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Equivalence Problems for Circuits over Sets of Natural Numbers

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Abstract

We investigate the complexity of equivalence problems for {∪,∩,−,+,×}-circuits computing sets of natural numbers. These problems were first introduced by Stockmeyer and Meyer (1973). We continue this line of research and give a systematic characterization of the complexity of equivalence problems over sets of natural numbers. Our work shows that equivalence problems capture a wide range of complexity classes like NL, C = L, P,Π P2 , PSPACE, NEXP, and beyond. McKenzie and Wagner (2003) studied related membership problems for circuits over sets of natural numbers. Our results also have consequences for these membership problems: We provide an improved upper bound for the case of {∪,∩,−,+,×}-circuits.

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References

  1. Àlvarez, C., Balcázar, J.L., Jenner, B.: Adaptive logspace reducibility and parallel time. Math. Syst. Theory 28(2), 117–140 (1995)

    Article  MATH  Google Scholar 

  2. Allender, E.: Making computation count: Arithmetic circuits in the nineties. SIGACT News 28(4), 2–15 (1997)

    Article  Google Scholar 

  3. Allender, E., Ogihara, M.: Relationships among PL, #L, and the determinant. RAIRO Theor. Inform. Appl. 30, 1–21 (1996)

    MATH  MathSciNet  Google Scholar 

  4. Breunig, H.: The complexity of membership problems for circuits over sets of positive numbers. In: Proceedings International Conference on Fundamentals of Computation Theory. Lecture Notes in Computer Science, vol. 4639, pp. 125–136. Springer, Berlin (2007)

    Chapter  Google Scholar 

  5. Bach, E., Shallit, J.: Algorithmic Number Theory. Efficient Algorithms of Foundations of Computing, vol. I. MIT Press, Cambridge (1996)

    MATH  Google Scholar 

  6. Ko, K.: Some observations on the probabilistic algorithms and NP-hard problems. Inf. Process. Lett. 14(1), 39–43 (1982)

    Article  MATH  Google Scholar 

  7. Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential time. In: Proceedings 13th Symposium on Switching and Automata Theory, pp. 125–129. IEEE Computer Society, Los Alamitos (1972)

    Chapter  Google Scholar 

  8. Mckenzie, P., Wagner, K.W.: The complexity of membership problems for circuits over sets of natural numbers. Comput. Complex. 16(3), 211–244 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Schönhage, A.: On the power of random access machines. In: ICALP, pp. 520–529 (1979)

  10. Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Proceedings 5th ACM Symposium on the Theory of Computing, pp. 1–9. ACM, New York (1973)

    Google Scholar 

  11. Travers, S.: The complexity of membership problems for circuits over sets of integers. Theor. Comput. Sci. 369(1), 211–229 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wagner, K.: The complexity of problems concerning graphs with regularities. In: Proceedings Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 176, pp. 544–552. Springer, Berlin (1984)

    Google Scholar 

  13. Wagner, K.W., Wechsung, G.: On the boolean closure of NP. In: Proceedings International Conference on Fundamentals of Computation Theory. Lecture Notes in Computer Science, vol. 199, pp. 485–493. Springer, Berlin (1985)

    Google Scholar 

  14. Yang, K.: Integer circuit evaluation is PSPACE-complete. In: IEEE Conference on Computational Complexity, pp. 204–213 (2000)

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  1. Julius-Maximilians-Universität Würzburg, Theoretische Informatik, Würzburg, Germany

    Christian Glaßer, Katrin Herr, Christian Reitwießner, Stephen Travers & Matthias Waldherr

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  1. Christian Glaßer
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  2. Katrin Herr
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  3. Christian Reitwießner
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  4. Stephen Travers
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  5. Matthias Waldherr
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Correspondence to Christian Glaßer.

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Glaßer, C., Herr, K., Reitwießner, C. et al. Equivalence Problems for Circuits over Sets of Natural Numbers. Theory Comput Syst 46, 80–103 (2010). https://doi.org/10.1007/s00224-008-9144-8

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  • DOI: https://doi.org/10.1007/s00224-008-9144-8

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