Abstract
Throughout the history of Formal Languages, one of the research directions has always been to describe computational completeness using only a small amount of possibly scarce resources. We review some of these results in the form of an essay.
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Notes
- 1.
Out of a series of 151 short videos showing interviews with Minsky within the Web of Stories - Life Stories of Remarkable People.
- 2.
If somebody wonders about how one of the ‘fathers of AI’ and the philosophy behind could write one of the first textbooks on Automata Theory, let Minsky himself make the link in the preface of his book: The abstract theory—as described in this book—tells us in no uncertain terms that the machines’ potential range is enormous, and that its theoretical limitations are of the subtlest and most elusive sort. There is no reason to suppose machines have any limitations not shared by man. But this role of Minsky created some peculiar situations until today; for instance, the said book is collected within the ‘methodological books’ in the part of our university library dedicated to psychology. Presumably, this was because one part of the book explains a mathematical model of neural networks.
- 3.
A further restricted form thereof Kuroda himself termed linear-bounded grammar, but our definition corresponds to what is nowadays called Kuroda normal form.
- 4.
- 5.
Observe that according to our definition, PCPs are collections of word pairs over a binary alphabet; of course, one could also consider such word pair collections over arbitrary alphabets, but we stick to this simpler case in our treatment.
- 6.
We use \(\overleftarrow{w}\) to denote the mirror (or reversal) of word w.
- 7.
For a discussion of programming languages tailored towards DNA computing, we refer to the study [49] conducted by Microsoft Research.
References
Bazgan, C., Brankovic, L., Casel, K., Fernau, H.: On the complexity landscape of the domination chain. In: Govindarajan, S., Maheshwari, A. (eds.) CALDAM 2016. LNCS, vol. 9602, pp. 61–72. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29221-2_6
Chomsky, N.: Three models for the description of language. IRE Trans. Inf. Theory 2(3), 113–124 (1956)
Chomsky, N.: On certain formal properties of grammars. Inf. Control 2, 137–167 (1959)
Cocke, J., Markstein, V.: The evolution of RISC technology at IBM. IBM J. Res. Dev. 34(1), 4–11 (1990)
Cocke, J., Minsky, M.: Universality of tag systems with \(P=2\). J. ACM 11(1), 15–20 (1964)
Dassow, J.: Remarks on the complexity of regulated rewriting. Fund. Inform. 7, 83–103 (1984)
Dassow, J.: A remark on limited 0L systems. J. Inf. Process. Cybern. EIK 24(6), 287–291 (1988)
Diekert, V., Kudlek, M.: Small deterministic turing machines. In: Gecseg, F., Peák, I. (eds.) Proceedings of 2nd Conference on Automata, Languages and Programming Systems, Salgótarján (Hungary) 1988, pp. 77–87. No. DM 88-4 in Technical report, Department of Mathematics, Karl Marx University of Economics (1988)
Ehrenfeucht, A., Karhumäki, J., Rozenberg, G.: The (generalized) Post correspondence problem with lists consisting of two words is decidable. Theoret. Comput. Sci. 21, 119–144 (1982)
Fernau, H.: Membership for 1-limited ET0L languages is not decidable. J. Inf. Process. Cybern. EIK 30(4), 191–211 (1994)
Fernau, H.: Membership for \(k\)-limited ET0L languages is not decidable. J. Autom. Lang. Comb. 1, 243–245 (1996)
Fernau, H.: Unconditional transfer in regulated rewriting. Acta Informatica 34, 837–857 (1997)
Fernau, H.: Nonterminal complexity of programmed grammars. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 202–213. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45132-3_13
Fernau, H.: Nonterminal complexity of programmed grammars. Theoret. Comput. Sci. 296, 225–251 (2003)
Fernau, H.: An essay on general grammars. J. Autom. Lang. Comb. 21, 69–92 (2016)
Fernau, H., Freund, R., Oswald, M., Reinhardt, K.: Refining the nonterminal complexity of graph-controlled, programmed, and matrix grammars. J. Autom. Lang. Comb. 12(1/2), 117–138 (2007)
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., Oladele, R.O.: New nonterminal complexity results for semi-conditional grammars. In: Manea, F., Miller, R.G., Nowotka, D. (eds.) CiE 2018. LNCS, vol. 10936, pp. 172–182. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94418-0_18
Fernau, H., Kuppusamy, L., Oladele, R.O., Raman, I.: Improved descriptional complexity results on generalized forbidding grammars. Discret. Appl. Math. (2021). https://doi.org/10.1016/j.dam.2020.12.027
Fernau, H., Kuppusamy, L., Raman, I.: Graph-controlled insertion-deletion systems generating language classes beyond linearity. In: Pighizzini, G., Câmpeanu, C. (eds.) DCFS 2017. LNCS, vol. 10316, pp. 128–139. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60252-3_10
