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**1 - 2**of**2**### Some Properties of MODm Circuits Computing

"... We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit w ..."

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We investigate the complexity of circuits consisting solely of modulo gates and obtain results which might be helpful to derive lower bounds on circuit complexity: (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii) We show some properties of such depth two circuits computing symmetric functions. As a consequence we might think of the strategy for deriving lower bounds on modular circuits: Suppose that a polynomial size constant depth modulo 2p circuit C computes a symmetric function. If we can show that the circuit obtained by applying the procedure given in (i) to the circuit C cannot satisfy the properties described in (ii), then we have a super-polynomial lower bound on the size of a constant depth modulo 2p circuit computing a certain symmetric function.

### Some Properties of Modulo m Circuit Computing Simple Functions

"... In this note, we investigate the complexity of circuits with modulo gates only in two ways. (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii ..."

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In this note, we investigate the complexity of circuits with modulo gates only in two ways. (i) We describe a procedure that converts a circuit with only modulo 2p gates, where p is a prime number, into a depth two circuit with modulo 2 gates at the input level and a modulo p gate at the output. (ii) We show some properties of such depth two circuits computing symmetric functions. Thus if we can show that, for any linear size constant depth modulo 6 circuit C, a circuit obtained from C by the procedure described in (i) can not satisfy the properties described in (ii), then we could have a superlinear lower bound on the size of a constant depth modulo 6 circuit computing some symmetric functions. Key words circuit complexity, modulo gate, constant depth circuit, symmetric function, lower bound 1 Introduction To derive a strong lower bound on the size complexity in various circuit models is a big challenge in theoretical computer science. In particular, constant depth circuits are wide...