# Learning one-variable pattern languages very efficiently on average, in parallel, and by asking queries

## Abstract

A pattern is a string of constant and variable symbols. The language generated by a pattern π is the set of all strings of constant symbols which can be obtained from π by substituting non-empty strings for variables. We study the learnability of one-variable pattern languages in the limit with respect to the *update time* needed for computing a new single guess and the *expected total learning time* taken until convergence to a correct hypothesis. The results obtained are threefold. First, we design a *consistent* and *set-driven* learner that, using the concept of descriptive patterns, achieves update time *O*(*n*^{2} log *n*), where *n* is the size of the input sample. The best previously known algorithm to compute descriptive one-variable patterns requires time *O*(*n*^{4} log *n*) (cf. Angluin [1]). Second, we give a parallel version of this algorithm requiring time *O*(log *n*) and *O*(*n*^{3}/log *n*) processors on an EREW-PRAM. Third, we devise a one-variable pattern learner whose *expected total learning time* is *O*(l^{2} log l) provided the sample strings are drawn from the target language according to a probability distribution *D* with expected string length l. The distribution *D* must be such that strings of equal length have equal probability, but can be arbitrary otherwise. Thus, we establish the first one-variable pattern learner having an expected total learning time that *provably* differs from the update time by a *constant factor* only.

Finally, we apply the algorithm for finding descriptive one-variable patterns to learn one-variable patterns with a polynomial number of superset queries with respect to the one-variable patterns as *query language*.

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### References

- 1.D. Angluin. Finding patterns common to a set of strings.
*Journal of Computer and System Sciences*, 21:46–62, 1980.CrossRefGoogle Scholar - 2.D. Angluin. Inductive inference of formal languages from positive data.
*Information and Control*, 45:117–135, 1980.CrossRefGoogle Scholar - 3.D. Angluin. Queries and concept learning.
*Machine Learning*, 2:319–342, 1988.Google Scholar - 4.T. Erlebach, P. Rossmanith, H. Stadtherr, A. Steger, and T. Zeugmann. Efficient Learning of One-Variable Pattern Languages from Positive Data. DOI-TR-128, Department of Informatics, Kyushu University, Dec. 12, 1996; http://www.i.kyushuu.ac.jp/ thomas/treport.html.Google Scholar
- 5.G. Filé. The relation of two patterns with comparable languages. In
*Proc. 5th Ann. Symp. Theoretical Aspects of Computer Science*, LNCS 294, pp. 184–192, Berlin, 1988. Springer-Verlag.Google Scholar - 6.S. Fortune and J. Wyllie. Parallelism in random access machines. In
*Proc. 10th Ann. ACM Symp. Theory of Computing*, pp. 114–118, New York, 1978, ACM Press.Google Scholar - 7.E. M. Gold. Language identification in the limit.
*Inf. Control*, 10:447–474, 1967.Google Scholar - 8.J. Jálá.
*An introduction to parallel algorithms*. Addison-Wesley, 1992.Google Scholar - 9.K. P. Jantke. Polynomial time inference of general pattern languages. In
*Proc. Symp. Theoretical Aspects of Computer Science*, LNCS 166, pp. 314–325, Berlin, 1984. Springer-Verlag.Google Scholar - 10.T. Jiang, A. Salomaa, K. Salomaa, and S. Yu. Inclusion is undecidable for pattern languages. In
*Proc. 20th Int. Colloquium on Automata, Languages and Programming*, LNCS 700, pp. 301–312, Berlin, 1993. Springer-Verlag.Google Scholar - 11.M. Kearns and L. Pitt. A polynomial-time algorithm for learning k-variable pattern languages from examples. In
*Proc. 2nd Ann. ACM Workshop on Computational Learning Theory*, pp. 57–71, Morgan Kaufmann Publ., San Mateo, 1989.Google Scholar - 12.K.-I. Ko and C.-M. Hua. A note on the two-variable pattern-finding problem. J. Comp. Syst. Sci., 34:75–86, 1987.Google Scholar
- 13.S. Lange and R. Wiehagen. Polynomial-time inference of arbitrary pattern languages.
*New Generation Computing*, 8:361–370, 1991.Google Scholar - 14.S. Lange and T. Zeugmann. Set-driven and rearrangement-independent learning of recursive languages.
*Mathematical Systems Theory*, 29:599–634, 1996.Google Scholar - 15.D. Osherson, M. Stob and S. Weinstein.
*Systems that learn: An introduction to learning theory for cognitive and computer scientists*. MIT Press, 1986Google Scholar - 16.L. Pitt. Inductive inference, DFAs and computational complexity. In
*Proc. Analogical and Inductive Inference*, LNAI 397, pp. 18–44, Berlin, 1989, Springer-Verlag.Google Scholar - 17.A. Salomaa. Patterns. EATCS Bulletin 54:46–62, 1994.Google Scholar
- 18.A. Salomaa. Return to patterns. EATCS Bulletin 55:144–157, 1994.Google Scholar
- 19.T. Shinohara and S. Arikawa. Pattern inference. In
*Algorithmic Learning for Knowledge-Based Systems*, LNAI 961, pp. 259–291, Berlin, 1995. Springer-Verlag.Google Scholar - 20.K. Wexler and P. Culicover.
*Formal Principles of Language Acquisition*. MIT Press, Cambridge, MA, 1980.Google Scholar - 21.T. Zeugmann. Lange and Wiehagen's pattern language learning algorithm: An average-case analysis with respect to its total learning time.
*Annals of Mathematics and Artificial Intelligence*, 1997, to appear.Google Scholar - 22.T. Zeugmann and S. Lange. A guided tour across the boundaries of learning recursive languages. In
*Algorithmic Learning for Knowledge-Based Systems*, LNAI 961, pp. 190–258, Berlin, 1995. Springer-Verlag.Google Scholar