A Degree-Decreasing Lemma for (MOD q - MOD p) Circuits

plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexity theory context as well as in the theory of parallel computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([13], [22], [9], [14], [15], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [9] generalized this resul

A Degree-Decreasing Lemma for (MOD q - MOD p) Circuits

Consider a (MOD q ; MOD p ) circuit, where the inputs of the bottom MOD p gates are degree-d polynomials of the input variables (p, q are different primes). Using our main tool --- the Degree 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 has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Th'erien [3], and generalizing the lower bound results of Yan and Parberry [21], Krause and Waack [13] and Krause and Pudl'ak [12]. Perhaps the most important application is an exponential lower bound for the size of (MOD q ; MOD p ) circuits computing the n fan-in AND, where the input of each MOD p gate at the bottom is an arbitrary integer valued function of cn variables (c ! 1) plus an arbitrary linear function of n input variables. We believe that the Degree Decreasing Lemma becomes a standard tool in mod..