A Combinatorial Limit to the Computing Power of VLSI Circuits

We introduce a property of Boolean functions, called transitivity which consists of integer, polynomial, and matrix products as well as of many interesting related computational problems. We show that the area of any circuit computing a transitive function grows quadratically with the circuit´s maximum data rate, expressed in bits/S. This result provides a precise analytic expression of an area-time tradeoff for a wide class of VLSI circuits. Furthermore (as shown elsewhere), this tradeoff is achievable. We have thus matching (to within a constant multiplicative factor) upper and lower complexity bounds for the three above products, in the VLSI circuits computational model.