D. Mix Barrington, H. Straubing, and D. Th'erien, "Nonuniform Automata over Groups", Information and Computation 89 (1990) 109-132.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a cyclic group H (see Section 2) in this case the group G being a finite Abelian group. Then in Section 3 a homomorphic cryptosystem is yielded for an arbitrary H, in this case the group G being a free product of certain Abelian groups produced in Section 2. In Section 4 we recall the result from [1] designing a polynomial size simulation of any boolean circuit B of the logarithmic depth over an arbitrary unsolvable group H (in particular, one can take H to be the symmetric group Sym(5) Combining this result with Theorem 1.3 provides an encrypted simulation of B over the group G: the output ....

....that the problem INVERSE(P ) is polynomial time reducible to the problem INVERSE(P ) Thus claim (b) follows from statement (i4) of Lemma 3.2. Theorem 1.3 is proved. 4 Encrypted simulating of boolean circuits Let B = B(X 1 ; X n ) be a boolean circuit and H be a group. Following [1] we say that a word X l 1 X l m m ; h 1 ; hm 2 H; l 1 ; l m 2 n; 20) is a simulation of size m of B in H if there exists a certain element h 2 H such that the equality m = h holds for any boolean vector (x 1 ; x n ) 2 f0; 1g . It is proved in [1] that ....

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D. M. Barrington, H. Straubing, D. Therien, Non-uniform automata over groups, Information and Computation, 132 (1990), 89--109.

.... of bichromatic edges in a three coloring of a given graph can be reduced to EQ G [14] other examples are described by Hstad [16] and Zwick [26] The general problem has also been studied due to applications to the fine structure of NC [3,14] specializing the framework of Barrington et al. [4,5]. Finally, the problem naturally gives rise to a number of well studied combinatorial enumeration problems: see, e.g. 6,13,24] and [23, pp. 110#. If G is Abelian and is a collection of equations over G, each of which can individually be satisfied, the trivial randomized approximation ....

D. A. M. Barrington, H. Straubing, D. Thrien, Non-uniform automata over groups, Information and Computation 89 (2) (1990) 109--132.

....using 4 additional levels of AND and OR gates. Therefore, MOD q is in ACC[p] contradicting Corollary 21. 12. Polynomials modulo a composite There are some striking upper bounds for circuits with MODm gates when m is composite. However, the lower bounds are unspectacular. A folklore theorem [51, 11] says AND is not represented by a low degree polynomial over Zm . Theorem 23 (Folklore) If g is a polynomial over Zm that represents AND(x 1 ; x n ) then the degree of g is n. Proof: Let g(1; 1) a. Then g(x 1 ; x n ) a 1in x i ; which has degree n. ....

....by a low degree polynomial over Zm . Theorem 23 (Folklore) If g is a polynomial over Zm that represents AND(x 1 ; x n ) then the degree of g is n. Proof: Let g(1; 1) a. Then g(x 1 ; x n ) a 1in x i ; which has degree n. Barrington, Straubing, and Therien [11] also noted that OR can be represented by a degree n= m Gamma 1) polynomial over Zm . When m is prime, this is tight. Theorem 24 (Barrington, Straubing, Therien) There is a degree (n= m Gamma 1) polynomial over Zm that represents OR. If m is prime, this degree is the best possible. Proof ....

D. A. M. Barrington, H. Straubing, and D. Th'erien. Non-uniform automata over groups. Inf. & Comp., 89(2):109--132, 1990.

....group on [4] The following fact about S 3 PBPs was proved by Barrington [B86] Fact 2 The class S 3 PBP is equivalent to mod 3 mod 2 circuit. Theorem 5.2 S 3 PBPs are exactly learnable using equivalence and membership queries. Proof Follows from Theorem 5.1. It was implicitly shown in [BST90] that A 4 PBP is equivalent to (mod 2 ; mod 2 ) mod 3 circuit, i.e. a depth two circuit consisting of mod 3 gates at the bottom level coming into two mod 2 gates at the second level. The output of the two mod 2 gates can then be combined using any Boolean operation. We will prove a more ....

David A. Mix Barrington, Howard Straubing, and Denis Th'erien. Non-uniform Automata over Groups. In Information and Computation, 89:109--132, 1990.

