Lower bounds for the complexity of restrictions of Boolean functions Original Research Article

The complexity of restrictions of Boolean functions in various computational models is studied. Lower bounds for the complexity of the most complicated restrictions to domains of fixed size are established. The complexity bounds are shown to be tight up to a constant multiplicative factor in the case of logic circuits. It is proved that for any Boolean function of n variables whose circuit size is greater than n2+ε, where ε is an arbitrary positive constant, there exists a domain in {0,1}n the restriction to which has a nonlinear circuit size, which differs by a constant factor from the circuit size of the most complicated partial Boolean function defined on this domain. Any Boolean function of n variables is proved to be uniquely determined by its values in at most n domains whose sizes are bounded from above by the product of the function complexity and n4.