Optimal versus stable in Boolean formulae

Abstract

We present some initial ideas of a new lower bound technique which captures, in a strong way, the structure of optimal circuits. The key observation is that optimal circuits are unstable not only w.r.t. deletion of gates but also w.r.t. renaming gates. Such an unstability allows one to extract some useful information about the inner structure of optimal circuits. We demonstrate the technique by new exponential lower bounds on the size of null-chain-free formulae over the basis {∧, ∨, ¬} approximating subsets of the Boolean n—cube, and in particular, for formulae computing ”semi-slice” functions, i.e. functions f such that, for some k<l ≤ m,

$$f = f \wedge T_k^n \wedge - T_l^n \vee T_m^n ,$$

where T ⁿ _k is the k—th threshold function.