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Some notes on the abstraction operation for multi-terminal binary decision diagrams

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Abstract

The starting point of this work are inaccurate statements found in the literature for Multi-terminal Binary Decision Diagrams (MTBDDs) (Bahar et al., Form. Methods Syst. Des. 10(2/3):171–206, 1997; Siegle, Behaviour analysis of communication systems: compositional modelling, compact representation and analysis of performability properties, 2002; Kuntz, Symbolic semantics and verification of stochastic process algebras, 2006) regarding the well-definedness of the MTBDD abstraction operation. The statements try to relate an operation ∗ on a set of terminal values M to the property that the abstraction over this operation does depend on the order of the abstracted variables. This paper gives a necessary and sufficient condition for the independence of the abstraction operation of the order of the abstracted variables in the case of an underlying monoid and it treats the more general setting of a magma.

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Acknowledgements

Special thanks to Markus Siegle: In an exercise to his lecture on distributed systems the question arose that was the spark to this paper. Further he first gave a simplified version of the proof of Lemma 2 that showed that the preimage does not have to be a Boolean algebra, a two element set is sufficient. We would also like to thank Cornelius Greither who enormously helped to clarify the statements (especially Lemma 5) and gave another nice example. We would further like to thank Alexander Gouberman for the fruitful mathematical discussions.

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Authors and Affiliations

  1. University of Applied Sciences, Landshut, Germany

    Ludwig Griebl

  2. University of the Federal Armed Forces, Munich, Germany

    Johann Schuster

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  1. Ludwig Griebl
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  2. Johann Schuster
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Correspondence to Johann Schuster.

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Griebl, L., Schuster, J. Some notes on the abstraction operation for multi-terminal binary decision diagrams. Form Methods Syst Des 44, 91–99 (2014). https://doi.org/10.1007/s10703-013-0198-0

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