Polynomial-size Circuits Research Articles

Computational Indistinguishability: A Sample Hierarchy

We consider the existence of pairs of probability ensembles that may be efficiently distinguished given k samples, but cannot be efficiently distinguished given k′<k samples. It is well known that in any such pair of ensembles it cannot be that both are efficiently computable (and that such phenomena cannot exist for nonuniform classes of distinguishers, say, polynomial-size circuits). It was also known that there exist pairs of ensembles that may be efficiently distinguished based on two samples, but cannot be efficiently distinguished, based on a single sample. In contrast, it was not known whether the distinguishing power increases when one moves from two samples to polynomially-many samples. We show the existence of pairs of ensembles which may be efficiently distinguished given k+1 samples but cannot be efficiently distinguished given k samples, where k can be any function bounded above by a polynomial in the security parameter. In the course of establishing the above result, we prove several technical lemmas regarding polynomials and graphs. We believe that these may be of independent interest.

Non-commutative arithmetic circuits: depth reduction and size lower bounds

We investigate the phenomenon of depth-reduction in commutative and non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant #LOGCFL. We show that AC 1 has no more power than arithmetic circuits of polynomial size and degree n O( log log n) (improving the trivial bound of n O( log n) ). Connections are drawn between TC 1 and arithmetic circuits of polynomial size and degree. Then we consider non-commutative computation. We show that over the algebra ( ∑ ∗ , max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O( log 2 n) depth (and even to O( log n) depth if unbounded-fanin gates are allowed). This establishes that OptLOGCFL is in AC 1. This is the first depth-reduction result for arithmetic circuits over a non-commutative semiring, and it complements the lower bounds of Kosaraju and Nisan showing that depth reduction cannot be done in the general non-commutative setting. We define new notions called “short-left-paths” and “short-right-paths” and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible. This class also can be characterized using the AuxPDA model. Finally, we characterize the languages generated by efficient circuits over the semiring ( 2 ∑ ∗ , union, concat) in terms of simple one-way machines, and we investigate and extend earlier lower bounds on non-commutative circuits.

Monotone Circuits for Connectivity Have Depth (log n)2-o(1)

We prove that a monotone circuit of size nd recognizing connectivity must have depth $\Omega((\log n)^2/\log d)$. For formulas this implies depth $\Omega((\log n)^2/\log\log n)$. For polynomial-size circuits the bound becomes $\Omega((\log n)^2)$ which is optimal up to a constant.

Forcing on bounded arithmetic II

Forcing method on Bounded Arithmetic was first introduced by J. B. Paris and A. Wilkie in [10]. Then in [1], [2] and [3], M. Ajtai used the method to get excellent results on the pigeon hole principle and the modulo p counting principle. The forcing method on Bounded arithmetic was further developed by P. Beame, J. Krajíček and S. Riis in [4], [7], [6], [8], [5], [12], [11], [13]. It should be noted that J. Krajíček and P. Pudlák used an idea of Boolean valued in [9] and also Boolean valued notion is efficiently used for model theoretic constructions in [7], [6], [8], [5].In our previous paper [14], we developed a Boolean valued version of forcing on Bounded Arithmetic using Boolean algebra which is generated by polynomial size circuits from Boolean variables and discussed its relation with NP = co-NP problem and P = NP problem. Especially we proved the following theorem and related theorems as Theorems 2, 3 and 4 in Section 2.Theorem. If M[G] is not a model of S2, then NP ≠ co-NP and therefore P ≠ NP.However in the proof of the Theorem, we used a consequence of P = NP. More precisely we used the following as a consequence of NP = co-NP, though it is a consequence of P = NP but not a consequence of NP = co-NP.Suppose that NP = co-NP holds. Then there exists an NP-complete predicate ∃x ≤ t(a) A(x,a) with sharply bounded A(x, a) and a sharply bounded B(y, a) such that ∃x ≤ t(a) A(x,a) ↔ ∀y ≤ s(a)B(y, a). Then there exists polynomial time computable functions f and g such that the following two sequents hold.

