V. Grolmusz and G. Tardos, "Lower Bounds for (MOD p-MOD m) Circuits", Proc. 39th IEEE FOCS, 1998. Home/Search Document Details and Download
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A Note on MOD p - MOD m Circuits - Straubing, Thérien (Correct)
.... 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 found by Grolmusz and Tardos [6], who showed that the only symmetric boolean functions computed by such circuits in subexponential size have a periodic spectrum with period mp ; where mp n: By the spectrum of a symmetric function f0; 1g; we mean the map f : f0; 1; ng f0; 1g such that ) They further ....
....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 note we show how to use the Fourier techniques to obtain different proofs of ....
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V. Grolmusz and G. Tardos, "Lower Bounds for (MOD p-MOD m) Circuits", Proc. 39th IEEE FOCS, 1998.
Some Properties of Modulo m Circuit Computing Simple Functions - Amano, Maruoka (Correct)
....the proof of this fact here since we could not find a published proof. Fact 3 Every Boolean function can be computed by a depth two (MOD 3 MOD 2 ) circuit. Proof It was known that the OR function of n variables, i.e. n i=1 x i , can be computed by a depth two (MOD 3 MOD 2 ) circuit. See [GT98] for the simple construction of size 2 n and see [KM91] for the slightly effective construction of size 2 n=2 1 . This implies that every conjunction of n literals, that is n i=1 l i where l i 2 fx i ; x i g, also can be computed by a depth two (MOD 3 MOD 2 ) circuit. Let f be an ....
....complexity of circuits with modulo gates using the techniques developed in this note. The following theorem says that even MOD 4 function requires exponential size ( MOD 3 ) l MOD 6 ) circuits for constant l. The generalization of the following theorem is recently proved by Grolmusz and Tardos[GT98] by using the random clustering technique. Theorem 8 Let f be a symmetric function on n variables such that f 6=MOD 213 k for some positive integer k. Suppose that a (MOD 3 1 1 1 MOD 3 MOD 6 ) circuit C of depth l computes f . Then the size of the circuit C is at least (2 3 k =2 0 ....
V. Grolmusz and G. Tardos, "Lower Bounds for (MOD p -- MOD m) Circuits", Proc. 39th FOCS, pp. 279--289, 1998.
A Note on MOD p - MOD m Circuits - Straubing, Therien (Correct)
.... 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 found by Grolmusz and Tardos [6], who showed that the only symmetric boolean functions computed by such circuits in subexponential size have a periodic spectrum with period mp t ; where mp t n: By the spectrum of a symmetric function f : f0; 1g n f0; 1g; we mean the map f : f0; 1; ng f0; 1g such that ....
....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 note we show how to use the Fourier techniques to obtain different proofs of ....
[Article contains additional citation context not shown here]
V. Grolmusz and G. Tardos, "Lower Bounds for (MOD p-MOD m) Circuits", Proc. 39th IEEE FOCS, 1998.
Languages Defined With Modular Counting Quantifiers - Straubing (2001) (1 citation) (Correct)
....three conjectures in this case. There are some results for Conjectures 1 and 2 for circuits with MODm gates on the input level, and a MOD p gate at the outputs, where p is prime. See Barrington, Th erien and Straubing [4] Barrington and Straubing [5] Krause and Pudlak [15] Grolmusz and Tardos [12]. 1.2 Connections with logic For a full account of the results cited in this subsection and the next one, see Straubing [22] We will use formulas of first order logic to define properties of strings over a finite alphabet A: The variables in these formulas denote positions in the string (that ....
V. Grolmusz and G. Tardos, "Lower Bounds for (MOD p-MOD m) Circuits", Proc. 39th IEEE FOCS, 1998.
Languages Defined With Modular Counting - Quantifiers Howard Straubing (Correct)
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V. Grolmusz and G. Tardos, "Lower Bounds for (MOD p-MOD m) Circuits", Proc. 39th IEEE FOCS, 1998.
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