On WLCDs and the Complexity of Word-Level Decision Diagrams—A Lower Bound for Division

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Several types of Decision Diagrams (DDs) have been proposed for the verification of Integrated Circuits. Recently, word-level DDs like BMDs, aBMDs, HDDs, KaBMDs and aPHDDs have been attracting more and more interest, e.g., by using aBMDs and aPHDDs it was for the first time possible to formally verify integer multipliers and floating point multipliers of “significant” bitlengths, respectively. On the other hand, it has been unknown, whether division, the operation inverse to multiplication, can be efficiently represented by some type of word-level DDs. In this paper we show that the representational power of any word-level DD is too weak to efficiently represent integer division. Thus, neither a clever choice of the variable ordering, the decomposition type or the edge weights, can lead to a polynomial DD size for division. For the proof we introduce Word-Level Linear Combination Diagrams (WLCDs), a DD, which may be viewed as a “generic” word-level DD. We derive an exponential lower bound on the WLCD representation size for integer dividers and show how this bound transfers to all other word-level DDs.