Simulating Threshold Circuits by Majority Circuits

We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size, depth-2 majority circuit. In general we show that a polynomial-size, depth-d threshold circuit can be simulated uniformly by a polynomial-size majority circuit of depth d + 1. Goldmann, Håstad, and Razborov showed in [Comput. Complexity, 2 (1992), pp. 277--300] that a nonuniform simulation exists. Our construction answers two open questions posed by them: we give an explicit construction, whereas they use a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (their simulation gives polynomial-size circuits only when d is constant).

New Collapse Consequences of NP Having Small Circuits

We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP (NP). As a consequence we get a deeper collapse of the polynomial-time hierarchy PH to ZPP(NP) under the assumption that NP has polynomial-size circuits. This improves on the well-known result in Karp and Lipton [ Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302--309] stating a collapse of PH to its second level $\Sigmap_2$ under the same assumption. Furthermore, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomial-size circuits. Finally, we investigate the circuit-size complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))-size circuits.

Observations on measure and lowness for Δ 2 P

Assuming that k≥2 and Δ k P does not have p-measure 0, it is shown that BP · Δ k P =Δ k P . This implies that the following conditions hold if Δ 2 P does not have p-measure 0. (i) AM ∩ co-AM is low for Δ 2 P . (Thus BPP and the graph isomorphism problem are low for Δ> 2 P .) (ii) If Δ 2 P ≠ PH, then NP does not have polynomial-size circuits.

On the computational power of depth-2 circuits with threshold and modulo gates

We investigate the computational power of depth-2 circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, threshold-MOD r circuits) and circuits with two levels of MOD gates ( MOD p-MOD q circuits). In particular, we will show the following results: 1. (i) For all prime numbers p and integers q, r, it holds that if p divides r but not q then all threshold-MOD q circuits for MOD r have exponentially many nodes. 2. (ii) For all integers r, all problems computable by depth-2 AND,OR,NOT circuits of polynomial size have threshold-MOD r circuits with polynomially many edges. 3. (iii) There is a problem computable by depth 3 AND,OR,NOT circuits of linear size and constant bottom fan-in which for all r needs threshold-MOD r circuits with exponentially many nodes. 4. (iv) For p, r different primes, and q ⩾ 2, k positive integers, where r does not divide q, every MOD p k -MOD q circuit for MOD r has exponentially many nodes. Results (i) and (iii) imply the first known exponential lower bounds on the number of nodes of threshold-MOD r circuits, r ≠ 2. They are based on a new method for estimating the minimum length of threshold realizations over predefined function bases, which, in contrast to previous related techniques (Goldmann et al., 1992; Bruck and Smolensky, 1990; Kailath et al., 1991; Goldmann, 1993; Grolmusz, 1993) works even if the weight of the realization is allowed to be unbounded, and if the bases are allowed to be nonorthogonal. The special importance of result (iii) consists of the fact that the known spectral-theoretically based lower bound methods for threshold-XOR circuits (Bruck and Smolensky, 1990; Kailath et al., 1991) can provably not be applied to AC 0 functions. Thus, by (ii), result (iii) is sharp. It gives a partial negative answer to the open question whether there exist simulations of AC 0 -circuits by small depth threshold circuits which are more efficient than that given by Yao's important result that ACC functions have depth-3 threshold circuits of quasipolynomial weight (Yao, 1990). Finally we observe that our method works also for MOD p-MOD q circuits, if p is a power of a prime ((iv) above); see (Barrington et al., 1990; Krause and Waack, 1991; Yan and Parberry, 1994) for related results. A preliminary version of this paper appeared in (Krause and Pudlák, 1993).