....if both MOD p and MOD q gates are allowed in the circuit for di erent primes p; q, or, if the modulus is a non prime power composite, e.g. 6. For example, it is consistent with our present knowledge that depth 3, linear size circuits with MOD 6 gates only, recognize an NP complete language (see [2]) It is not dicult to see that constant depth circuits with MOD p gates only, p prime) cannot compute even very simple functions: the n fan in OR or AND functions, since they can only compute constant degree polynomials of the input variables over GF p (see [13] But depth 2 circuits with MOD ....

.... cannot compute even very simple functions: the n fan in OR or AND functions, since they can only compute constant degree polynomials of the input variables over GF p (see [13] But depth 2 circuits with MOD 2 and MOD 3 gates, or MOD 6 gates can compute the n fan in OR and AND functions [8] [2]. Consequently, these circuits are more powerful than circuits with MOD p gates only. The sketch of the construction: we take a MOD 3 gate at the top of the circuit, and 2 n MOD 2 gates on the next level, where each subset of the n input variables is connected to exactly one MOD 2 gate, then ....

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D. A. M. Barrington, H. Straubing, and D. Th erien, Non-uniform automata over groups, Information and Computation, 89 (1990), 109-132. Grolmusz-Tardos: Lower Bounds for (MOD p { MOD m) Circuits 14

....: Z p k # G be the injective homomorphism defined by h(1) g; then, by Lemma 4, h(P ) g accepts MOD p j . 2 4. 2 The programs over monoids model The programs over monoids model of computation is a generalization of the branching program, which was introduced in [9] We refer the reader to [3] for a description of this model of computation. The polynomial program model encompasses this model in that a program over a monoid M is a polynomial program of degree one over M , and vice versa 3 . A number of nice characterizations of circuit complexity classes in terms of programs over ....

....for R (Theorems 14 and 17) 6.2 Nilpotent Groups We now show that for any accepting polynomial program P over a nilpotent group, there is an polynomial program of degree at most a constant times the degree of P accepting the same set over Z p , for some prime p. This has been observed [3, 15] in the special case of degree one polynomial programs; we give an alternative proof. Let NC 0 [MOD p] denote the class of functions decidable by NC 0 circuits with oracle gates for the MOD p function, where the MOD p gates may have unbounded fan in (but where all other gates must have ....

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D. A. Mix Barrington, H. Straubing, and D. Therien. Non-uniform automata over groups. Information and Computation, 89:109--132, 1990.

....machinery from the theory of non uniform automata, it is shown in x3.2 that for nilpotent groups G, EQN G can be recognized in polynomial time. This gap between nilpotent groups and non solvable groups seems to be the same as that which manifests itself in the study of non uniform automata (cf. [2]) and the fine structure of NC 1 . 2 Systems of equations over finite groups We begin by studying the complexity of solving systems of equations. For completeness, we recall that over an Abelian group, systems can be solved in polynomial time. 2.1 Abelian groups The fact that solving linear ....

....can recognize L f0; 1g n if there is a subset S G for which L = f 1 A (S) The theory of such automata has been admirably developed, motivated both by their connection with the fine structure of NC 1 and the satisfying algebraic perspective they offer the theory of finite automata. See [2]. For a fixed group G = fg 1 ; g k g, observe now that an equation w 1 w 2 w l = over variables v 1 ; vn may be transformed into an automaton program, with the property that there is an input accepted by the program if and only if there is a solution to the equation. ....

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D. A. M. Barrington, H. Straubing, and D. Th erien. Non-uniform automata over groups. Information and Computation, 89(2):109--132, Dec. 1990.

....f : L n L can be represented over L. As a consequence, we have the following loop analogue of Barrington s theorem [3] Theorem 7.6 If G is a groupoid that contains a nonsolvable loop then, the problem of evaluating an expression over G is complete for NC 1 under AC 0 reductions. In [4] Barrington, Straubing and Th erien conjectured that the word problem over a solvable group is not complete for NC 1 . However, in Section 8, we will define a class of solvable loops for which the problem of evaluating an expression is complete for NC 1 , showing that the above conjecture ....

