The Complexity of SPP Formula Minimization

Abstract

Circuit minimization is a useful procedure in the field of logic synthesis. Recently, it was proven that the minimization of (∨, ∧, ¬) formulae is hard for the second level of the polynomial hierarchy [BU08]. The complexity of minimizing more specialized formula models was left open, however. One model used in logic synthesis is a three-level model in which the third level is composed of parity gates, called SPPs. SPPs allow for small representations of Boolean functions and have efficient heuristics for minimization. However, little was known about the complexity of SPP minimization. Here, we show that SPP minimization is complete for the second level of the Polynomial Hierarchy under Turing reductions.

On the Optimal OBDD Representation of 2-XOR Boolean Affine Spaces

“…Their algebraic representation is an AND of XORs of literals, also called CEX expression. CEX expressions are often exploited in logic synthesis for three level logic minimization, e.g., for SPP and 2-SPP forms [1], [9], [18], [23], [24]; or for handling regular functions, e.g., autosymmetric functions [2], [18]. In all these contexts, the compact ROBDD representation of CEX expressions is crucial for guaranteeing reduced synthesis times.…”