Skip to main content
SpringerLink
Log in

Generating Boolean μ-expressions

  • Published:
Acta Informatica Aims and scope Submit manuscript

Abstract

In this paper, we consider the class of Boolean μ-functions, which are the Boolean functions definable by μ-expressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formulaE into an equivalent μ-expression-if possible-in time linear in ‖E‖ times\(2^{n_m } \), where ‖E‖ is the size ofE andn m is the number of variables that occur more than once inE. As an application, we obtain a polynomial time algorithm for Mundici's problem of recognizing μ-functions fromk-formulas [17]. Furthermore, we show that recognizing Boolean μ-functions is co-NP-complete for functions essentially dependent on all variables and we give a bound close to co-NP for the general case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Eiter. Exact Transversal Hypergraphs and Application to Boolean μ-Functions. Technical Report CD-TR 92/38, CD-Laboratory for Expert Systems, TU Vienna, Austria, June 1992. Journal of Symbolic Computation (to appear)

    Google Scholar 

  2. M. Garey and D. S. Johnson. Computers and Intractability — A Guide to the Theory of NP-Completeness. W.H. Freeman, New York, 1979

    Google Scholar 

  3. V. A. Gurvich. On Repetition-Free Boolean Functions. Uspekhi Math. Nauk 32: 183–184, 1977. [Russian]

    Google Scholar 

  4. V. A. Gurvich. On the Normal Form of Positional Games. Soviet Math. Dokl. 25: 572–575, 1982. English translation of Dokl. Akad. Nauk SSSR 264: 30–33 (1982)

    Google Scholar 

  5. V. A. Gurvich. Some Properties and Applications of Complete Edge-Chromatic Graphs and Hypergraphs. Soviet Math. Dokl. 30: 803–807, 1984. English translation of Dokl. Akad. Nauk SSSR 279: 1306–1310 (1984)

    Google Scholar 

  6. V. A. Gurvich. Criteria for Repetition-Freeness of Functions in the Algebra of Logic. Soviet Math. Dokl. 43: 721–726, 1991. English translation of Dokl. Akad. Nauk SSSR 318 (1991)

    Google Scholar 

  7. V. A. Gurvich. Positional Structures and Chromatic Graphs. Soviet Math. Dokl. 45: 168–172, 1992. English translation of Dokl. Akad. Nauk SSSR 322: 828–831 (1992)

    Google Scholar 

  8. V. A. Gurvich and L. Libkin. Trees as Semilattices. Discrete Mathematics, 1994 (to appear)

  9. H. Hunt III and R. Stearns. Monotone Boolean Functions, Distributive Lattices, and the Complexities of Logics, Algebraic Structures, and Computation Structures (Preliminary Report). In Proceedings STACS-86 vol. 210 of LNCS, pp. 277–287, 1986

  10. H. Hunt III and R. Stearns. The Complexity of Very Simple Boolean Formulas With Applications. SIAM J. Comp. 19(1): 44–70 (1990)

    Google Scholar 

  11. D. S. Johnson. A Catalog of Complexity Classes. volume A of Handbook of Theoretical Computer Science, chapter 2. Elsevier Science Publishers B. V. (North-Holland), 1990

  12. J. Kadin.P NP[O(logn)] and Sparse Turing-Complete Sets forNP. Journal of Computer and System Sciences39: 282–298, 1989

    Google Scholar 

  13. M. Krentel. The Complexity of Optimization Problems. Journal of Computer and System Sciences 36: 490–509, 1988

    Google Scholar 

  14. A. V. Kuznetsov. On Repetition-Free Contact Schemes and Repetition-Free Superpositions of the Functions in the Algebra of Logic. Trudy Mat. Inst. Steklov. 51: 186–225, 1958. [Russian]

    Google Scholar 

  15. Y. Matiyasewich. Possible Nontraditional Methods for Establishing the Deducibility of Propositional Formulas. Voprosy Kibernet. (Moscow), 113: 87–90, 1987. [Russian]

    Google Scholar 

  16. D. Mundici. Functions Computed by Monotone Boolean Formulas With No Repeated Variables. Theoretical Computer Science 66: 113–114, 1989

    Google Scholar 

  17. D. Mundici. Reducibility of Monotone Formulas to μ-Formulas. In 3rd Workshop Computer Science Logic (CSL), LNCS 440, pp. 267–270, 1989

  18. D. Mundici. Solution of Rota's Problem on the Order of Series-Parallel Networks. Advances in Applied Mathematics 12: 455–463, 1991

    Google Scholar 

  19. L. Pitt and L. Valiant. Computational Limitations on Learning from Examples. Journal of the ACM 35: 965–984, 1988

    Google Scholar 

  20. B. A. Trakhtenbrot. On the Theory of Repetition-Free Contact Schemes. Trudy Mat. Inst. Steklov. 51: 226–269, 1958. [Russian]

    Google Scholar 

  21. L. Valiant. A Theory of the Learnable. Commun. ACM 27(11): 1134–1142, 1984

    Google Scholar 

  22. K. Wagner. Bounded query classes. SIAM J. Comp. 19(5): 833–846, 1990

    Google Scholar 

  23. I. Wegener. The Complexity of Boolean Functions. Wiley-Teubner series in computer science. Wiley & Sons Ltd and Teubner, Stuttgart, 1987

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information Systems Department, Technical University of Vienna, Paniglgasse 16, A-1040, Vienna, Austria

    Thomas Eiter

Authors
  1. Thomas Eiter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eiter, T. Generating Boolean μ-expressions. Acta Informatica 32, 171–187 (1995). https://doi.org/10.1007/BF01177746

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01177746

Keywords

Search

Navigation