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.: On the generative power of graph-controlled insertion-deletion systems with small sizes. J. Autom. Lang. Comb. 22, 61–92 (2017)
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 Informatique théorique et Applications/Theor. Inform. Appl. 52(1), 1–21 (2018)
Fernau, H., Kuppusamy, L., Raman, I.: Properties of language classes between linear and context-free. J. Autom. Lang. Comb. 23(4), 329–360 (2018)
Fernau, H., Kuppusamy, L., Raman, I.: Computational completeness of simple semi-conditional insertion-deletion systems of degree (2, 1). Nat. Comput. 18(3), 563–577 (2019)
Fernau, H., Kuppusamy, L., Raman, I.: Descriptional complexity of matrix simple semi-conditional grammars. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds.) DCFS 2019. LNCS, vol. 11612, pp. 111–123. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-23247-4_8
Fernau, H., Kuppusamy, L., Raman, I.: On matrix ins-del systems of small sum-norm. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds.) SOFSEM 2019. LNCS, vol. 11376, pp. 192–205. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10801-4_16
Fernau, H., Kuppusamy, L., Raman, I.: On path-controlled insertion-deletion systems. Acta Informatica 56(1), 35–59 (2019)
Fernau, H., Kuppusamy, L., Raman, I.: On the power of generalized forbidding insertion-deletion systems. In: Jirásková, G., Pighizzini, G. (eds.) DCFS 2020. LNCS, vol. 12442, pp. 52–63. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-62536-8_5
Fernau, H., Kuppusamy, L., Raman, I.: Generalized forbidding matrix grammars and their membrane computing perspective. In: Freund, R., Ishdorj, T.-O., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) CMC 2020. LNCS, vol. 12687, pp. 31–45. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77102-7_3
Fernau, H., Kuppusamy, L., Verlan, S.: Universal matrix insertion grammars with small size. In: Patitz, M.J., Stannett, M. (eds.) UCNC 2017. LNCS, vol. 10240, pp. 182–193. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58187-3_14
Fernau, H., Stephan, F.: Characterizations of recursively enumerable languages by programmed grammars with unconditional transfer. J. Autom. Lang. Comb. 4(2), 117–142 (1999)
Freund, R.: A general framework for sequential grammars with control mechanisms. In: Hospodár, M., Jirásková, G., Konstantinidis, S. (eds.) DCFS 2019. LNCS, vol. 11612, pp. 1–34. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-23247-4_1
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. EPTCS, vol. 31, pp. 88–98 (2010)
Freund, R., Păun, G.: On the number of non-terminal symbols in graph-controlled, programmed and matrix grammars. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 214–225. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45132-3_14
Geffert, V.: Problémy zložitosti generatívnych systémov (in Slovak). Ph.D. thesis, Katedra teoretickej kybernetiky, Matematicko-fyzikálnej fakulty UK, Bratislava (1987)
Geffert, V.: How to generate languages using only two pairs of parentheses. J. Inf. Process. Cybern. EIK 27(5/6), 303–315 (1991)
Geffert, V.: Normal forms for phrase-structure grammars. RAIRO Informatique théorique et Applications/Theor. Inform. Appl. 25, 473–498 (1991)
Jantzen, M., Kudlek, M., Lange, K.-J., Petersen, H.: Dyck\(_1\)-reductions of context-free languages. In: Budach, L., Bukharajev, R.G., Lupanov, O.B. (eds.) FCT 1987. LNCS, vol. 278, pp. 218–227. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-18740-5_45
Kari, L.: On insertions and deletions in formal languages. Ph.D. thesis, University of Turku, Finland (1991)
Kari, L.: DNA computing: arrival of biological mathematics. Math. Intell. 19(2), 9–22 (1997)
Kari, L., Daley, M., Gloor, G., Siromoney, R., Landweber, L.F.: How to compute with DNA. In: Rangan, C.P., Raman, V., Ramanujam, R. (eds.) FSTTCS 1999. LNCS, vol. 1738, pp. 269–282. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-46691-6_21
Kari, L., Păun, Gh., Thierrin, G., Yu, S.: At the crossroads of DNA computing and formal languages: characterizing recursively enumerable languages using insertion-deletion systems. In: Rubin, H., Wood, D.H. (eds.) DNA Based Computers III, DIMACS Series in Discrete Mathematics and Theretical Computer Science, vol. 48, pp. 329–338. AMS (1999)
Krassovitskiy, A., Rogozhin, Y., Verlan, S.: Computational power of insertion-deletion (P) systems with rules of size two. Nat. Comput. 10, 835–852 (2011)
Kudlek, M.: Small deterministic Turing machines. Theoret. Comput. Sci. 168(2), 241–255 (1996)
Kuroda, S.Y.: Classes of languages and linear-bounded automata. Inf. Control 7, 207–223 (1964)
