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

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Abstract

Several types of Decision Diagrams (DDs) have been proposed for the verification of Integrated Circuits. Recently, word-level DDs like BMDs, *BMDs, HDDs, K*BMDs and *PHDDs have been attracting more and more interest, e.g., by using *BMDs and *PHDDs 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.

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Authors and Affiliations

  1. Institute of Computer Science, Albert-Ludwigs-University, D 79110, Freiburg im Breisgau, Germany

    Christoph Scholl, Bernd Becker & Thomas Weis

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  1. Christoph Scholl
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  2. Bernd Becker
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  3. Thomas Weis
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Scholl, C., Becker, B. & Weis, T. On WLCDs and the Complexity of Word-Level Decision Diagrams—A Lower Bound for Division. Formal Methods in System Design 20, 311–326 (2002). https://doi.org/10.1023/A:1014702331828

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