## Decision problems for semi-Thue systems with a few rules.(English)Zbl 1078.03033

Summary: We show that the accessibility problem, the common descendant problem, the termination problem and the uniform termination problem are undecidable for 3-rules semi-Thue systems. As a corollary we obtain the undecidability of the Post correspondence problem for 7 rules.

### MSC:

 03D03 Thue and Post systems, etc. 68Q42 Grammars and rewriting systems 03D35 Undecidability and degrees of sets of sentences
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### References:

 [1] S.I. Adjan, G.S. Makanin, Investigations on algorithmic problems in algebra (in Russian), Trudy Mat. Inst. Steklov. 168 (1984) 197-217, English translation in Proceedings of the Steklov Institute of Mathematics, Vol. 168, 1986, pp. 207-226. [2] Blondel, V.D.; Canterini, V., Undecidable problems for probabilistic automata of fixed dimension, Theory comput. syst., 36, 3, 231-245, (2003) · Zbl 1039.68061 [3] Book, R.; Otto, F., String rewriting systems, texts and monographs in computer science, (1993), Springer Berlin [4] Cassaigne, J.; Harju, T.; Karhumäki, J., On the undecidability of freeness of matrix semigroups, Internat. J. algebra comput., 9, 3-4, 295-305, (1999) · Zbl 1029.20027 [5] Cassaigne, J.; Karhumäki, J., Examples of undecidable problems for 2-generator matrix semi-groups, Theoret. comput. sci., 204, 29-34, (1998) · Zbl 0913.68068 [6] Claus, V., Some remarks on PCP(k) and related problems, Bull. EATCS, 12, 54-61, (1980) [7] Collins, D.J., Word and conjugacy problems in groups with only a few defining relations, Z. math. logik grundlag. math., 15, 4, 305-323, (1969) · Zbl 0188.02903 [8] Dauchet, M., Simulation of Turing machines by a left-linear rewrite rule, (), 109-120 [9] Davis, M., Computability and unsolvability, (1958), McGraw-Hill New York, reprinted by Dover Publications, New York, 1982 · Zbl 0080.00902 [10] Dershowitz, N., Termination of rewriting, J. symbolic comput., 3, 69-116, (1987) · Zbl 0637.68035 [11] Dershowitz, N.; Jouannaud, J.P., Rewrite systems, (), 243-320, (Chapter 2) · Zbl 0900.68283 [12] Ehrenfeucht, A.; Karhumäki, J.; Rozenberg, G., The (generalized) post correspondence problem with lists consisting of two words is decidable, Theoret. comput. sci., 21, 2, 119-144, (1982) · Zbl 0493.68076 [13] Geser, A., Decidability of termination of grid string rewriting rules, SIAM J. comput., 31, 4, 1156-1168, (2002), (electronic) · Zbl 1008.68063 [14] Geser, A.; Middeldorp, A.; Ohlebusch, E.; Zantema, H., Relative undecidability in term rewriting. I. the termination hierarchy, Inform. and comput., 178, 1, 101-131, (2002) · Zbl 1012.68096 [15] Geser, A.; Zantema, H., Non-looping string rewriting, Theor. inform. appl., 33, 3, 279-301, (1999) · Zbl 0951.68054 [16] Halava, V.; Harju, T.; Hirvensalo, M., Generalized post correspondence problem for marked morphisms, Internat. J. algebra comput., 10, 6, 757-772, (2000) · Zbl 0971.68124 [17] Hooper, P., The undecidability of the Turing machine immortality problem, J. symbolic logic, 31, 2, 219-234, (1966) · Zbl 0173.01201 [18] Huet, G., Confluent reductionsabstract properties and applications to term rewriting systems, J. assoc. comput. Mach., 27, 4, 797-821, (1980) · Zbl 0458.68007 [19] M. Jantzen, Confluent String Rewriting, EATCS Monograph, Vol. 14, Springer, Berlin, 1988. · Zbl 1097.68572 [20] Kobayashi, Y.; Katsura, M.; Shikishima-Tsuji, K., Termination and