....in w by any element in Gamma1 (c) Then, for any x 1 Delta Delta Delta x n 2 f0; 1g n , we have that f(x 1 ; x n ) 1 if and only if v( x 1 ) x n ) 2 Gamma1 (b) Clearly, the reduction from f to v is a simple projection. 8 Solvable loops It is conjectured in [4] that the problem of evaluating a word over a solvable group is not complete for NC 1 . However, we can construct a solvable loop of order 10 for which the problem of evaluating an expression is complete for NC 1 . Let Z 5 be the cyclic group of order five, and let G be the loop of order five ....

D.A. Barrington H. Straubing and D. Th'erien, Non-Uniform Automata Over Groups, Information and Computation 89 (1990), pp. 109-132.

....any element in Gamma1 (c) Then, for any x 1 Delta Delta Delta x n 2 f0; 1g n , we have that f(x 1 ; x n ) 1 if and only if v( x 1 ) x n ) 2 Gamma1 (b) Clearly, the reduction from f to v is a simple projection. 2 107 4. 9 Solvable loops It is conjectured in [7] that the problem of evaluating a word over a solvable group is not complete for NC 1 . However, we can construct a solvable loop of order 10 for which the problem of evaluating an expression is complete for NC 1 . Let Z 5 be the cyclic group of order five, and let G be the loop of order five ....

D.A. Barrington, H. Straubing and D. Th'erien, Non-Uniform Automata Over Groups, Information and Computation 89 (1990) pp.109-132.

....lower bound result known to apply in all cases. Theorem 1.2 is due to Cai and Lipton [7] we include it here because it is a direct corollary of our main theorem. In fact, the present work owes a lot to the methods that Cai and Lipton used to establish this theorem. It is conjectured (see [4]) that if G is solvable, then exponential length programs are required to compute the AND function, whereas there is a polynomial upper bound for nonsolvable groups. Theorem 1.3(a) improves on a Omega Gamma n log log n= log log log n) lower bound due to Pudl ak [13] Alon and Maass [1] ....

D. Mix Barrington, H. Straubing and D. Th'erien, Non-uniform automata over groups, Information and Computation 89, 109-132 (1990).

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D. Mix Barrington, H. Straubing, and D. Th'erien, "Nonuniform Automata over Groups", Information and Computation 89 (1990) 109-132.

....groups is NP complete [9] On the other hand, if a groupoid lacks this expressive power, all these problems may be signi cantly easier. Languages recognized by solvable groups have simple combinatorial descriptions [18, 21] and circuits over them can be evaluated quickly in parallel [2, 3]. Similarly, cellular automata de ned with polyabelian operations can be predicted much more quickly than by explicit simulation [14] Thus the algebraic properties of a groupoid are intimately linked to its computational complexity. This paper is organized as follows. Section 2 gives an ....

D.A. Barrington, H. Straubing and D. Therien, \Non-uniform automata over groups." Information and Computation 89 (1990) 109-132.

....an M program. In particular, there is a polynomial length program computing the AND function over the group S 3 A 5 (and in fact an explicit construction in [Bar89] shows how this can be achieved over A 5 alone) However, any program computing AND over the subgroup S 3 has exponential length [BST90]. This does not ruin the possibility that the polynomial length property is preserved under division, as PLP is provably false for S 3 A 5 , but the example shows that an argument to prove the closure property will crucially depend on PLP holding for the larger monoid. How do the two notions ....

....group but do not have any non nilpotent subgroup. Note that this is not the same as DA G nil . 3. The case of groups In this section we will prove that the conjecture holds for the restricted case of groups. This can be seen by putting together results that have implicitly appeared earlier in [BST90], but we present here an alternative proof. Let G be a group and g; h be elements of G: the group element g h gh is said to be a commutator of weight 2 and is denoted by [g; h] More generally, all elements of the group will be said to be commutators of weight 1, and commutators of weight k ....

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David A. Mix Barrington, Howard Straubing, and Denis Therien. Nonuniform automata over groups. Information and Computation, 89(2):109{ 132, December 1990.