Multiple Product Modulo Arbitrary Numbers

Let n binary numbers of length n be given. The Boolean function “Multiple Product” MP n asks for (some binary representation of ) the value of their product. It has been shown (K.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems, SIAM J. Discrete Math. 7 , 285–292 (1994)) that this function can be computed in polynomial-size threshold circuits of depth 4. For many other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits of depth 2. In this paper, we investigate the complexity of computing MP n modulo m by depth-2 threshold circuits. It turns out that for all but a few integers m , exponential size is required. In particular, it is shown that for m ∈{2, 4, 8}, polynomial-size circuits exist, for m ∈{3, 6, 12, 24}, the question remains open and in all other cases, exponential-size circuits are required. The result still holds if we allow m to grow with n .

Oracles and Queries That Are Sufficient for Exact Learning

We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected polynomial time with equivalence queries and the aid of anNP-oracle. We also show that circuits are exactly learnable in deterministic polynomial time with equivalence queries and a ∑P3-oracle. The hypothesis class for the above learning algorithms is the class of circuits of larger—but polynomially related—size. Also, the algorithms can be adapted to learn the class of DNF formulas with hypothesis class consisting of depth-3 ∧-∨-∧ formulas (by the work of Angluin this is optimal in the sense that the hypothesis class cannot be reduced to DNF formulas, i.e., depth-2 ∨-∧ formulas). We also investigate the power of anNP-oracle in the context of learning withmembershipqueries. We show that there are deterministic learning algorithms that use membership queries and anNP-oracle to learn: monotone boolean functions in time polynomial in the DNF size and CNF size of the target formula; and the class ofO(logn)-DNF∩O(logn)-CNF formulas in time polynomial inn. We also show that, with anNP-oracle and membership queries, there is a randomized expected polynomial time algorithm that learns any class that is learnable from membership queries with unlimited computational power. Using similar techniques, we show the following both for membership and for equivalence queries (when the hypotheses allowed are precisely the concepts in the class); any class learnable with unbounded ?computational-power is learnable in deterministic polynomial time with a ∑:p5-oracle. Furthermore, we identify the combinatorial properties that completely determine learnability in this information-theoretic sense. Finally we point out a consequence of our result in structural complexity theory showing that if everyNPset has polynomial-size circuits then the polynomial hierarchy collapses to ZPPNP.

Completeness and Weak Completeness under Polynomial-Size Circuits

This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform reductions that are computed by polynomial-size circuits (P/Poly-Turing reductions and P/Poly-many–one reductions). A tight, exponential lower bound on the space-bounded Kolmogorov complexities of weakly P/Poly-Turing-complete problems is established. A Small Span Theorem for P/Poly-Turing reductions in ESPACE is proven and used to show thateveryP/Poly-Turing degree—including the complete degree—has measure 0 in ESPACE. (In contrast, it is known that almost every element of ESPACE is weakly P-many–one complete.) Every weakly P/Poly-many–one-complete problem is shown to have a dense, exponential, nonuniform complexity core. More importantly, the P/Poly-many–one-complete problems are shown to beunusually simpleelements of ESPACE, in the sense that they obey nontrivialupperbounds on nonuniform complexity (size of nonuniform complexity cores and space-bounded Kolmogorov complexity) that are violated by almost every element of ESPACE.

