n, try to parse x = 12m01r0y, for some integer r, m andy ∈ 0, 1∗ with |y| = 22mr. If this fails, output 0. Otherwise, construct the2m × 2m matrix M whose (left-to-right, top-to-bottom) entries are given byconsecutive r bit blocks of y. Then using standard tools8, try to compute anonzero integer 2m-vector c such that Mc = 0. Output c if this succeeds, 0otherwise. We define α to be the absolute integer constant such that for all largeenough n, F runs within time nα. This implies that for all large enough n, forany x ∈ 0, 1n, |F (x)| ≤ nα.

By Lemma 5, (An) is weakly-definable in CH/poly. Clearly, since γ(n) isassumed to be a monotone growing function, for any constant c′ ≥ 1, it is satisfiedthat `(n)c

′/γ(`(n)) = nO(1), as for all but finitely many n, n ≤ `(n) ≤ n4γ(n).Hence we can apply Lemma 4. We get that F (An) is an integer sequence weakly-definable in CH/poly. We have that F (An) encodes a 2m-vector of integers.Indexing this vector by e ∈ 0, 1m, we let cm(e) be the integer encoded by theeth entry, i.e. cm(e) = F (An)e, and let fm be the polynomial given by fixingthese integer coefficients in Equation (1). We have the following properties:

– ∀∞n, cm(e) has bit size at most `(n)α ≤ 24αγ(n) logn ≤ 24αm.– L := (1n, e, j, b) : e ∈ 0, 1m, j ∈ 0, 14αm, and cm(e)j = b, where m =dγ(n) log ne ∈ CH/poly.

4.3 Finishing Up using the Completeness of Permanent

First we prove the following claim:

Claim 2 fm = perm′(M ′) for m′ = 2o(m), where M ′ is a matrix whose entriesare in x1, x2, . . . , xm ∪ −1,− 1

2 , 0,12 , 1 ∪ 2

20, 221

, . . . , 224αm.

Proof. Recall the bit sizes of cm(e) are bounded by 24αm. Let cm(e)idenote the ith bit of cm(e), and let sm(e) be the sign bit of cm(e).Write fm =

∑2m−1e=0

((−1)sm(e)

∑24αm−1i=0 cm(e)i2i

)xe11 x

e22 . . . xemm . Since L ∈

nonuniform-TC0 ⊆ nonuniform-NC1, we have a Boolean formula Cm for com-puting cm(e)i, given i in binary of length 4αm bits and e ∈ 0, 1m and n inunary of size poly(4αm+ n+ logm) = nO(1). The depth of Cm is O(log n) andthe fan-in can be assumed to be bounded by two for each gate.

8If M is of full rank 0 must be returned. Otherwise, one way to compute c, whileavoiding divisions so the result is integer, would be to first compute a set of independentrows of M , say with indices in I. Then form an invertible matrix M ′ by incrementallyextending this set with standard basis vectors. Now a nonzero integer solution canbe selected to be any column with index 6∈ I of the adjugate adj(M ′) of M ′. Thelatter matrix is defined entirely in terms of determinants of minors of M ′ and satisfiesM ′adj(M ′) = det(M ′)I.

Fact 1 Since γ(n) is an unbounded monotone function and m = dγ(n) log ne,we can say that the family Cm of Boolean formulas has depth o(m).

Hardcoding the unary input 1n into the formula Cm, and performing astraightforward arithmetization where we replace (x ∨ y) by x+ y−xy, (x ∧ y)by xy, and ¬x by (1 − x), we obtain an arithmetic circuit Ψ ′m on inputsα1, α2, . . . , αm, β1, β2, . . . , β4αm of depth o(m) that coincides with Cm on 0, 1-inputs. Similarly, we obtain an arithmetic formula Ψ of depth o(m) that coincideswith (−1)sm(e)Cm(e, i) on 0, 1-inputs. By duplicating nodes, we can assume Ψ isa formula of depth o(m) and size 2o(m). Hence

fm =∑

e∈0,1m

∑i∈0,14αm

Ψm(e, i)2nat(i)

xe11 xe22 . . . xemm ,

where nat(i) ∈ N denote the integer represented by the binary string i ∈0, 14αm. It is now easy to write f as an exponential sum of the form

fm =∑

b∈0,1s(m)