....[9, 19] and has been exploited since the 50 s to classify these languages and analyze their combinatorial structure. Using the formalism of programs over monoids, introduced by Barrington, these ideas were recently extended to obtain algebraic characterizations of small classes of circuits [3, 4, 17]. A surprising link between algebra and communication complexity was discovered by Szegedy in [23] He proved that the membership question for a language can be decided using constant communication in the 2 party model of Yao if and only if that language can be recognized by a program over a ....

D. A. M. Barrington, H. Straubing, and D. Th'erien. Non-uniform automata over groups. Information and Computation, 89(2):109--132, Dec. 1990.

....of constant depth polynomial size circuits constructed with OR and AND gates of unbounded fan in corresponds to computations over aperiodic monoids. These theorems can be trivially modi ed to be phrased in terms of semigroups instead of monoids. More results along those lines were presented in [4] and [18] The central notion in these investigations is that of a program over a nite semigroup. A program, similarly to a morphism, is used to translate an input string from some alphabet A into a sequence of semigroup elements: the di erence is that the program may query each input bit ....

D. Mix Barrington, H. Straubing and D. Therien, Non-uniform Automata over Groups, Inform. and Comput. 89 (1990) 109-132.

....layer of MODm gates connected to the inputs, followed by a fixed number of layers of MOD p k gates, where p is prime. We may always assume that p does not divide m; for if pjm; then we can construct an equivalent circuit with MODm gates at the inputs. Barrington, Straubing and Th erien [2] showed that such circuits require exponential size to compute AND; Krause and Pudl ak [7] and Barrington and Straubing [1] showed that such circuits require exponential size to compute MOD q ; when q is a prime different from p that does not divide m: The definitive result in this direction was ....

....: ng f0; 1g such that ) They further showed that any such periodic function with t = O(log log n) can be computed by quasipolynomial size circuits, thus completely characterizing the symmetric functions computable by quasipolynomialsize circuits of this special form. The proofs in [2] and [1] use Fourier expansions over finite fields, while those in [7] and [6] are more combinatorial and rely on probabilistic arguments. In particular, 6] employs a new method, which is a kind of modular analogue of the random restriction techniques of Furst, Saxe and Sipser [5] In the present ....

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D. Mix Barrington, H. Straubing, and D. Th'erien, "Nonuniform Automata over Groups", Information and Computation 89 (1990) 109-132.

....and Th erien Herv e Caussinus 1 D epartement d informatique et de recherche op erationnelle, Universit e de Montr eal C.P. 6128, Succ. Centre ville, Montr eal (Qu ebec) CANADA H3C 3J7 e mail: caussinus iro.umontreal.ca Abstract We show that the result of Barrington, Straubing and Th erien [5] provides, as a direct corollary, an exponential lower bound for the size of depth two MOD 6 circuits computing the AND function. This problem was solved, in a more general way, by Krause and Waack [8] We point out that all known lower bounds rely on the special form of the MOD 6 gate occurring ....

....and al. We denote by C n a Boolean circuit with n inputs, this circuit computing a function from f0; 1g n to f0; 1g. The size of C n is the number of gates in the circuit. The theorem of Barrington, Straubing and Th erien can be stated as follows in the circuit model framework: Theorem 1 ([5]) Let m 1 ; m u be a constant number of integers. Let p be a prime number and k 1 ; k v be a constant number of integers. Let B n be a constant depth circuit, with general MODm i gates at the input level and MOD p k i gates elsewhere. If B n computes the AND function then its size ....

D. A. Mix Barrington, H. Straubing, and D. Th'erien. Non uniform automata over groups. Information and Computation, 89:109--132, 1990. 3

....Decreasing Lemma we show that this circuit can be converted to a (MOD q , MOD p ) circuit with linear polynomials on the input level with the price of increasing the size of the circuit. This result implies special cases of the Constant Degree Hypothesis of Barrington, Straubing and Th erien [3], and implies also a generalization of the lower bound results of Yan and Parberry [21] Krause and Waack [12] and Krause and Pudl ak [11] Perhaps the most important application is an exponential lower bound for the size of (MOD q , MOD p ) circuits computing the fan in n AND, where the input of ....

