Abstract
A word w is obtained by an ordered n-pattern interpretation of a word x if there are n homomorphisms h 1,h2,ċċċ, hn such that w=h1(x)h2(x)ċs hn(x). This ordered multiple pattern interpretation is naturally extended to languages. We show a strong relationship between the family of languages obtained by ordered multiple pattern interpretations of regular, linear, and context-free languages and the family of regular, linear, and context-free simple matrix languages. Concepts of ambiguity and inherent ambiguity of ordered multiple pattern interpretation are defined and it is shown that these properties are not decidable on the class of context-free languages. Then, we investigate arbitrary multiple pattern interpretations of the same classes of languages in the Chomsky hierarchy. We show that the classes of languages obtained in this way are recognizable in polynomial time provided that all components of the pattern interpretation are injective homomorphisms. We also present a series of open problems.
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References
Angluin, D. Finding patterns common to a set of strings. Journal of Computer and System Sciences, 21:46–62, 1980.
Borsley, R. Syntactic Theory. A Unified Approach. Edward Arnold, London, 1999.
Braine, M. Children's first word combinations. Monographs of the Society for Research in Child Development, 41 (1976) (Serial No. 164).
Brown, B. and L. Leonard. Lexical influences on children's early positional patterns. Journal of Child Language, 13:219–229, 1986.
Chomsky, N. Aspects of the Theory of Syntax. MIT Press, Cambridge MA, 1965.
Chomsky, N. The Logical Structure of Linguistic Theory. Plenum, New York, 1975. (Orig.: 1955.) Chomsky, N. Lectures on Government and Binding. Foris, Dordrecht, 1981.
Chomsky, N. Knowledge of Language. Its Nature, Origin and Use. Praeger, New York, 1986.
Dassow, J. and G. Păun. Regulated Rewriting in Formal Language Theory. Akademie, Berlin, 1989.
Garey, M. and D. Johnson. Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco CA, 1979.
Honary, B. and G. Markarian. Trellis Decoding of Block Codes. Kluwer, Dordrecht, 1997.
Ibarra, O. Simple matrix languages. Information and Control, 17:359–394, 1970.
Jantke, K.P. Algorithmic learning from incomplete information. In Machines, Languages and Complexity, J. Dassow and J. Kelemen (eds.), Lecture Notes in Computer Science 381. Springer, Berlin, 188–207, 1989.
Jiang, T., E. Kinber, A. Salomaa, K. Salomaa, and S. Yu. Pattern languages with and without erasing. International Journal of Computer Mathematics, 50:147–163, 1994.
Jiang, T. and B. Ravikumar. A note on the space complexity of some decision problems for finite automata. Information Processing Letters, 40:25–31, 1991.
Joshi, A.K. and Y. Schabes. Tree-adjoining grammars. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa (eds.), vol. 3. Springer, Berlin, 69–123, 1997.
Joshi, A.K., K. Vijay-Shanker, and D. Weir. The convergence of mildly context-sensitive grammatical formalisms. In Foundational Issues in Natural Language Processing, P. Sells, S. Shieber and T. Wasow (eds.). MIT Press, Cambridge MA, 31-81, 1991.
Kolb, H.P. and U. Mönnich. Introduction. In The Mathematics of Syntactic Structure, H.P. Kolb and U. Mönnich (eds.). Mouton de Gruyter, Berlin, 1999.
Kudlek, M. and V. Mitrana. Normal forms of grammars, finite automata, abstract families, and closure properties of macrosets. In Multiset Processing: Mathematical, Computer Science, and Molecular Computing Points of View, C.S. Calude, G. Păun, G. Rozenberg and A. Salomaa (eds.), Lecture Notes in Computer Science 2235. Springer, Berlin, 135–146, 2001.
Kuich, W. and H. Maurer. On the inherent ambiguity of simple tuple languages. Computing, 7:194–203, 1971.
Langton, C.G. (ed.) Artificial Life. Santa Fe Institute Studies in the Science of Complexity VI. Addison-Wesley, Reading MA, 1989.
Maurer, H. and W. Kuich. Tuple languages. In Proceedings of the ACM International Computing Symposium, Bonn, 882–891, 1970.
Owens, R.E. Language Development: An Introduction. Allyn and Bacon, Boston, MA, 2001.
Pine, J.M. and E.V.M. Lieven. Reanalysing rote-learned phrases: individual differences in the transition to multiword speech. Journal of Child Language, 20:551–571, 1993.
Restivo, A. and S. Salemi. Words and patterns. In Developments in Language Theory, 5th International Conference DLT 2001, Vienna, W. Kuich, G. Rozenberg, A. Salomaa (eds.), Lecture Notes in Computer Science 2295. Springer, Berlin, 117–129, 2002.
Rozenberg, G. and A. Salomaa (eds.). Handbook of Formal Languages, 3 vols. Springer, Berlin, 1997.
Vogler, H. Principle languages and principle-based parsing. In The Mathematics of Syntactic Structure, H.P. Kolb and U. Mönnich (eds.). Mouton de Gruyter, Berlin, 83–111, 1999.
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Kudlek, M., Martín-vide, C. & Mitrana, V. Multiple Pattern Interpretations. Grammars 5, 223–238 (2002). https://doi.org/10.1023/A:1022175314408
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DOI: https://doi.org/10.1023/A:1022175314408