On Circuit Lower Bounds from Derandomization

: We present an alternate proof of the result by Kabanets and Impagliazzo (2004) that derandomizing polynomial identity testing implies circuit lower bounds. Our proof is simpler, scales better, and yields a somewhat stronger result than the original argument.


Introduction
It is well-known that the standard approach for derandomizing BPP, namely via quick pseudorandom generators for circuits, requires proving superpolynomial circuit lower bounds for EXP (see [4], for example).For the larger class promise-BPP (in fact, for MA), Impagliazzo, Kabanets and Wigderson [3] showed that any derandomization-using pseudorandom generators or not-implies superpolynomial circuit lower bounds for NEXP.Building on that result, Kabanets and Impagliazzo [5] proved that any derandomization of BPP implies superpolynomial circuit lower bounds of some kind.More specifically, they showed that if polynomial identities over the integers can be decided deterministically (or even nondeterministically) in subexponential time, then either NEXP does not have Boolean circuits of polynomial size or the permanent over Z does not have arithmetic circuits of polynomial size.Note that BPP contains the language of all polynomial identities over the integers [2], so any derandomization of BPP implies a deterministic algorithm for deciding polynomial identities over the integers.
The proof in [5] hinges on the result from [3] that NEXP ⊆ P/poly ⇒ NEXP = MA, which itself involves the use of multi-prover interactive proofs for EXP (through the result that EXP ⊆ P/poly ⇒ EXP = MA [1]), worst-case to average-case reductions for EXP (as in [1]), and the Nisan-Wigderson hardness-based pseudorandom generator construction [9,10].The proof in [5] also uses Toda's Theorem that the polynomial-time hierarchy reduces to counting [11].The work on typically-correct derandomization by Kinne, Van Melkebeek, and Shaltiel [7] includes an alternate proof that avoids the most involved ingredient of the original proof, namely the result from [3], but still uses Toda's Theorem.In this note we present a yet simpler proof that avoids both [3] and Toda's Theorem.Our proof is completely elementary modulo the use of the #P-completeness of the permanent [12].The only other nontrivial ingredients are that NTIME (2 O(n) ) has a complete problem under linear-time reductions, and Kannan's circuit lower bounds for Σ p 2 [6].Apart from its simplicity and possible didactic value, our argument also yields a somewhat stronger result than the original one.It essentially gives the same lower bound for NEXP ∩ coNEXP as [5] does for NEXP.More importantly, our argument scales better than the original one, although the one in [7] scales even a bit better.We defer the discussion of these issues until after the presentation of the original and our new argument.

Notation
We adopt the following notation for the rest of this note.PIT stands for "polynomial identity testing" and is formalized as the language of all arithmetic circuits that compute the zero polynomial over Z, where arithmetic circuits consist of internal nodes representing addition, subtraction, and multiplication, and leaves representing variables and the constants 0 and 1. PERM denotes the permanent of matrices over Z. NSUBEXP is a shorthand for ε>0 NTIME(2 n ε ).For any function s(n) we denote by SIZE(s(n)) the class of languages L such that L at length n can be decided by a Boolean circuit of size s(n) for all but finitely many input lengths n.We denote by ASIZE(a(n)) the class of families (p n ) n∈N of polynomials over Z where p n has n variables and can be computed by an arithmetic circuit of size a(n) for all but finitely many n ∈ N.
Using the above notation we can state the result from [5] as follows.

The arguments
We now present the argument from [5] and our new argument for Theorem 2.1.The arguments have a similar structure and share a common part.They can both be cast as indirect diagonalization argumentsassume the theorem fails and obtain a contradiction with a diagonalization result.We break up the arguments into two steps, where the first step corresponds to the common part.In order to maximize the commonality, we present the argument from [5] slightly differently1 than in that paper.
Step 1 Use the hypothesis that PERM has small arithmetic circuits, the checkability properties of PERM, and the hypothesis that PIT can be derandomized, to obtain an NSUBEXP algorithm for PERM and problems that efficiently reduce to PERM (such as all of Σ p 2 ).
Step 2 Use the hypothesis that NEXP has small Boolean circuits and the algorithm from Step 1 to derive a contradiction with a diagonalization result.
Step 1 is common to both arguments.Step 2 in [5] uses the result from [3] that NEXP ⊆ P/poly ⇒ NEXP = MA, whereas our alternate does not.We expand on Step 1 in Section 3.1, on Step 2 of [5] in Section 3.2, and on the new Step 2 in Section 3.3.−i,− j denote the same for k − 1 but starting from the matrix M with the i-th row and j-th column deleted.

