Abstract
Consider a (MOD_q,MOD_p) circuit, where the inputs of the bottom MOD_p gates are degree-d polynomials with integer coefficients 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 \emphlinear 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 Thérien, and generalizing the lower bound results of Yan and Parberry, Krause and Waack, and Krause and Pudlák. 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 \empharbitrary integer valued function of cn variables (c<1) plus an arbitrary linear function of n input variables.
Highlights
Boolean circuits are one of the most interesting models of computation
Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function
Since the modular gates are very simple to define, and they are immune to the random restriction techniques in lower bound proofs for the PARITY function, the following natural question was asked by several researchers: How powerful will become the Boolean circuits if — beside the standard AND, OR and NOT gates — MODm gates are allowed in the circuit? Here a MODm gate outputs 1 iff the sum of its inputs is in a set A ⊂ {0, 1, 2, . . . , m − 1} modulo m
Summary
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). Since the modular gates are very simple to define, and they are immune to the random restriction techniques in lower bound proofs for the PARITY function, the following natural question was asked by several researchers: How powerful will become the Boolean circuits if — beside the standard AND, OR and NOT gates — MODm gates are allowed in the circuit? We believe that – in the light of the result of Yao, Beigel and Tarui – our result may have further important consequences in modular circuit theory
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