Γm(x1, x2, . . . , xm, b, 220, 221

, . . . , 224αm−1),

for 2o(m) size arithmetic formula Γm and s(m) = Θ(m). Namely, take

Γm(x1, x2, . . . , xm, α1, α2, . . . , αm, β1, β2, . . . , β4αm, δ1, δ2, . . . , δ4αm) =

Ψm(α1, α2, . . . , αm, β1, β2, . . . , β4αm)4αm∏j=1

(βjδj + 1− βj)m∏j=1

(αjxj + 1− αj)

Then ∑e∈0,1m

∑i∈0,14αm

Γm(x, e, i, 220, 221

, . . . , 224α−1) =

∑e∈0,1m

∑i∈0,14αm

Ψm(e, i)4αm∏j=1

(ij22j−1+ 1− ij)

m∏j=1

(ejxj + 1− ej) =

∑e∈0,1m

∑i∈0,14αm

Ψm(e, i)

4αm∏j=1

22j−1ij

xe11 xe22 . . . xemm =

∑e∈0,1m

∑i∈0,14αm

Ψm(e, i)(

2P4αmj=1 2j−1ij

)xe11 x

e22 . . . xemm =

∑e∈0,1m

∑i∈0,14αm

Ψm(e, i)2nat(i)xe11 xe22 . . . xemm = fm.

By Proposition 1, the above implies fm = perm′(M ′) for m′ = 2o(m),where M ′ is a matrix whose entries are in x1, x2, . . . , xm ∪ −1,− 1

2 , 0,12 , 1 ∪

220, 221

, . . . , 224αm−1. This proves the claim. ut

We can now finish the proof of Theorem 2. Applying Proposition 2 to theconstant-free arithmetic circuit Φm′ computing perm′ gives a constant-free arith-metic circuit of size s := O((m′)2c0) = 2o(m) computing am′ · perm′ , for someam′ ∈ Z\0, together with a constant-free arithmetic circuit of size 2o(m) com-puting am′ . Consider f ′m = 2m

′am′ ·fm. Observe that f ′m is a nonzero polynomial

in m << N variables that vanishes on HN . Hence we have that

Fact 2 f ′m does not have constant-free arithmetic circuits of size N .

By Claim 2, we have that f ′m = am′ · perm′(2M ′). We can put together 2M ′ us-ing O((m′)2) = 2o(m) constant-free circuitry. Combining with the constant-freearithmetic circuits computing am′ , we obtain s′ := 2o(m) size constant-free arith-metic circuits for f ′m Recalling that N = b2εγ(n) lognc and m = dγ(n) log ne, wehave that for all but finitely many n, s′ < N . We have arrived at a contradictionwith Fact 2. ut

5 Proving a Weak Derandomization HypothesisUnconditionally

We don’t know how to prove Working Hypothesis 1, but as we will see in Sec-tion 6, neither do we need to for obtaining the sought after lower bound forthe permanent as in the conclusion of Theorem 2! What we can establish is thefollowing theorem. Note that there is no restriction to constant depth.

Theorem 3. There exists an integer sequence an(i) of polynomial bit size with0 ≤ i < n, such that an(i) is weakly-definable in the polynomial hierarchy, andfor which the following holds:

– For any arithmetic circuit Φ of size n over m ≤ n variables and constants inΓn := 220

, 221, . . . , 22n∪−1, 0, 1, if Φ(x1, x2, . . . , xm) computes a nonzero

polynomial, then Φ(an(0), an(1), . . . , an(m− 1)) 6= 0.

