Special issue: Formal verification of cyber-physical systems

Two problems are studied. (1) How much external or internal information cost is required to compute a Boolean-valued function with an error at most 1/2−ξ for small ξ>0? A lower bound Ω(ξ2) and an upper bound O(ξ) are established. (2) How much external information cost can be saved to compute a function with a small error ϵ>0 comparing to the case when no error is allowed? A lower bound Ω(ϵ) and an upper bound O(h(ϵ)) are established, the lower bound can be improved to Ω(h(ϵ)) for product distributions. Except the O(h(ϵ)) upper bound, the other three bounds are tight. For distribution μ that is equally distributed on (0,0) and (1,1), it is shown that ICμext(XOR,ϵ)=1−2ϵ holds for every 0≤ϵ≤1/2. This equality seems to be the first example of exact information complexity when ϵ>0.