A common lemma
We have that C correctly computes the permanent of m × m matrices over Z iff for each 1 ≤ k ≤ m, the polynomial is identically zero, as well as the polynomial C0 = C(X (0) ) − 1, where X denotes an m × m matrix of variables (x i j ) m i, j=1 .By introducing one more variable x 0 , those conditions can be expressed equivalently as whether the following polynomial is identically zero: C = ∑ m k=0 Ck • x k 0 .The straightforward implementation of C given C yields an arithmetic circuit that consists of O(m2 ) copies of C and some simple additional circuitry.That arithmetic circuit is in PIT iff C correctly computes the permanent on m × m matrices over Z.
Given Claim 3.2, we prove Lemma 3.1 as follows.
Proof of Lemma 3.1.Assuming the hypotheses of the lemma, we will show that coNP ⊆ NSUBEXP, which implies that Σ . ( Assuming that coNP ⊆ NSUBEXP, we can replace the predicate ( * ) in ( 1) by an NSUBEXP computation on the combined input x, y , which turns ( * * ) into an NSUBEXP computation on input x.
For any language L ∈ coNP there exists a function f ∈ #P such that for any input x, x ∈ L iff f (x) = 0. Valiant's proof that the permanent is #P-complete [12] implies that for any f ∈ #P there exists a polynomial-time computable mapping g onto square matrices with entries in {−1, 0, 1} such that f (x) = 0 iff PERM(g(x)) = 0. Thus, it suffices to develop an NSUBEXP-algorithm to decide whether the permanent of a given m × m matrix M with entries in {−1, 0, 1} is zero over the integers.We use the following procedure.
1. Guess a polynomial-sized candidate arithmetic circuit C for PERM on matrices of dimension m.

Verify the correctness of C. Halt and reject if the test fails.
3. Use the circuit C to determine the permanent of M and accept iff the result is 0.
The circuit in Step 1 exists by virtue of the hypothesis that PERM has polynomial-size arithmetic circuits.The combination of Claim 3.2 and the hypothesis that PIT ∈ NSUBEXP shows how to do Step 2 in NSUBEXP.In order to execute Step 3 in deterministic polynomial time, we can evaluate the arithmetic circuit C on the given input M and perform the arithmetic modulo m! + 1.The latter quantity exceeds PERM(M) in absolute value, so the outcome of the computation is zero iff PERM(M) is.Overall, the above 3-step procedure correctly decides whether PERM(M) = 0 in NSUBEXP.

The original argument
Assume by way of contradiction that the hypothesis of Theorem 2.1 holds but that (i) and (ii) fail.We have that NEXP ⊆ MA ⊆ NSUBEXP . ( The first inclusion in (2) follows from [3] by our second hypothesis (NEXP ⊆ SIZE(poly(n))).The second inclusion follows from Lemma 3.1 by our first hypothesis (PIT ∈ NSUBEXP) and third hypothesis (PERM ∈ ASIZE(poly(n))) and the fact that MA ⊆ Σ p 2 .The two inclusions in (2) combined yield a contradiction to the nondeterministic time hierarchy theorem. 2

The new argument
Assume by way of contradiction that the hypothesis of Theorem 2.1 holds but that (i) and (ii) fail.We have that for some constant c.The first inclusion in (3) follows from Lemma 3.1 by our first hypothesis (PIT ∈ NSUBEXP) and our third hypothesis (PERM ∈ ASIZE(poly(n))).The second inclusion follows from our second hypothesis (NEXP ⊆ SIZE(poly(n))) and the fact that NTIME(2 O(n) ) has a complete problem under linear-time reductions,3 e. g., { M, x,t | M is a nondeterministic Turing machine that accepts x in at most t steps} .
All together (3) yields a contradiction to the result of Kannan's that for every fixed constant c there exists a language in Σ p 2 that does not have Boolean circuits of size n c [6].

