M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pages 61-69, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....not have the power to produce such AC circuits before the computation begins. To overcome this problem we construct a first order formula which is in a sense complete for all first order formulas. Of course, Sipser proved that there is a strict alternation hierarchy for first order formulas [Sip83]. Thus there can be no single first order formula that is truly complete for all first order formulas. What we mean is that every first order formula, can be simulated by the formula iterated d 1 times where d is the depth of , evaluated on a structure that is a polynomial padding of the ....

M. Sipser, "Borel Sets and Circuit Complexity," ACM Symp. Theory Of Comput. (1983), 61--69. 10

....of a circuit is the number of AND and OR gates appearing in it. The bottom fan in of a depth d circuit is the maximum number of inputs of a gate at level d. For more detail on the basics of constant depth circuits, consult the survey by Boppana and Sipser [11] 4.1 The Functions Definition 4. 1 [30, 11] Let integers d and m 1 , m d be given, and let there be variables x i 1 , i d for 1 i j m j . d = #m1 K x i 1 , i d Where if d is even, and if d is odd. The Sipser function f d with m 1 = dm log m 2. m1 , m d ,k d 1 , with m 1 = m ....

M. Sipser. Borel sets and circuit complexity. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (STOC), pages 61-- 69, 1983.

....weight wt(x) of an assignment x 2 f 0; 1 g equals the number of bits that are set to 1. The circuit is monotone if it has no NOT gates, and anti monotone if all wires from an input go to a NOT gate, and these are the only NOT gates in the circuit. A pure Sigma t circuit as defined by Sipser [Sip83] consists of t levels of large gates that alternate and with a single gate at the top (i.e. the output) and with the bottom level gates connected to the input gates x 1 ; x n and their negations x 1 ; x n . A pure Pi t circuit is similarly defined with a large gate at the ....

....by depth d unbounded fan in Boolean circuits of polynomial size having a single OR gate at the output, as described in the survey by Boppana and Sipser [BS90] Let Pi d stand for the complements of these languages, which are recognized by depth d circuits with an AND gate at the output. Sipser [Sip83] showed that for all d 1, Sigma 6= Pi . It is not surprising that this carries over to the parameterized setting to show that the G[t] hierarchy is proper, but it is noteworthy that it extends to our nondeterministic classes: Theorem 4.1 For all t 1, N [t] ae N [t 1] Proof. Suppose N ....

M. Sipser. Borel sets and circuit complexity. In the Proceedings of the 15th ACM Symposium on the Theory of Computing (1983), 61--69.

....not have the power to produce such AC circuits before the computation begins. To overcome this problem we construct a first order formula # which is in a sense complete for all first order formulas. Of course, Sipser proved that there is a strict alternation hierarchy for first order formulas [Sip83]. Thus there can be no single first order formula that is truly complete for all first order formulas. What we mean is that every first order formula, #, can be simulated by the formula # iterated d 1 times where d is the depth of #, evaluated on a structure that is a polynomial padding of the ....

M. Sipser, "Borel Sets and Circuit Complexity," ACM Symp. Theory Of Comput. (1983), 61--69.

....by programs over semigroups of dot depth one, and we prove that this class forms a p variety. Along the way, we will give a simple direct argument that the hierarchy de ned by k interleaves strictly with the one de ned by AC k , i.e. 1 1 AC 2 thus re ning a result of [16]. This result can also be derived from a special case of a more general theorem in [7] The fact that semigroups of dot depth one form a p variety allows us to generalize a result of Straubing [19] and P eladeau [13] on regular languages expressible by 1 formulas with arbitrary numerical ....

....to compute any function of t positions of the input string, for some xed t. By using the fact that any function of a constant number of input positions can be written as either a constant size OR of AND s or as a constant size AND of OR s, it is easy to see that BC k k BC k 1 . In [16], Sipser showed that the BC k hierarchy is in nite and H astad [9] later strengthened the result by showing that the separation is exponential. In fact, H astad s result even shows that k 1 . This implies that the hierarchy is in nite, and the separation is exponential. In this ....

M. Sipser, Borel sets and circuit complexity, in: Proceedings of the 15th ACM Symposium on Theory of Computing (1983) 61-69.

....the polynomial time hierarchy PH relativized to an oracle A. By taking our second point of view, their work defines the type 2 polynomial time hierarchy PH, in which each member relation takes an oracle A as an argument, in addition to a string argument. Yao [30] using results from Sipser [25] and Furst, Saxe, and Sipser [11] constructed an oracle A in which all levels in PH are distinct. It follows that all levels in PH are absolutely distinct. Generic sets were introduced by Cohen [4] as a tool for proving independence results in set theory. A general treatment of complexity ....