On randomized versus deterministic computation

In contrast to deterministic or nondeterministic computation, it is a fundamental open problem in randomized computation how to separate different randomized time classes (at this point we do not even know how to separate linear randomized time from O(n log n randomized time) or how to compare them relative to corresponding deterministic time classes. In other words, we are far from understanding the power of random coin tosses in the computation, and the possible ways of simulating them deterministically. In this paper we study the relative power of linear and polynomial randomized time compared with exponential deterministic time. Surprisingly, we are able to construct an oracle A such that exponential time (with or without the oracle A) is simulated by linear time Las Vegas algorithms using the oracle A. For Las Vegas polynomial time ( ZPP) this will mean the following equalities of the time classes: ZPP A = EXPTIME A = EXPTIME (= DTIME(2 poly )). Furthermore, for all the sets M ⊆ ∑ ∗ , M ⩽ UR A ̄ a ́ t M ϵ EXPTIME (⩽ UR being unfaithful polynomial random reduction, cf. [10]). Thus A ̄ is ⩽ UR complete for EXPTIME, but interestingly not NP-hard under (deterministic) polynomial reduction unless EXPTIME = NEXPTIME. We also prove, for the first time, that randomized reductions are exponentially more powerful than deterministic or nondeterministic ones (cf. [2]). Moreover, a set B is constructed such that Monte Carlo polynomial time ( BPP) under the oracle B is exponentially more powerful than deterministic time with nondeterministic oracles, more precisely, BPP B = Δ 2 EXPTIME B = Δ 2 EXPTIME (= DTIME(2 poly ) NTIME( n) ). This strengthens considerably a result of Stockmeyer [17] about the polynomial time hierarchy that for some decidable oracle B, BPP B n ́ Δ 2P B . Under our oracle BPP B is exponentially more powerful than Δ 2 P B , and B does not add any power to Δ 2 EXPTIME. One of the consequences of this result is that under oracle B, Δ 2 EXPTIME has polynomial size circuits.

Polynomial-Time Membership Comparable Sets

This paper studies a notion called polynomial-time membership comparable sets. For a function g, a set A is polynomial-time g-membership comparable if there is a polynomial-time computable function f such that for any $x_{ 1}, \dotsc , x_{m}$ with $m \geq g(\max \{|x_{1}|, \dotsc, |x_{m}|\})$, f outputs $b \in \{0,1\}^{m}$ such that $(A (x_{1}), \dotsc , A (x_{m})) \neq b$. The following is a list of major results proven in the paper: 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and any constant $c > 0$, if a set is $\leq_{f(n)-tt}^{p}$-reducible to a P-selective set:, then the set is polynomial-time $(1 + c) \log f(n)$-membership comparable. 4. For any $\mathcal{C}$ chosen from $\{ {\text{PSPACE}},{\text{UP}},{\text{FewP}},{\text{NP}},{\text{C}}_ {=} {\text{P}},{\text{PP}},{\text{MOD}}_{2} {\text{P}},{\text{MOD}}_{3} {\text{P}}, \dotsc \} $, if $\mathcal{C} \subseteq {\text{P-mc}}(c \log n)$ for some $c < 1$, then $\mathcal{C} = {\text{P}}$. As a corollary of the last two results, it is shown that if there is some constant $c < 1$ such that all $\mathcal{C}$ are polynomial-time $n^{c}$-truth-table reducible to some P-selective sets, then $\mathcal{C} = P$, which resolves a question that has been leftft open for a long time.

When Won′t Membership Queries Help?

We investigate cryptographic limitations on the power of membership queries to help with concept learning. We extend the notion of prediction-preserving reductions to prediction with membership queries. We exhibit a number of reductions and show several prediction problems to be complete for different complexity classes. We show that assuming the intractability of (1) quadratic residues module a composite, (2) inverting RSA encryption, or (3) factoring Blum integers, there is no polynomial time prediction algorithm with membership queries for Boolean formulas, constant depth threshold circuits, 3 μ-Boolean formulas, finite unions or intersections of DFAs, 2-way DFAs, NFAs, or CFGs. Also, we show that if there exist one-way functions that cannot be inverted by polynomial-sized circuits, then CNF or DNF formulas and convex polytopes intersected with the Boolean hypercube are either polynomial time predictable without membership queries, or they are not polynomial time predictable even with membership queries; so, in effect, membership queries will not help with predicting CNF or DNF formulas.

If NP has polynomial-size circuits, then MA = AM

It is shown that the assumption of NP having polynomial-size circuits implies (apart from a collapse of the polynomial-time hierarchy as shown by Karp and Lipton) that the classes AM and MA of Babai's Arthur-Merlin hierarchy coincide. This means that also a certain inner collapse of the remaining classes of the polynomial-time hierarchy occurs.

