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,\({\rm \Pi^P_{2}}\), 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allender, E.: Making computation count: Arithmetic circuits in the nineties. SIGACT NEWS 28(4), 2–15 (1997)
Breunig, H.: The complexity of membership problems for circuits over sets of positive numbers. In: Proceedings 16th International Symposium on Fundamentals of Computation Theory. LNCS, Springer, Heidelberg (to appear, 2007)
Ko, K.: Some observations on the probabilistic algorithms and NP-hard problems. Information Processing Letters 14(1), 39–43 (1982)
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 Press, Los Alamitos (1972)
McKenzie, P., Wagner, K.W.: The complexity of membership problems for circuits over sets of natural numbers. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 571–582. Springer, Heidelberg (2003)
Schönhage, A.: On the power of random access machines. In: ICALP, pp. 520–529 (1979)
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 Press, New York (1973)
Sudborough, I.H.: On the tape complexity of deterministic context-free languages. Journal of the ACM 25(3), 405–414 (1978)
Travers, S.: The complexity of membership problems for circuits over sets of integers. Theoretical Computer Science 369(1), 211–229 (2006)
Wagner, K.: The complexity of problems concerning graphs with regularities. In: Proceedings Mathematical Foundations of Computer Science. LNCS, vol. 176, pp. 544–552. Springer, Heidelberg (1984)
Wagner, K.W., Wechsung, G.: On the boolean closure of NP. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 485–493. Springer, Heidelberg (1985)
Yang, K.: Integer circuit evaluation is PSPACE-complete. In: IEEE Conference on Computational Complexity, pp. 204–213. IEEE Computer Society Press, Los Alamitos (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Glaßer, C., Herr, K., Reitwießner, C., Travers, S., Waldherr, M. (2007). Equivalence Problems for Circuits over Sets of Natural Numbers. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds) Computer Science – Theory and Applications. CSR 2007. Lecture Notes in Computer Science, vol 4649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74510-5_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-74510-5_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74509-9
Online ISBN: 978-3-540-74510-5
eBook Packages: Computer ScienceComputer Science (R0)