Proof. Let Cn be the set9 of all arithmetic circuits of size n over m ≤ nvariables using constants in Γn. For some constant c > 0, we can bound|Cn| ≤ 2cn

2, provided n is large enough. Note that circuits in Cn can com-

pute polynomials with degree at most 2n. Let S = 1, 2, . . . , 2n3+n. Ifwe pick s1, . . . , sn independently and uniformly at random from S, thenby Lemma 1, for any nonzero polynomial f computed by a circuit in Cn,Pr[f(s1, s2, . . . , sn) = 0] ≤ 2n/2n

3+n = 2−n3. Hence by the union bound,

Pr[∃ nonzero f computed by a circuit ∈ Cn, f(s1, s2, . . . , sn) = 0] ≤ 2cn2−n3

<1. This means there exist at least one s ∈ Sn, such that for any nonzeropolynomial f computed by a circuit from Cn, f(s1, s2, . . . , sn) 6= 0. For s =(s1, s2, . . . , sn) ∈ Sn define the predicate P(s) by

∀Ψ ∈ Cn, (if Ψ(s1, s2, . . . , sn) = 0, then ∀t ∈ Sn, Ψ(t1, t1, . . . , tn) = 0). (2)9If f is computed by a circuit in the class Cn, then it is defined over m ≤ n variables,

but in our notation we will still write f(s1, s2, . . . , sn) with it being understood thatsm+1, sm+2, . . . , sn are not being used.

Observe that if P(s) is true, then for any nonzero f computed by a circuitfrom Cn, f(s1, s2, . . . , sn) 6= 0. Also observe that for both universal quantifiers in(2), they range over sets whose elements can be described by poly(n) size strings,assuming encodings of circuits using type names of size O(log n) as defined inSection 3. Hence if we can argue that given Ψ ∈ Cn and u ∈ Sn, we can decidewhether Ψ(u1, u2, . . . , un) ?= 0 in PH, then the predicate P(s) is PH-decidable.Consequently, we can define an to be the lexicographically least element s of Sn

such that P(s) holds. This makes an computable in PPH by binary search. Thisimplies that an is weakly-definable in PH.

We complete the proof by showing that given Ψ ∈ Cn and u ∈ Sn, we candecide whether Ψ(u1, u2, . . . , un) ?= 0 in PH. The only problem that may arisewhen evaluating Ψ(u1, u2, . . . , un) is that intermediate values get too large. Forthis we employ the idea of Ibarra and Moran [IM83] by evaluating modulo aprime number in some large range.

Namely, consider primes p in the range 1, 2, . . . , 2n2. Each such prime takesn2 many bits. We can evaluate Ψ(u1, u2, . . . , un) mod p in polynomial time10.By the Prime Number Theorem, the number of primes in this range is at least

2n2

log 2n2 > 22n, provided n is large enough. By a simple induction, it follows that

the value |Ψ(u1, u2, . . . , un)| can be bounded by 222n. Hence it cannot be that all

primes ≤ 2n2

divide Ψ(u1, u2, . . . , un), if Ψ(u1, u2, . . . , un) 6= 0. In other words,

Ψ(u1, u2, . . . , un) = 0⇔

∀m ∈ 1, 2, . . . , 2n2, if m is prime, then Ψ(u1, u2, . . . , un) mod p = 0.

Agrawal, Kayal and Saxena [AKS02] proved primality testing is in polynomialtime. Hence we get that the above is a Π1P-predicate. ut

6 Proof of Theorem 1

It is sufficient to show the following claim:

Claim 3 For any unbounded monotone function γ(n) = o(log log n), there doesnot exist a polynomial p(n) such that ∀∞n, pern has size p(n) depth d constant-free arithmetic circuits using divisions by constants, where in addition these cir-cuits are n1/γ(n)-succinct.

Proof. We argue by contradiction. Let γ(n) be given as in the claim, and supposethat for some polynomial p(n), for all large enough n, pern has size p(n) depth dconstant-free arithmetic circuits using divisions by constants, where in additionthese circuits are n1/γ(n)-succinct.