Sipser, M. (1983), Borel sets and circuit complexity, in "Proceedings, 15th ACM Symposium on Theory of Computing", pp. 61--69.

.... theory, see, e.g. HU79, BDG95, BC94, Pap94] For more background on the models we use, we refer the reader to the different chapters in [RS97] Our Turing machines are standard multi tape machines, see [HU79] For the definition of sublinear time classes we use indexing machines, introduced in [Sip83] These machines cannot directly access their input tape, but instead have to write down a number in binary on a so called index tape. When they enter a specified read state with bin(i) on the index tape, they are supplied with the ith input symbol (or a particular blank symbol, if i exceeds the ....

M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Symposium on Theory of Computing, pages 61--69. ACM Press, 1983.

.... theory, see, e.g. HU79, BDG95, BC94, Pap94] For more background on the models we use, we refer the reader to the different chapters in [RS97] Our Turing machines are standard multi tape machines, see [HU79] For the definition of sublinear time classes we use indexing machines, introduced in [Sip83] These machines cannot directly access their input tape, but instead have to write down a number in binary on a so called index tape. When they enter a specified read state with bin(i) on the index tape, they are supplied with the ith input symbol (or a particular blank symbol, if i exceeds the ....

M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Symposium on Theory of Computing, pages 61--69. ACM Press, 1983.

....theory. For example, Kadin [25] has proven that if some sparse set is p T complete for NP then PH = P NP jj . Hemachandra and Wechsung [20] have shown that the theory of randomness (in the form of the resource bounded Kolmogorov complexity theory of Adleman [1] Hartmanis [18] and Sipser [39]) is deeply tied to the question of whether P NP jj = P NP , i.e. whether parallel and sequential access to NP coincide. Buss and Hay [10] have shown that P NP jj exactly captures the class of sets acceptable 4 via multiple rounds of parallel queries to NP and also exactly captures the ....

M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 61-69. ACM Press, 1983.

....Let k 1. For every Sigma P;1 k predicate there is a polynomial q such that for every x, there exists a Sigma k (q(jxj) circuit C ;x , having the property that for any set A, C ;x d ae A = 1 if and only if (A; x) is true. Also, for each variable v z (v z ) in C ;x , jzj q(jxj) Sipser [Sip83] introduced a family of functions f n k , n 1, k 1, computed by special types of circuits. Hastad later introduced a modified family of f n k functions [Has87] and Ko, still later, introduced a further modification of the f n k functions in constructing oracles that separate and collapse ....

M. Sipser. Borel sets and circuit complexity. In Proc. 15th ACM Symposium on Theory of Computing, pages 61--69, 1983.

....D p;A k 6 PP S p;A k Gamma2 . 1 Introduction There is a strong connection between lower bounds for boolean circuits (consisting of AND, OR, and NOT gates) and relativization results about the polynomial time hierarchy. This fact was first established by Furst, Saxe, and Sipser [5] Sipser [13] later defined a family of functions that are computable by linear size circuits of depth k, and showed that they require super polynomial size boolean circuits of depth k Gamma 1. Yao [14] and Hstad [8, 9] improved Sipser s result by showing that the same functions actually require exponential ....

....that P NP[log] PP, and later Beigel, Reingold, and Spielman [3] proved the even stronger P PP[log] PP. A relativization of these results show that our result is almost tight. 2 A lower bound for perceptrons We begin this section by defining the function f m k , first defined by Sipser [13], which can be computed by linear size circuits of depth k. Then we show the main theorem, which states that perceptrons of depth k with bounded fan in that compute this function must be large. As a corollary we get that perceptrons of depth k Gamma 1 computing f m k must be large. 2 A ....

Michael Sipser. Borel sets and circuit complexity. In Proceedings of 15th Annual ACM Symposium on Theory of Computing, pages 61--69, 1983.

....that is a tree. At the leaves of the tree there are unnegated variables. The i th level from the bottom consists of gates if i is even and otherwise it consists of gates. The fanin at the top and bottom levels is N and at all other levels it is N 2 . This function was used by Sipser in [18] who showed that it requires superpolynomial size depth k Gamma 1 circuits over the basis f; g. It will be convenient to also consider the functions f k , the negations of f k . Clearly f k is computed by a circuit very similar to the circuit computing f k . The only difference being that ....

M. Sipser. Borel sets and circuit complexity. Proceedings of 15th Annual ACM Symposium on Theory of Computing, pages 61--69, 1983.

