Complexity Based on Partitioning of Boolean Circuits and their Relation to Multivalued Circuits

We present a new complexity measure for Boolean functions based on partitions of combinatorial circuits into subcircuits, and give upper and lower bounds on the complexity for Boolean functions. Roughly speaking, for a function g whose range is the set of positive integers, g(n)-partition of a circuit is a partition of a circuit into subcircuits such that. 1) each subcircuit has at most g(n) output gates, through which gates in the subcircuit are connected to gates in another succeeding subcircuits and 2) each subcircuit has at most two preceding subcircuits where subcircuit N1 (N2) is said to precede (succeed) subcircuit N2 (N1) when there is a line directed from a gate in N1 to a gate in N2. Our main result, which is stated in terms of a lower bound theorem and an upper bound theorem, is a precise version of the following statement: For "almost all" n-argument Boolean functions fn, the minimum nu mber of subcircuits over g n)-partition of a circuit to compute fn is given as (2n/g(n)22g(n)) unless g(n) grows much more slowly than n as n increases. To prove the theorems, we regard a Boolean circuit, together with g(n)-partition of it, as the multivalued circuit composed of multivalued gates corresponding to the subcircuits obtained from the g(n)-partition.