Formal Methods in System Design

, Volume 8, Issue 3, pp 273-282

First online:

Mod-2-OBDDs—A data structure that generalizes EXOR-sum-of-products and ordered binary decision diagrams

  • Jordan GergovAffiliated withMax-Planck-Institut für Informatik
  • , Christoph MeinelAffiliated withFachbereich IV-Informatik, Universität Trier

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We present a data structure for Boolean manipulation-the Mod-2-OBDDs-that considerably extends ESOPs (EXOR-sum-of-products) as well as OBDDs (ordered binary decision diagrams). There are Boolean functions of practical interest which have exponential size optimal ESOPs (even multilevel EXOR-expressions) and/or OBDDs that can be represented by (low degree) polynomial size Mod-2-OBDDs.

We show that Boolean manipulation tasks such as apply operation, quantification, composition can be performed with Mod-2-OBDDs at least as efficient as with OBDDs. Indeed, since the size of a minimal Mod-2-OBDD-representation of a Boolean function is, in general, smaller (sometimes even exponentially smaller) than the size of an optimal OBDD-representation, the increase in efficiency is considerable. Moreover, EXOR-operations as well as complementations can be performed in constant timeO (1).

However, the price of constant time EXOR-apply operations is the canonicity of the Mod-2-OBDD-representation. In order to allow in spite of this fact efficient analysis of Mod-2-OBDDs we present a fast probabilistic equivalence test with one-sided error probability for Mod-2-OBDDs (and, hence, for ESOPs) which performs only linear many arithmetic operations.


data structures for Boolean functions BDDs verification EXOR expressions