Abstract
We consider a class of length-preserving string rewriting systems and show that the set of encodings of pairs of strings < s, f > such that f can be derived from s using the rewriting rules can be accepted by finite automata. As a consequence, we show the existence of a linear time algorithm for determining the solvability of a given k x n peg-solitaire board, for any fixed k. This result is in contrast to the recent results of [UEHA] and [AVIS] that the same problem is NP-hard for n × n boards. We look at some related string rewriting systems and find conditions under which the encodings of the pairs < s, f > where f can be derived from s is regular.
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References
D. Avis and A. Deza 1996. Solitaire Cones, Technical Report, School of Computer Science, McGill University.
E. Berlekamp, J. Conway and R. Guy 1982. Winning Ways for Your Mathematical Plays: Vol. 1 Academic Press.
J. D. Beasley 1985. Ins and Outs of Peg Solitaire, Oxford University Press, Recreations in Mathematics Series.
R. Book and F. Otto 1993. String Rewriting Systems Springer-Verlag Graduate Texts in Computer Science.
R. Downey and M. Fellows, 1992. Fixed-parameter Intractability. Proceedings of 7th Annual Conf. on Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos, CA, 1992.
M. Frazier and C.D. Page, 1993. Prefix Grammars: An alternative characterization of regular languages, Information Processing Letters. Vol. 51, pp. 67–71.
M. Gardner, Peg Solitaire, in Unexpected Hanging and other Mathematical Diversions, Simon and Schuster, New York.
J. Hopcroft and J. Ullman 1979. Introduction to Automata Theory, Languages and Computation, Addison-Wesley Inc., Reading, Mass. (1979).
L. Kari, G. Paun, G. Rozenberg, A. Salomaa and S. Yu 1996. DNA Computing, Matching Systems and Universality, Turku Center for Computer Science, TUCS Technical Report No. 49.
Z. Manna 1974. Mathematical Theory of Computation, McGraw-Hill book company.
K. Sutner 1989. Linear Cellular Automata and the Garden-of-Eden, Mathematical Intelligencer Vol. 11, No. 2, Springer-Verlag, New York.
R. Uehara and S. Iwata 1990. Generalized Hi-Q is NP-complete, Trans IEICE 73.
E. Chang, S. Phillips and J. Ullman 1991. A Programming and Problem Solving Seminar, Stanford University, Computer Science Department Technical Report No. STAN-CS-91-1350.
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© 1997 Springer-Verlag Berlin Heidelberg
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Ravikvmar, B. (1997). Peg-solitaire, string rewriting systems and finite automata. In: Leong, H.W., Imai, H., Jain, S. (eds) Algorithms and Computation. ISAAC 1997. Lecture Notes in Computer Science, vol 1350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63890-3_26
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DOI: https://doi.org/10.1007/3-540-63890-3_26
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