Abstract
It has recently been shown, that hyper-arithmetical sets can be represented as the unique solutions of language equations over sets of natural numbers with operations of addition, subtraction and union. It is shown that the same expressive power, under a certain encoding, can be achieved by systems of just two equations, X + A = B and (X + X) + C = (X − X) + D, without using union. It follows that the problems concerning the solutions of systems of the general form are as hard as the same problems restricted to these systems with two equations, it is known that the question for solution existence is \(\Sigma^1_1\) complete.
This is a preview of subscription content, log in via an institution to check access.
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
Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. Journal of the ACM 9, 350–371 (1962)
Jeż, A.: Conjunctive grammars can generate non-regular unary languages. International Journal of Foundations of Computer Science 19(3), 597–615 (2008)
Jeż, A., Okhotin, A.: Equations over sets of natural numbers with addition only. In: STACS 2009, Freiburg, Germany, February 26-28, pp. 577–588 (2009)
Jeż, A., Okhotin, A.: Least and Greatest Solutions of Equations over Sets of Integers. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 441–452. Springer, Heidelberg (2010)
Jeż, A., Okhotin, A.: On the Computational Completeness of Equations over Sets of Natural Numbers. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 63–74. Springer, Heidelberg (2008)
Jeż, A., Okhotin, A.: Representing hyper-arithmetical sets by equations over sets of integers. Theory of Computing Systems (accepted)
Kunc, M.: The power of commuting with finite sets of words. Theory of Computing Systems 40(4), 521–551 (2007)
Kunc, M.: What Do We Know About Language Equations? In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 23–27. Springer, Heidelberg (2007)
Lehtinen, T., Okhotin, A.: On equations over sets of numbers and their limitations. International Journal of Foundations of Computer Science 22(2), 377–393 (2011)
Lehtinen, T., Okhotin, A.: On Language Equations XXK = XXL and XM = N over a Unary Alphabet. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 291–302. Springer, Heidelberg (2010)
Leiss, E.L.: Unrestricted complementation in language equations over a one-letter alphabet. Theoretical Computer Science 132, 71–93 (1994)
Okhotin, A.: Conjunctive grammars. Journal of Automata, Languages and Combinatorics 6(4), 519–535 (2001)
Okhotin, A.: Decision problems for language equations. Journal of Computer and System Sciences 76(3-4), 251–266 (2010)
Okhotin, A.: Unresolved systems of language equations: expressive power and decision problems. Theoretical Computer Science 349(3), 283–308 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lehtinen, T. (2012). Equations X + A = B and (X + X) + C = (X − X) + D over Sets of Natural Numbers. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_54
Download citation
DOI: https://doi.org/10.1007/978-3-642-32589-2_54
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32588-5
Online ISBN: 978-3-642-32589-2
eBook Packages: Computer ScienceComputer Science (R0)