Abstract
Variants of the union and concatenation operations on formal languages are investigated, in which Boolean logic in the definitions (that is, conjunction and disjunction) is replaced with the operations in the two-element field GF(2) (conjunction and exclusive OR). Union is thus replaced with symmetric difference, whereas concatenation gives rise to a new GF(2)-concatenation operation, which is notable for being invertible. All operations preserve regularity, and their state complexity is determined. Next, a new class of formal grammars based on GF(2)-operations is defined, and it is shown to have the same computational complexity as ordinary grammars with union and concatenation.
This research was carried out as a summer project at the Sirius Education Centre, Sochi, Russia. Elizaveta Sazhneva was supported by “Native Towns”, a social investment programme of PJSC “Gazprom Neft”.
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References
Albrecht, M.R., Bard, G.V., Hart, W.: Algorithm 898: efficient multiplication of dense matrices over GF(2). ACM Trans. Math. Softw. 37(1), 9:1–9:14 (2010)
Brent, R.P., Goldschlager, L.M.: A parallel algorithm for context-free parsing. Aust. Comp. Sci. Comm. 6(7), 1–10 (1984)
Damm, C.: Problems complete for \(\oplus \)L. Inf. Process. Lett. 36(5), 247–250 (1990)
Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9(3), 350–371 (1962)
Jeż, A.: Conjunctive grammars generate non-regular unary languages. Int. J. Found. Comput. Sci. 19(3), 597–615 (2008)
Leiss, E.L.: Unrestricted complementation in language equations over a one-letter alphabet. Theor. Comput. Sci. 132(2), 71–84 (1994)
Maslov, A.N.: Estimates of the number of states of finite automata. Sov. Math. Dokl. 11, 1373–1375 (1970)
Okhotin, A.: Conjunctive and Boolean grammars: the true general case of the context-free grammars. Comput. Sci. Rev. 9, 27–59 (2013)
Okhotin, A.: Parsing by matrix multiplication generalized to Boolean grammars. Theor. Comput. Sci. 516, 101–120 (2014)
Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, pp. 257–289. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-01492-5_7. Chap. 7
Rossmanith, P., Rytter, W.: Observations on log(n) time parallel recognition of unambiguous cfl’s. Inf. Process. Lett. 44(5), 267–272 (1992)
Rytter, W.: On the recognition of context-free languages. In: Computation Theory - Proceedings of 5th Symposium, Zaborów, Poland, 3–8 December 1984, pp. 318–325 (1984)
Sudborough, I.H.: A note on tape-bounded complexity classes and linear context-free languages. J. ACM 22(4), 499–500 (1975)
Valiant, L.G.: General context-free recognition in less than cubic time. J. Comput. Syst. Sci. 10(2), 308–315 (1975)
Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994)
van Zijl, L.: Magic numbers for symmetric difference NFAs. Int. J. Found. Comput. Sci. 16(5), 1027–1038 (2005)
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Bakinova, E., Basharin, A., Batmanov, I., Lyubort, K., Okhotin, A., Sazhneva, E. (2018). Formal Languages over GF(2). In: Klein, S., Martín-Vide, C., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2018. Lecture Notes in Computer Science(), vol 10792. Springer, Cham. https://doi.org/10.1007/978-3-319-77313-1_5
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