.... instance, the de nition of alternating Turing machine given in [CKS81] does not allow an interesting notion of sublinear time complexity, whereas augmenting this model with random access to the input provides a useful model for studying circuit complexity classes such as NC 1 and AC 0 [Ruz81, Sip83] How can one de ne a useful notion of interactive proof system for deterministic log time In attempting to answer this and related questions, we take as our starting point the work of Baier and Wagner [BW98a] where it was shown that (single prover and multi prover) interactive proof systems ....

M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Symposium on Theory of Computing, pages 61-69. ACM Press, 1983. 21

....[P oly] is the natural probabilistic version of NP . One vote for AM [2] is the recent result by Nisan and Wigderson [NW] is that the class of languages which are in NP B with probability 1 for a random oracle B is equal to AM [2] 1. 3 Outline of Our Proof Furst, Saxe, Sipser [FSS] and Sipser [S] were the first to show that oracle separation results involving classes such as the levels of the polynomial time hierarchy could be achieved by proving lower bounds for constant depth circuits. Since then improved bounds and subsequent separations have been achieved by Yao [Y] and Hastad [H1] ....

....is to give a circuit formulation of Sigma B g(n) 1 and of the AM protocol recognizing L(B) which we do in Sections 3 and 5 respectively. 3. Relativized Complexity and Circuits Let us first state the connection between Pi B g(n) and g(n) depth circuits. This was first established in [FSS] and [S]. Definition: A Pi B g(n) Sigma B g(n) machine is an alternating Turing machine which runs in polynomial time, has at most g(n) alternations along any computation branch, starts with an ( alternation, and makes polynomial length queries to an oracle for B: Definition: A language L is ....

Sipser M., "Borel Sets and Circuit Complexity," Proc. of the 15th ACM Symposium on Theory of Computing, pp 61-69, Boston, 1983.

....of length log(n Gamma 1) padded with leading zeros if necessary) Our model of computation is the oracle alternating multitape Turing machine with random access to the input. This model, originally defined by Ruzzo [29] and used by Barrington, Immerman and Straubing [5] Buss [9] and Sipser 5 [31] among others, is a modification of the model of Chandra, Kozen and Stockmeyer [10] to allow sublinear time bounds. These machines are equipped with an address tape on which to write a number in binary. When the machine enters a distinguished state with a number p written on its address tape, the ....

M. Sipser. Borel sets and circuit complexity. In 15th Annual ACM Symposium on the Theory of Computing, pages 61--69, 1983.

....denotes logarithms to base 2. By similar methods it is possible to prove that there is a family of functions f n k of n inputs which have linear size circuits of depth k but require exponential size circuits when restricted to depth k Gamma 1. These functions f n k were introduced by Sipser in [S]. Sipser proved superpolynomial lower bounds for the size of the circuits when the depth was restricted to be k Gamma 1. Yao claimed exponential lower bounds for the same situation. 1.3 Small depth circuits and Relativized Complexity. Lower bounds for small depth circuits have some interesting ....

....of an oracle separating PSPACE from the polynomial time hierarchy. Yao [Y] was the first to prove sufficiently good lower bounds to obtain the separation for an oracle A. Cai [C] extended his methods to prove that a random oracle separated the two complexity classes with probability 1. In [S] Sipser proved the corresponding theorem that the same lower bounds for the functions f n k would imply the existence of oracles separating the different levels in the polynomial hierarchy. The lower bounds claimed by Yao gives the first oracle achieving this separation. Our bounds are of course ....

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Sipser M. "Borel Sets and Circuit Complexity", Proceedings of 15th Annual ACM Symposium on Theory of Computing, 1983, 61-69.

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M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pages 61-69, 1983.

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M. Sipser. Borel sets and circuit complexity. In Proceedings of the ACM Symposium on Theory of Computing, pages 61--69, 1983.

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Sipser, M. 1983. Borel sets and circuit complexity. In Proc. 15th Annual ACM Symposium on the Theory of Computing, pp. 61--69.

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M. Sipser. Borel sets and circuit complexity. In Proc. 15th Annual ACM Symposium on the Theory of Computing, pages 61--69, 1983.

No context found.

M. Sipser. Borel sets and circuit complexity. In 15th Annual ACM Symposium on the Theory of Computing, pages 61--69, 1983.

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M. Sipser. Borel sets and circuit complexity. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (STOC), pages 61-- 69, 1983.

No context found.

M. Sipser. Borel sets and circuit complexity. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 61--69, 1983.

No context found.

M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Symposium on Theory of Computing, pages 61--69. ACM Press, 1983. 21

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M. Sipser. Borel sets and circuit complexity. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pages 61-69, 1983.

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