Language Equations with Symmetric Difference
Systems of language equations used by Ginsburg and Rice (“Two families of languages related to ALGOL”, JACM, 1962) to represent context-free grammars are modified to use the symmetric difference operation instead of union. Contrary to a natural expectation that these two types of equations should have incomparable expressive power, it is shown that equations with symmetric difference can express every recursive set by their unique solutions, every recursively enumerable set by their least solutions and every co-recursively-enumerable set by their greatest solutions. The solution existence problem is Π1-complete, the existence of a unique, a least or a greatest solution is Π2-complete, while the existence of finitely many solutions is Σ3-complete.
KeywordsBoolean Operation Turing Machine Expressive Power Great Solution Input Alphabet
Unable to display preview. Download preview PDF.
- 9.Hartmanis, J.: Context-free languages and Turing machine computations. In: Proceedings of Symposia in Applied Mathematics, vol. 19, pp. 42–51. AMS (1967)Google Scholar
- 13.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, pp. 239–251. Springer, Heidelberg (2003); Full journal version submittedGoogle Scholar
- 14.Okhotin, A.: Sistemy yazykovykh uravnenii i zamknutye klassy funktsii algebry logiki (Systems of language equations and closed classes of logic algebra functions) in Russian. In: Proceedings of the Fifth International conference Discrete models in the theory of control systems, pp. 56–64 (2003)Google Scholar
- 19.Okhotin, A., Yakimova, O.: On language equations with complementation, TUCS Technical Report No 735, Turku, Finland (December 2005)Google Scholar
- 20.Post, E.L.: The two-valued iterative systems of mathematical logic (1941)Google Scholar
- 21.Yablonski, S.V., Gavrilov, G.P., Kudryavtsev, V.B.: Funktsii algebry logiki i klassy Posta. Functions of logic algebra and the classes of Post (1966) (in Russian)Google Scholar