....if both MOD p and MOD q gates are allowed in the circuit for different primes p,q, or, if the modulus is a non prime power composite, e.g. 6. For example, it is consistent with our present knowledge that depth 3, linear size circuits with MOD 6 gates only, recognize the Hamiltonian graphs (see [3]) The existing lower bound results use diverse techniques from Fourier analysis, communication complexity theory, group theory and several forms of random restrictions (see [3] 11] 17] 18] 16] 8] 6] 7] 2] 10] It is not difficult to see that constant depth circuits with MOD p ....

[Article contains additional citation context not shown here]

D. A. M. Barrington, H. Straubing, and D. Th erien. Non-uniform automata over groups. Information and Computation, 89:109--132, 1990.

....Decreasing Lemma we show that this circuit can be converted to a (MOD q ; MOD p ) circuit with linear polynomials on the input level with the price of increasing the size of the circuit. This result implies special cases of the Constant Degree Hypothesis of Barrington, Straubing and Therien [3], and implies also a generalization of the lower bound results of Yan and Parberry [21] Krause and Waack [12] and Krause and Pudlak [11] Perhaps the most important application is an exponential lower bound for the size of (MOD q ; MOD p ) circuits computing the fan in n AND, where the input of ....

....if both MOD p and MOD q gates are allowed in the circuit for different primes p;q, or, if the modulus is a non prime power composite, e.g. 6. For example, it is consistent with our present knowledge that depth 3, linear size circuits with MOD 6 gates only, recognize the Hamiltonian graphs (see [3]) The existing lower bound results use diverse techniques from Fourier analysis, communication complexity theory, group theory and several forms of random restrictions (see [3] 11] 17] 18] 16] 8] 6] 7] 2] 10] It is not difficult to see that constant depth circuits with MOD p ....

[Article contains additional citation context not shown here]

D. A. M. Barrington, H. Straubing, and D. Therien. Non-uniform automata over groups. Information and Computation, 89:109--132, 1990.

....[8, 18] and has been exploited since the 50 s to classify these languages and analyze their combinatorial structure. Using the formalism of programs over monoids, introduced by Barrington, these ideas were recently extended to obtain algebraic characterizations of small classes of circuits [3, 4, 16]. A surprising link between algebra and communication complexity was discovered by Szegedy in [22] He proved that the membership question for a language can be decided using constant communication in the 2 party model of Yao if and only if that language can be recognized by a program over a ....

D. A. M. Barrington, H. Straubing, and D. Th'erien. Non-uniform automata over groups. Information and Computation, 89(2):109--132, Dec. 1990.

....and results in the case of three types of classes defined by circuits of constant depth, polynomial size, and unbounded fan in, for three different types of gates. These are AND and OR gates [FSS84] AND, OR, and MOD p gates [Ra87, Sm87] and modular gates alone with certain other restrictions [BST90]. Most of this survey was prepared for the McGill University Workshop in Theoretical Computer Science, held in February 1989. 2. Introduction It would be nice to show that the CLIQUE problem is not solved by a family of boolean circuits where circuit size grows polynomially with input size. This ....

....this map of a subset of M . 3 One can ask which languages can be recognized by program families over which monoids, but for most monoids this is not particularly interesting because all languages can be recognized by some program (the notable exceptions are if the monoid is a nilpotent group [BST90] or a particularly small aperiodic monoid [Th89] However, we can define complexity classes by bounding the length of the programs in the family as a function of the input size n. When we restrict this length to be polynomial, the classes which emerge in many cases are previously studied circuit ....

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D. A. M. Barrington, H. Straubing, and D. Th'erien, "Non-uniform automata over groups", Information and Computation, to appear. Also COINS Technical Report 89-56, University of Massachusetts.

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D. A. M. Barrington, H. Straubing, D. Thrien, Non-uniform automata over groups, Information and Computation 89 (2) (1990) 109132.

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D. A. M. Barrington, H. Straubing, D. Thrien, Non-uniform automata over groups, Information and Computation 89 (2) (1990) 109132.

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D.A. Mix Barrington, H. Straubing, and D. Th'erien. Non-uniform automata over groups. Information and Computation, 89:109--132, 1990.

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D.A. Barrington, H. Straubing and D. Th' erien, Non-Uniform Automata Over Groups, Information and Computation 89, 2 (1990), pp. 109-132.

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