Abstract
A matrix ins-del system is described by a set of insertion-deletion rules presented in matrix form, which demands all rules of a matrix to be applied in the given order. These systems were introduced to model very simplistic fragments of sequential programs based on insertion and deletion as elementary operations as can be found in biocomputing. We are investigating such systems with limited resources as formalized in descriptional complexity. A traditional descriptional complexity measure of such a system is its ins-del size. Summing up the according numbers, we arrive at the sum-norm. We show that matrix ins-del systems with sum-norm 4 and (i) maximum length 3 with only one of insertion or deletion being performed under a one-sided context, or (ii) maximum length 2 with both insertion and deletion being performed under a one-sided context, can describe all recursively enumerable languages. We also show that if a matrix ins-del system of size s can describe the class of linear languages \(\mathrm {LIN}\), then without any additional resources, matrix ins-del systems of size s also describe the regular closure of \(\mathrm {LIN}\).
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Notes
- 1.
The symbol (*) marks situations where (parts of) a proof were omitted.
- 2.
The technical condition on MAT ins-del systems is not that severe, as we can always take a new start symbol and first generate any finite set with the resources at hand.
- 3.
There is one subtlety with the case when \(\lambda \in L(G)\): in that case, \(\lambda \) should be added as an axiom of \(\varGamma \).
References
Dassow, J., Păun, G.: Regulated Rewriting in Formal Language Theory. EATCS Monographs in Theoretical Computer Science, vol. 18. Springer, Heidelberg (1989)
Fernau, H., Kuppusamy, L.: Parikh images of matrix ins-del systems. In: Gopal, T.V., Jäger, G., Steila, S. (eds.) TAMC 2017. LNCS, vol. 10185, pp. 201–215. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-55911-7_15
Fernau, H., Kuppusamy, L., Raman, I.: Computational completeness of path-structured graph-controlled insertion-deletion systems. In: Carayol, A., Nicaud, C. (eds.) CIAA 2017. LNCS, vol. 10329, pp. 89–100. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60134-2_8
Fernau, H., Kuppusamy, L., Raman, I.: On the computational completeness of graph-controlled insertion-deletion systems with binary sizes. Theor. Comput. Sci. 682, 100–121 (2017). Special Issue on Languages and Combinatorics in Theory and Nature
Fernau, H., Kuppusamy, L., Raman, I.: Computational completeness of simple semi-conditional insertion-deletion systems. In: Stepney, S., Verlan, S. (eds.) UCNC 2018. LNCS, vol. 10867, pp. 86–100. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-92435-9_7
Fernau, H., Kuppusamy, L., Raman, I.: Investigations on the power of matrix insertion-deletion systems with small sizes. Nat. Comput. 17(2), 249–269 (2018)
Fernau, H., Kuppusamy, L., Raman, I.: On describing the regular closure of the linear languages with graph-controlled insertion-deletion systems. RAIRO Inf. théor. et Appl./Theor. Inf. Appl. 52(1), 1–21 (2018)
Fernau, H., Kuppusamy, L., Raman, I.: Properties of language classes between linear and context-free. J. Autom. Lang. Combin. 23(4), 329–360 (2018)
Freund, R., Kogler, M., Rogozhin, Y., Verlan, S.: Graph-controlled insertion-deletion systems. In: McQuillan, I., Pighizzini, G. (eds.) Proceedings Twelfth Annual Workshop on Descriptional Complexity of Formal Systems, DCFS, vol. 31 of EPTCS, pp. 88–98 (2010)
Geffert, V.: How to generate languages using only two pairs of parentheses. J. Inf. Process. Cybern. EIK 27(5/6), 303–315 (1991)
Kari, L., Thierrin, G.: Contextual insertions/deletions and computability. Inf. Comput. 131(1), 47–61 (1996)
Krassovitskiy, A., Rogozhin, Y., Verlan, S.: Computational power of insertion-deletion (P) systems with rules of size two. Nat. Comput. 10, 835–852 (2011)
Kuppusamy, L., Mahendran, A.: Modelling DNA and RNA secondary structures using matrix insertion-deletion systems. Int. J. Appl. Math. Comput. Sci. 26(1), 245–258 (2016)
Păun, G., Rozenberg, G., Salomaa, A.: DNA Computing: New Computing Paradigms. Springer, Heidelberg (1998). https://doi.org/10.1007/978-3-662-03563-4
Petre, I., Verlan, S.: Matrix insertion-deletion systems. Theor. Comput. Sci. 456, 80–88 (2012)
Stabler, E.: Varieties of crossing dependencies: structure dependence and mild context sensitivity. Cogn. Sci. 28, 699–720 (2004)
Verlan, S.: Recent developments on insertion-deletion systems. Comput. Sci. J. Moldova 18(2), 210–245 (2010)
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Fernau, H., Kuppusamy, L., Raman, I. (2019). On Matrix Ins-Del Systems of Small Sum-Norm. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_16
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