derivational complexity of confluent one-rule string-rewriting systems, Theoret. comput. sci., 262, 1-2, 583-632, (2001) · Zbl 0992.68120 [21] Kurth, W., One-rule semi-thue systems with loops of length one two or three, RAIRO inform. theor. appl., 30, 5, 415-429, (1996) · Zbl 0867.68064 [22] Lallement, G., The word problem for thue rewriting systems, (), 27-38 [23] Markov, A.A., Impossibility of certain algorithms in the theory of associative systems, Dokl. akad. nauk. SSSR, 55, 7, 587-590, (1947), (in Russian), reprinted in his Selected Papers, Vol. 2, MTsNMO Publisher, Moscow, 2003, pp. 3-7 (in Russian) [24] Markov, A.A., Impossibility of certain algorithms in the theory of associative systems, Dokl. akad. nauk. SSSR, 58, 353-356, (1947), (in Russian), reprinted in his Selected Papers, Vol. 2, MTsNMO Publisher, Moscow, 2003, pp. 13-17 (in Russian) · Zbl 0030.19401 [25] Yu.V. Matiyasevich, Simple examples of undecidable associative calculi, Dokl. AN SSSR 173 (16) (in Russian), English translation in Soviet Math. Dokl. 8 (1967) 555-557. · Zbl 0189.01102 [26] Matiyasevich, Yu.V., Simple examples of unsolvable canonical calculi, Trudy mat. inst. Steklov, 93, 50-88, (1967), (in Russian), English translation in Proc. Steklov Inst. Math. 93 (1967) 61-110 [27] Matiyasevich, Yu.V., On investigations on some algorithmic problems in algebra and number theory, trudy matematicheskogo instituta im. V. A. steklova, 168, 218-235, (1984), (in Russian), English translation in Proc. Steklov Inst. Math. 168(3) (1986) 227-252 · Zbl 0597.03020 [28] Matiyasevich, Yu., Word problem for thue systems with a few relations, in: term rewriting (font romeux, 1993), (), 39-53 [29] Matiyasevich, Yu.; Sénizergues, G., Decision problems for semi-thue systems with a few rules, (), 523-531 [30] McNaughton, R., Semi-thue systems with an inhibitor, J. automat. reason., 26, 4, 409-431, (2001) · Zbl 0981.68073 [31] Post, E.L., Formal reductions of the general combinatorial decision problem, Amer. J. math., 65, 197-215, (1943) · Zbl 0063.06327 [32] Post, E.L., Recursive unsolvability of a problem of thue, J. symbolic logic, 12, 1, 1-11, (1947), reprinted in: M. Davis (Ed.), Solvability, Provability, Definability: The Collected Works of Emil L. Post, Birkhäuser, Boston a.o., 1994, pp. 503-513 · Zbl 1263.03030 [33] Sénizergues, G., Some undecidable termination problems for semi-thue systems, Theoret. comput. sci., 142, 257-276, (1995) · Zbl 0873.68054 [34] G. Sénizergues, On the termination-problem for one-rule semi-Thue systems, in: RTA 96, Lecture Notes in Computer Science 1103 (1996) 302-316. [35] Steinby, M.; Thomas, W., Trees and term rewriting in 1910on a paper by axel thue, Bull. eur. assoc. theoret. comput. sci. EATCS (, 72), 256-269, (2000) [36] A. Thue. Probleme über veränderungen von zeichenreihen nach gegebenen regeln, Skr. Vid. Kristiania, I Mat. Naturv. Klasse 10 (1914) 34. Reprinted in his Selected Mathematical Papers, Universitetsforlaget, Oslo, 1977, pp. 493-524. [37] A. Thue, Die lösung eines spezialfalles eines generellen logichen problems, Kra. Videnskabs-Selskabets Skrifter. I. Mat. Nat. Kl.nr 8. Reprinted in his Selected Mathematical Papers, Universitetsforlaget, Oslo, 1977, pp. 273-310. [38] Zantema, H.; Geser, A., A complete characterization of termination of $$0^p 1^q \rightarrow 1^r 0^s$$, Appl. algebra eng. comm. comput., 11, 1, 1-25, (2000) · Zbl 0962.68086
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