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Automation and Remote Control

, Volume 63, Issue 1, pp 125–138 | Cite as

A Unifying Approach to Edge-valued and Arithmetic Transform Decision Diagrams

  • C. Moraga
  • T. Sasao
  • R. Stanković
Article

Abstract

This paper shows that binary decision diagrams (BDDs) and their generalizations are not only representations of switching and integer-valued functions, but also Fourier-like series expansions of them. Furthermore, it shows that edge-valued binary decision diagrams (EVBDDs) are related to arithmetic transform decision diagrams (ACDDs), which are the integer counterparts of the functional decision diagrams (FDDs). Finally, it shows that the complexity of multi-terminal binary decision diagrams (MTBDDs), EVBDDs and ACDDs of a function f depends on the structure of the truth-vector of f, partial arithmetic transform spectra of f and the arithmetic transform spectrum of f, respectively.

Keywords

Mechanical Engineer System Theory Series Expansion Unify Approach Binary Decision 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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REFERENCES

  1. 1.
    Akers, S.B., Binary Decision Diagrams, IEEE Trans.Computers, 1978, vol. C-27, no. 6, pp. 509–516.Google Scholar
  2. 2.
    Becker, B., Drechsler, R., and Theobald, M., On the Implementation of a Package for Efficient Representation and Manipulation of Functional Decision Diagrams, Proc.IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expansion in Circuit Design, Kebschull, U., Schubert, E., and Rosenstiel, W., Eds., Hamburg, Germany, 1993, pp. 162–169.Google Scholar
  3. 3.
    Besslich, Ph.W. and Trachtenberg, E.A., Binary Input/Ternary Output Switching Circuits Designed via the Sign Transform, Proc.22nd Int.Symp.on Multiple-Valued Logic, Sendai, Japan, 1992, pp. 348–354.Google Scholar
  4. 4.
    Bahar, R.I., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., and Somenzi, F., Algebraic Decision Diagrams and Their Applications, Int.Conf.on CAD, 1993, pp. 188–191.Google Scholar
  5. 5.
    Bryant, R.E., Graph-Based Algorithms for Boolean Functions Manipulation, IEEE Trans.Comput., 1986, vol. C-35, no. 8, pp. 667–691.Google Scholar
  6. 6.
    Bryant, R.E. and Chen, Y-A., Verification of Arithmetic Functions with Binary Moment Decision Diagrams, unpublished paper, 1994, CMU-CS–94–160.Google Scholar
  7. 7.
    Clarke, E.M., McMillan, K.L., Zhao, X., Fujita, M., and Yang, J., Spectral Transforms for Large Boolean Functions with Application to Technology Mapping, 30th A CM/IEEE Design.Autom.Conf., 1993.Google Scholar
  8. 8.
    Clarke, E.M., Zhao, X., Fujita, M., Matsunaga, Y., and McGeer, R., FastWalsh Transform Computation with Binary Decision Diagram, Proc.IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expansion in Circuit Design, Kebschull, U., Schubert, E., and Rosenstiel, W., Eds., Hamburg, Germany, 1993, pp. 82–85.Google Scholar
  9. 9.
    Clarke, E.M., McMillan, K.L., Zhao, X., and Fujita, M., Spectral Transforms for Extremely Large Boolean Functions, Proc.IFIP WG 10.5 Workshop on Applications of the Reed-Muller Expansion in Circuit Design, Kebschull, U., Schubert, E., and Rosenstiel, W., Eds., Hamburg, Germany, 1993, pp. 86–90.Google Scholar
  10. 10.
    Clarke, E.M., Fujita, M., McGeer, P.O., McMillan, K.L., and Yang, J.C.-Y., Multi-Terminal Binary Decision Diagrams: An Efficient Data Structure for Matrix Representation, Int.Workshop on Logic Synthesis, 1993.Google Scholar
  11. 11.
    Falkowski, B.J. and Chang, C.H., Efficient Algorithms for the Calculation of Walsh Spectrum from OBDD and Synthesis of OBDD from Walsh Spectrum for Incompletely Specified Boolean Functions, 37th Midwest Symp.on Circuits and Systems, Lafayette, Louisiana, U.S.A., 1994.Google Scholar
  12. 12.
    Falkowski, B.J. and Chang, C.H., Efficient Algorithm for the Calculation of Arithmetic Spectrum from OBDD and Synthesis of OBDD from Arithmetic Spectrum for Incompletely Specified Boolean Functions, IEEE Int.Symp.on Circuits and Systems ISCAS94, U.S.A., 1994.Google Scholar
  13. 13.
    Heidtmann, K.D., Arithmetic Spectrum Applied to Fault Detection for Combinatorial Networks, IEEE Trans.Comput., 1991, vol. 40, no. 3, pp. 320–324.Google Scholar
  14. 14.
    Hurst, S.L., Logical Processing of Digital Signals, London: Crane Russak and Edward Arnold, 1978.Google Scholar
  15. 15.
    Hurst, S.L., Miller, D.M., and Muzio, J.C., Spectral Techniques in Digital Logic, Bristol: Academic, 1985.Google Scholar