Lakin, M.R., Phillips, A.: Domain-specific programming languages for computational nucleic acid systems. ACS Synth. Biol. 9(7), 1499–1513 (2020)
Margenstern, M., Păun, Gh., Rogozhin, Y., Verlan, S.: Context-free insertion-deletion systems. Theoret. Comput. Sci. 330(2), 339–348 (2005)
Masopust, T., Meduna, A.: Descriptional complexity of generalized forbidding grammars. In: Geffert, V., Pighizzini, G. (eds.) 9th International Workshop on Descriptional Complexity of Formal Systems - DCFS, pp. 170–177. University of Kosice, Slovakia (2007)
Matveevici, A., Rogozhin, Y., Verlan, S.: Insertion-deletion systems with one-sided contexts. In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 205–217. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74593-8_18
Minsky, M.L.: Recursive unsolvability of post’s problem of “tag” and other topics in theory of Turing machines. Ann. Math. 74(3), 437–455 (1961)
Minsky, M.L.: Steps toward artificial intelligence. Proc. IRE 49, 8–30 (1961)
Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice Hall, Hoboken (1967)
Neary, T.: Undecidability in binary tag systems and the Post correspondence problem for five pairs of words. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science, STACS. LIPIcs, vol. 30, pp. 649–661. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)
Neary, T., Woods, D.: The complexity of small universal Turing machines: a survey. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 385–405. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-27660-6_32
Paramasivan, M.: Operations on graphs, arrays and automata. Ph.D. thesis, Fachbereich IV, Universität Trier, Germany (2017)
Păun, Gh.: Six nonterminals are enough for generating each R.E. language by a matrix grammar. Int. J. Comput. Math. 15(1–4), 23–37 (1984)
Păun, Gh.: A variant of random context grammars: semi-conditional grammars. Theoret. Comput. Sci. 41, 1–17 (1985)
Păun, Gh., Rozenberg, G., Salomaa, A.: DNA Computing: New Computing Paradigms. Springer, Heidelberg (1998). https://doi.org/10.1007/978-3-662-03563-4
Post, E.L.: Formal reductions of the general combinatorial decision problem. Am. J. Math. 65(2), 197–215 (1943)
Post, E.L.: A variant of a recursively unsolvable problem. Bull. Am. Math. Soc. 52(4), 264–268 (1946)
Révész, G.E.: Comment on the paper “error detection in formal languages’’. J. Comput. Syst. Sci. 8(2), 238–242 (1974)
Rovan, B.: A framework for studying grammars. In: Gruska, J., Chytil, M. (eds.) MFCS 1981. LNCS, vol. 118, pp. 473–482. Springer, Heidelberg (1981). https://doi.org/10.1007/3-540-10856-4_115
Rozenberg, G., Vermeir, D.: On the effect of the finite index restriction on several families of grammars; Part 2: context dependent systems and grammars. Found. Control Eng. 3(3), 126–142 (1978)
Savitch, W.J.: How to make arbitrary grammars look like context-free grammars. SIAM J. Comput. 2(3), 174–182 (1973)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 & 623–656 (1948)
Shannon, C.E.: A universal Turing machine with two internal states. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, Annals of Mathematics Studies, vol. 34, pp. 157–165. Princeton University Press, Princeton (1956)
Takahara, A., Yokomori, T.: On the computational power of insertion-deletion systems. Nat. Comput. 2(4), 321–336 (2003)
Verlan, S.: Recent developments on insertion-deletion systems. Comput. Sci. J. Moldova 18(2), 210–245 (2010)
Verlan, S., Fernau, H., Kuppusamy, L.: Universal insertion grammars of size two. Theoret. Comput. Sci. 843, 153–163 (2020)
Vu, M., Fernau, H.: Insertion-deletion systems with substitutions I. In: Anselmo, M., Della Vedova, G., Manea, F., Pauly, A. (eds.) CiE 2020. LNCS, vol. 12098, pp. 366–378. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51466-2_33
Vu, M., Fernau, H.: Insertion-deletion with substitutions II. In: Jirásková, G., Pighizzini, G. (eds.) DCFS 2020. LNCS, vol. 12442, pp. 231–243. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-62536-8_19
Vu, M., Fernau, H.: Adding matrix control: insertion-deletion systems with substitutions III. Algorithms 14(5) (2021). https://doi.org/10.3390/a14050131
Wang, H.: A variant to Turing’s theory of computing machines. J. ACM 4(1), 63–92 (1957)
Wätjen, D.: \(k\)-limited 0L systems and languages. J. Inf. Process. Cybern. EIK 24(6), 267–285 (1988)
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Fernau, H. (2021). Parsimonious Computational Completeness. In: Moreira, N., Reis, R. (eds) Developments in Language Theory. DLT 2021. Lecture Notes in Computer Science(), vol 12811. Springer, Cham. https://doi.org/10.1007/978-3-030-81508-0_2
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