Strengthening and parameterized statement
Kannan actually showed that is not in SIZE(n c ) for any constant c.Incorporating that fact into our argument leads to a strengthening that essentially 4 allows us to replace NEXP in part (i) of Theorem 2.1 by NEXP ∩ coNEXP.
Both the original and the new argument can trade the running time of the nondeterministic algorithm for PIT and the size of the arithmetic circuits for PERM in part (ii).However, due to the use of the implication NEXP ⊆ SIZE(poly(n)) ⇒ NEXP = MA from [3], the original argument does not accommodate changes to either the right-hand side or the left-hand side of (i), whereas the new argument allows us to play with both sides.On the left-hand side, the proof in [3] can only handle time bounds that are at least exponential. 5This is true even when the running time of the nondeterministic algorithm for PIT is polynomial, in which case our argument only needs the time bound on the left-hand side of (i) to be superpolynomial.On the right-hand side, the proof in [3] can only handle circuit sizes that are polynomial;6 our proof gives nontrivial results for circuit sizes ranging from linear to linear-exponential.
We can further improve the parameterized version of Theorem 2.1 by slightly modifying our argument and incorporating Toda's Theorem [11].The strengthening and improved parameterization are captured in the following statement, which appears in [7].We use (N∩coN)TIME(τ) as a shorthand for NTIME(τ)∩ coNTIME(τ).Theorem 4.1 (Theorem 9 in [7]).Let γ(n) denote the maximum circuit complexity of Boolean functions on n inputs.There exists a constant c > 0 such that the following holds for any functions a(n), s(n), and t(n) such that a(n) and s(n) are constructible, a(n) and t(n) are monotone, and n ≤ s The instantiation of Theorem 4.1 to the setting of Theorem 2.1 yields the following statement.
We refer to [7] for the detailed analysis and formal proof of Theorem 4.1.Here we merely present a sketch of the modifications to our proof of Theorem 2.1 and an explanation of how Toda's Theorem helps.We point out that the focus in [7] lies on so-called typically-correct derandomization.In particular, [7] shows that Theorem 2.1 holds even under the weaker hypothesis that for every positive ε there exists a nondeterministic algorithm that runs in time 2 n ε and decides PIT correctly on all but 2 n ε of the inputs of length n for all but finitely many n.
The weakness of the argument without Toda's Theorem stems from the use of Kannan's result that Σ p 2 ⊆ SIZE(n c ) for any constant c, a result that does not scale as well as one might hope (see [8] for a discussion).The issue disappears when we go higher up in the polynomial-time hierarchy, where Kannan's argument generalizes to Σ for any constructible bound t(n).The proof of Lemma 3.1 shows that its statement holds even when we replace Σ p 2 by P #P , and the argument scales optimally.The fact that P #P is closed under complementation automatically leads to a simulation in (N ∩ coN)TIME(•) in the generalization of Lemma 3.1.Combining all the ingredients in a similar way as in our proof of Theorem 2.1 yields the statement of Theorem 4.1.

Lemma 3 . 1 .
If PIT ∈ NSUBEXP and PERM ∈ ASIZE(poly(n)), then Σ p 2 ⊆ NSUBEXP.The proof of Lemma 3.1 relies on the following claim, which shows how checking the correctness of a purported arithmetic circuit for the permanent reduces to PIT.Claim 3.2.There exists a polynomial-time algorithm that takes an arithmetic circuit C and an integer m, and produces an arithmetic circuit C such that C computes the permanent of m × m matrices over Z iff C ∈ PIT.Claim 3.2 follows from viewing the Laplace expansion of the permanent as a downward self-reduction.We include a proof for completeness.Proof of Claim 3.2.We use the following notation.Let M be an m × m matrix M. For 0 ≤ k ≤ m and 1 ≤ i, j ≤ k, we let M (k) denote the matrix obtained by taking the m × m identity matrix and replacing the top left k × k submatrix by the corresponding submatrix of M. Let M (k−1)

p 2 ⊆
NSUBEXP.The latter implication follows because for every L ∈ Σ p 2 there exists L ∈ coNP and a positive integer c such that for any input x:x ∈ L ⇔ (∃y ∈ {0, 1} |x| c ) x, y ∈ L

p 3 TIME
((s(n)) 2 log a s(n)) ⊆ SIZE(s(n)) and Σ p 4 TIME(s(n) log a s(n)) ⊆ SIZE(s(n)) for some constant a and any constructible bound s(n) less than the maximum circuit complexity.By Toda's Theorem there exists a constant b and a problem A ∈ #P such that Σ p 4 TIME(t(n)) ⊆ DTIME A ((t(n)) b )