Abstract
Systems of equations of the form X = Y + Z and X = C, in which the unknowns are sets of natural numbers, “+” denotes elementwise sum of sets S + T = m + n ∣ m ∈ S, n ∈ T, and C is an ultimately periodic constant, have recently been proved to be computationally universal (Jeż, Okhotin, “Equations over sets of natural numbers with addition only”, STACS 2009). This paper establishes some limitations of such systems. A class of sets of numbers that cannot be represented by unique, least or greatest solutions of systems of this form is defined, and a particular set in this class is constructed.
Supported by the Academy of Finland under grant 118540.
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References
Charatonik, W.: Set constraints in some equational theories. Information and Computation 142(1), 40–75 (1998)
Jeż, A., Okhotin, A.: Conjunctive grammars over a unary alphabet: undecidability and unbounded growth. Theory of Computing Systems (to appear)
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.: Equations over sets of natural numbers with addition only. In: STACS 2009, Freiburg, Germany, February 26-28, 2009, pp. 577–588 (2009)
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)
Leiss, E.L.: Unrestricted complementation in language equations over a one-letter alphabet. Theoretical Computer Science 132, 71–93 (1994)
McKenzie, P., Wagner, K.: The complexity of membership problems for circuits over sets of natural numbers. Computational Complexity 16(3), 211–244 (2007)
Okhotin, A.: Conjunctive grammars. Journal of Automata, Languages and Combinatorics 6(4), 519–535 (2001)
Okhotin, A.: Decision problems for language equations with Boolean operations. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719. Springer, Heidelberg (2003)
Okhotin, A.: Unresolved systems of language equations: expressive power and decision problems. Theoretical Computer Science 349(3), 283–308 (2005)
Okhotin, A., Rondogiannis, P.: On the expressive power of univariate equations over sets of natural numbers. In: IFIP Intl. Conf. on Theoretical Computer Science, TCS 2008, Milan, Italy, September 8-10, 2008, vol. 273, pp. 215–227. IFIP (2008)
Okhotin, A., Yakimova, O.: On language equations with complementation. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 420–432. Springer, Heidelberg (2006)
Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: STOC 1973, pp. 1–9 (1973)
Tao, T., Vu, V.: Additive Combinatorics. Cambridge University Press, Cambridge (2006)
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Lehtinen, T., Okhotin, A. (2009). On Equations over Sets of Numbers and Their Limitations. In: Diekert, V., Nowotka, D. (eds) Developments in Language Theory. DLT 2009. Lecture Notes in Computer Science, vol 5583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02737-6_29
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DOI: https://doi.org/10.1007/978-3-642-02737-6_29
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