10As a technical detail, we remark that for large numbers like 22n , which appear as aO(logn) type name in the encoding of Ψ , we can compute (22n mod p) by performingn repeated squarings computed modulo p.

By Lemma 2, we get that we can compute pern(M) over Z, where entriesof M are in 0, 1 by O(n1/γ(n))-succinct TC0 circuits. By Toda’s Theorem andValiant’s completeness result for pern, any L ∈ PH can be decided in poly-nomial time with a single query to the 0, 1-permanent. We can think of thisas a three stage process: first apply a function Q ∈ FP to the input x, thenobtain per(Q(x)), finally apply R ∈ FP to produce the output R(per(Q(x)).Since FP ⊆ #P, Lemma 3 implies that for some constant b, we obtain nb/γ(n)-succinct TC0-circuits for Q and R. Combining all three levels of TC0 circuitsyields TC0-circuits for L, where for some constant c depending on L, this familyis nc/γ(n)-succinct. The constant c can be picked larger than b to accommodatefor the increase in size of joining the three representations. Wlog. assume c > 1.

In particular, for the integer sequence an(i) of Theorem 3, we get thatuBit(an) can be decided by nc/γ(n)-succinct TC0 circuits, for some constantc > 1. This means that Working Hypothesis 1 is satisfied for γ′(n) := c−1γ(n)instead of γ(n). Since Theorem 2 holds for any unbounded monotone growingfunction γ(n) where ∀∞γ(n) < log log n, we can apply Theorem 2 with γ′(n)instead, to yield that there does not exist a polynomial q(n) and integer d′ con-stant such that, for all large enough n, pern has size q(n) depth d′ constant-freearithmetic circuits using divisions by constants, where in addition these circuitsare n1/γ′(n)-succinct. Since 1/γ′(n) > 1/γ(n), this contradicts the assumptionthat for all large enough n, pern has size p(n) depth d constant-free arithmeticcircuits using divisions by constants, where in addition these circuits are n1/γ(n)-succinct.

ut

7 Advice Elimination

In this section TC0 refers to the DLOGTIME uniform version of the class.QTC0

is the quasi-polynomial analogue. QP is deterministic quasi-polynomial time,and BPQP is probabilistic quasi-polynomial time.

We show how to extend Allender’s lower bound for the permanent to a lowerbound against a small amount of advice. The technique used is advice elimination- we show that if the permanent is in TC0 with a small amount (polylogarithmic)of advice, then the permanent is in fact in QTC0.

Allender [All99] showed that the Permanent does not have uniform thresholdcircuits of “fractional sub-exponential” size, where a fractional sub-exponentialfunction s is one such that for any integer k, sk(n) = o(2n). The following is aconsequence.

Theorem 4. The permanent is not in QTC0.

In order to show our advice elimination result, we use the following lemma,which is implicit in work by Trevisan and Vadhan [LS07], and follows from thefact that the permanent has instance checkers [LFKN92].

Lemma 6. If the permanent is in BPP/O(polylog(n)), then the permanent isin BPQP.

We now show how to do the advice elimination when the lower bound isagainst very weak circuits, i.e., uniform constant-depth threshold circuits. Ourproof has three stages. First we use Lemma 6 to show that if the permanent is inTC0 with polylogarithmic advice, then it is in BPQP. Next, we use the downwardself-reducibility property for SAT to show that the assumption also implies thatBPP ⊆ QP and hence by a translation argument the permanent is in QP. Finally,we use local checkability for the P-complete Circuit Value Problem (CVP) toget the collapse all the way down to QTC0. Indeed, our advice elimination worksagainst any “reasonable” class of uniform circuits between AC0 and P, but wewill not state the result in its full generality.

Theorem 5. If the permanent is in TC0/O(polylog(n)), then the permanent isin QTC0.

