Universal boolean functional vectors

Abstract

In the simplest setting, one represents a boolean function using expressions over variables, where each variable corresponds to a function input. So-called parametric representations, used to represent a function in some restricted subspace of its domain, break this correspondence by allowing inputs to be associated with functions. This can lead to more succinct representations, for example when using binary decision diagrams (BDDs). Here we introduce Universal Boolean Functional Vectors (UBFVs), which also break the correspondence, but done so such that all input vectors are accounted for. Intelligent choice of a UBFV can have a dramatic impact on BDD size; for instance we show how the hidden weighted bit function can be efficiently represented using UBFVs, whereas without UBFVs BDDs are known to be exponential for any variable order. We show several industrial examples where the UBFV approach has a huge impact on proof performance, the killer app being floating point addition, wherein the wide case-split used in the state-of-the-art approach is entirely done away with, resulting in 70-fold reduction in proof runtime. We give other theoretical and experimental results, and also provide two approaches to verifying the crucial universality aspect of a proposed UBFV. Finally, we suggest several interesting avenues of future research stemming from this program.