Fast Parallel Band Matrix Arithmetic

Fast parallel algorithms are presented for computation of the determinant, adjoint, characteristic polynomial, and rank of band matrices, and for the solution of systems of linear equations with band matrices as coefficient matrices. The algorithms can be implemented using arithmetic-boolean circuits of polynomial size and depth O (log n log m ), or depth O (log n log m + log n log log n ) for computations over small finite fields, where n is the order and m the band width of the matrix given as input. They can be implemented for computations over number fields and finite fields using log space uniform boolean circuits of depth O (log n log m + log n log log n ) and polynomial size, for input size n and band width m . While they are not processor efficient, these algorithms can be implemented using circuits of asymptotically smaller depth than previous algorithms for these problems.

A Note on polynomial-size circuits with low resource-bounded Kolmogorov complexity

It is well known that the class P/poly can be characterized in terms of polynomial-size circuits. We obtain a characterization of the class P/log using polynomial-size circuits with low resource-bounded Kolmogorov complexity.

Explicit Constructions of Depth-2 Majority Circuits for Comparison and Addition

All Boolean variables here range over the two-element set $\{ - 1,1 \}$. Given n Boolean variables $x_1 , \ldots ,x_n $, a nonmonotone MAJORITY gate (in the variables $x_i $) is a Boolean function whose value is the sign of $\Sigma _{i = 1}^n \varepsilon _i x_i $, where each $ \varepsilon _i $ is either 1 or $ - 1$. The COMPARISON function is the Boolean function of two n-bits integers X and Y whose value is $ - 1$ if and only if $X\geqq Y$. An explicit sparse polynomial whose sign computes this function is constructed. Similar polynomials are constructed for computing all the bits of the summation of the two numbers X and Y. This supplies explicit constructions of depth-2 polynomial-size circuits computing these functions, which use only nonmonotone MAJORITY gates. These constructions are optimal in terms of the depth and can be used to obtain the best-known explicit constructions of MAJORITY circuits for other functions like the product of two n-bit numbers and the maximum of nn-bit numbers. A crucial ingredient is the construction of a discrete version of a sparse “delta polynomial”—one that has a large absolute value for a single assignment and extremely small absolute values for all other assignments.

BPP has subexponential time simulations unlessEXPTIME has publishable proofs

We show thatBPP can be simulated in subexponential time for infinitely many input lengths unless exponential time We also show thatBPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, we showBPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages inMA-P. The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low-degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize. One of the ingredients of our proof is a lemma that states that ifEXPTIME has polynomial size circuits thenEXPTIME=MA. This extends previous work by Albert Meyer.

On the Computational Complexity of Small Descriptions

For a set L that is polynomial-time reducible (in short, $ \leqslant _{\text{T}}^{\text{P}} $-reducible) to some sparse set, the authors investigate the computational complexity of such sparse sets relative to L and prove the first lower bounds on the complexity of recognizing such sets. Sets A and B are constructed such that both of them are $ \leqslant _{\text{T}}^{\text{P}} $-reducible to some sparse set, but A (respectively, B) is $ \leqslant _{\text{T}}^{\text{P}} $-reducible to no sparse set in ${\text{P}}^A $ (respectively, ${\text{NP}}^B \cap {\text{co-NP}}^B $); that is, the complexity of sparse sets to which A (respectively, B) is $ \leqslant _{\text{T}}^{\text{P}} $-reducible is more than ${\text{P}}^A $ (respectively, ${\text{NP}}^B \cap {\text{co-NP}}^B $). From these results or application of the proof technique the authors obtain (1) lower bounds for the relative complexity of generating polynomial-size circuits for some sets in ${{\text{P}} / {{\operatorname{poly}}}}$ and (2) separations of the classes of sets equivalent or reducible to sparse sets under various polynomial-time reducibilities.