Proof. If the permanent is in TC0/O(polylog(n)), then the permanent is inBPP/O(polylog(n)), since TC0 ⊆ BPP. Using Lemma 6, we have that the per-manent is in BPQP. Now using, the assumption again, we have that SAT ∈P/O(polylog(n)), since SAT m-reduces to the permanent by Toda’s theorem[Tod91], and TC0 ⊆ P. SAT ∈ P/O(polylog(n)) implies that NP ⊆ QP [BU92],and hence PH ⊆ QP. Since BPP is in the polynomial hierarchy [Lau83,Sip83],we get that BPP ⊆ QP, from which it follows by translation that BPQP ⊆ QPand hence the permanent is in QP.

Now using the assumption a third time, we have that CVP ∈TC0/O(polylog(n)), since TC0 ⊆ P and CVP m-reduces to the permanent via anAC0 reduction [Zan91,All99]. We will show that this implies that CVP ∈ QTC0

and hence QP ⊆ QTC0. The basic idea is to use local checkability of CVP toverify that the answer given by a certain advice string for an instance is correct.More specifically, suppose < C, x > is the input. Given an advice string a, wewould like to verify that the uniform TC0 circuit with advice a gives the cor-rect answer on < C, x >. To do this, it is enough to check that for every gateg of C with input gates g1 and g2 (which might be constants or literals), theanswer given by the TC0 circuit with advice a on < g, x > is consistent with theanswers given on < g1, x > and < g2, x > (or with the values for g1 and/or g2specified by x, if g1 and/or g2 are constants or literals). Such a check is what wecall a local check. Note that these are also questions about CVP. We assumean encoding of CVP in which inputs can be padded so that it makes sense touse advice a for all questions about smaller sub-circuits. Note that if an advicestring a is the correct one for length | < C, x > | , then all the local checks oninput < C, x > succeed, whereas if an advice string gives the wrong answer for< C, x > at least one of the local checks must fail. All the local checks for all(quasi-polynomially many) possible advice strings can be performed in parallelby uniform TC0 circuits. If there is some advice string for which all the localchecks succeed and the advice-taking circuit outputs 1, then the uniform circuitoutputs 1, otherwise it outputs 0. The uniform circuits are of depth at most 2greater than the advice-taking TC0 circuits presumed to compute CVP, andthey are of quasi-polynomial size, hence we have that P ⊆ QTC0. Whence by

a translation argument, QP ⊆ QTC0. Together with our earlier conclusion thatthe permanent is in QP, we have that the permanent is in QTC0.

Using Theorem 5 in conjunction with Theorem 4 and Lemma 2, we have thefollowing corollary.

Corollary 2. The permanent does not have O(polylog(n))-succinct constant-depth arithmetic circuits of polynomial size.

Since the advice elimination argument requires multiple use of the assump-tion, we do not know how to use it to rule out no(1)-succinct circuits, which iswhat is done in the main result of this paper.

8 Conclusion

Let us briefly discuss some open problems. One obvious task would be to improvethe parameters of our result. As a matter of fact we have already obtainedsome recent improvements using a different framework involving high degreeunivariate polynomial identity testing [JS11]. Still, using low-degree multivariatepolynomial identity testing as the target for derandomization may be easier, butthere is the problem of connecting this up to lower bounds for permanent. Inthis paper we established this connection for the succinct circuit model.

A main question is whether more sophisticated methods for constructinghitting sets can be developed against arithmetic circuit, which of course leadsus into the general area of polynomial identity testing. However, to pose a moredirected question here, can Theorem 3 to be improved beyond the constant-freemodel to give a hitting set against circuits that use constants from some field Fof characteristic zero, e.g. the complex numbers, at unit cost ?

Acknowledgments We thank the anonymous referees for their valuable com-ments. We thank Pavel Hrubes for pointing out to us that without divisiongates a lower bound can be obtained for succinct circuits of constant depth bya reduction to the Razborov-Smolensky lower bound.

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