  16. 16.
    Karpovsky, M.G., Finite Orthogonal Series in the Design of Digital Devices, New York: Wiley, 1976.Google Scholar
  17. 17.
    Kukharev, G.A., Shmerko, V.P., and Yanushkievich, S.N., Technique of Binary Data Parallel Processing for VLSI, Minsk: Vysheyshaja Shcola, 1991.Google Scholar
  18. 18.
    Kurepa, Dj.R., Ensembles ordonnfies et ramififies, Paris, 1935.Google Scholar
  19. 19.
    Kurepa, Dj.R., Sets-Logics-Machines, Proc.Int.Symp.Theory of Switching, Harvard University, Cambridge, 1957, part. 1, pp. 137–146.Google Scholar
  20. 20.
    Lai, Y.F., Pedram, M., and Vrudhula, S.B.K., EVBDD-Based Algorithms for Integer Linear Programming, Spectral Transformation, and Functional Decomposition, IEEE Trans.Computer-Aided Design, 1994, vol. 13, no. 8, pp. 959–975.Google Scholar
  21. 21.
    Lee, C.Y., Representation of Switching Circuits by Binary Decision Diagrams, Bell Syst.Tech.J., 1959, vol. 38, pp. 985–999.Google Scholar
  22. 22.
    Malyugin, V.D., On a Polynomial Realization of Cortege of the Boolean Functions, Vestn.Akad.Nauk SSSR, 1982, vol. 265, no. 6.Google Scholar
  23. 23.
    Malyugin, V.D. and Sokolov, V.V., Intensive Logical Calculations, Avtom.Telemekh., 1993, no. 4, pp. 160–167.Google Scholar
  24. 24.
    Malyugin, V.D., Stankovific, R.S., and Stankovific, M., Calculations of the Coefficients of Polynomial Representations of Switching Functions Through Binary Decision Diagrams, Proc.Preventive Engineering and Information Technologies, Nifis, Yugoslavia, 1994, pp. 10–1–10–4.Google Scholar
  25. 25.
    Malyugin, V.D. and Veits, A.V., Intensive Calculations in Parallel Logic, Proc.5th Int.Workshop on Spectral Techniques, 1994, Beijing, China, pp. 63–64.Google Scholar
  26. 26.
    Muzio, J.C., Stuck Fault Sensitivity of Reed-Muller and Arithmetic Coefficients, in Theory and Applications of Spectral Techniques, Moraga, C., Ed., Dortmund, 1989, pp. 36–45.Google Scholar
  27. 27.
    Picard, C.F., Theorie des questionnaires, Paris: Gauthier-Villars, 1965.Google Scholar
  28. 28.
    Logic Synthesis and Optimization, Sasao, T., Ed., Boston: Kluwer, 1993.Google Scholar
  29. 29.
    Sasao, T., AND-EXOR Expressions and Their Optimizations, in Logic Synthesis and Optimization, Sasao, T., Ed., Boston: Kluwer, 1993, pp. 287–312.Google Scholar
  30. 30.
    Sasao, T. and Debnath, D., An Exact Minimization Algorithm for Generalized Reed-Muller Expansions, IEICE Trans.Fundamentals, 1996, vol. E79-A, no. 12, pp. 2123–2130.Google Scholar
  31. 31.
    Sasao, T., Representations of Logic Functions by Using EXOR Operators, in Representations of Discrete Functions, Sasao, T. and Fujita, M., Eds., Boston: Kluwer, 1996, pp. 29–54.Google Scholar
  32. 32.
    Stankovific, R.S., A Note on the Relation between Reed-Muller Expansions and Walsh Transform, IEEE Trans.Electromagnetic Compatibility, 1982, vol. EMC-24, no. 1, pp. 68–70.Google Scholar
  33. 33.
    Stankovific, R.S., Edge-Valued Decision Diagrams Based on Partial Reed-Muller Transforms, Proc.Reed-Muller Colloquium UK'95, Bristol, England, 1995, pp. 9–1–9–13.Google Scholar
  34. 34.
    Stankovific, R.S. and Moraga, C., Some Remarks about the Relationship among the Reed-Muller, Algebraic and Walsh Transform of Switching Functions, Yugoslav Conference for ETAN, Belgrade, Yugoslavia, 1993.Google Scholar
  35. 35.
    Stankovific, R.S. and Moraga, C., Edge-Valued Decision Diagrams for Switching Functions Based Upon the Partial Reed-Muller Transforms, Proc.2nd Int.Conf.Application of Computer Systems Szczecin, Poland, 1995, pp. 7–16.Google Scholar
  36. 36.
    Stankovic, R.S., Sasao, T., and Moraga, C., Spectral Transform Decision Diagrams, in Representations of Discrete Functions, Sasao, T. and Fujita, M., Ed., Boston: Kluwer, 1996, pp. 55–92.Google Scholar
  37. 37.
    Thayse, A., Davio, M., and Deschamps, J.-P., Optimization of Multiple-Valued Decision Diagrams, ISMVL-8, 1978, pp. 171–177.Google Scholar
  38. 38.
    Varma, D. and Trachtenberg, E.A., Efficient Spectral Techniques for Logic Synthesis, in Logic Synthesis and Optimization, Sasao, T., Ed., Boston: Kluwer, 1993, pp. 215–232.Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2002

Authors and Affiliations

  • C. Moraga
    • 1
  • T. Sasao
    • 2
  • R. Stanković
    • 3
  1. 1.University of DortmundDortmundGermany
  2. 2.Kyushu Institute of TechnologyIizukaJapan
  3. 3.NišYugoslavia

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