Learning Languages in a Union

In inductive inference, a machine is given words in a language and the machine is said to identify the language if it correctly names the language. In this paper we study classes of languages where the unions of up to a fixed number (n say) of languages from the class are identifiable. We distinguish between two different scenarios: in one scenario, the learner need only to name the language which results from the union; in the other, the learner must individually name the languages which make up the union (we say that the unioned language is discerningly identified). We define three kinds of identification criteria based on this and by the use of some naturally occurring classes of languages, demonstrate that the inferring power of each of these identification criterion decreases as we increase the number of languages allowed in the union, thus resulting in an infinite hierarchy for each identification criterion. A comparison between the different identification criteria also yielded similar hierarchies. We show that for each n, there exists a class of disjoint languages where all unions of up to n languages from this class can be discerningly identified, but there is no learner which identifies every union of n+1 languages from this class. We give sufficient conditions for classes of languages where the unions can be discerningly identified. We also present language classes which are complete with respect to weak reduction (in terms of intrinsic complexity) for our identification criteria.

[1]  Manuel Blum,et al.  A Machine-Independent Theory of the Complexity of Recursive Functions , 1967, JACM.

[2]  Hiroki Arimura,et al.  Finding Minimal Generalizations for Unions of Pattern Languages and Its Application to Inductive Inference from Positive Data , 1994, STACS.

[3]  Arun Sharma,et al.  On the intrinsic complexity of language identification , 1994, COLT '94.

[4]  Sanjay Jain,et al.  Intrinsic Complexity of Learning Geometrical Concepts from Positive Data , 2001, COLT/EuroCOLT.

[5]  Dana Angluin,et al.  Inductive Inference of Formal Languages from Positive Data , 1980, Inf. Control..

[6]  Mark A. Fulk.  Prudence and Other Conditions on Formal Language Learning , 1990, Inf. Comput..

[7]  Rolf Wiehagen,et al.  Language Learning from Texts: Degrees of Intrinsic Complexity and Their Characterizations , 2000, J. Comput. Syst. Sci..

[8]  Takeshi Shinohara,et al.  The correct definition of finite elasticity: corrigendum to identification of unions , 1991, COLT '91.

[9]  Mark A. Fulk.  A study of inductive inference machines , 1986 .

[10]  Joseph B. Kruskal,et al.  The Theory of Well-Quasi-Ordering: A Frequently Discovered Concept , 1972, J. Comb. Theory, Ser. A.

[11]  Keith Wright.  Identification of unions of languages drawn from an identifiable class , 1989, COLT '89.

[12]  Daniel N. Osherson,et al.  Systems That Learn: An Introduction to Learning Theory for Cognitive and Computer Scientists , 1990 .

[13]  Carl H. Smith,et al.  On the intrinsic complexity of learning recursive functions , 1999, COLT '99.

[14]  Manuel Blum,et al.  Toward a Mathematical Theory of Inductive Inference , 1975, Inf. Control..

[15]  Stephen Kwek,et al.  On Learning Unions of Pattern Languages and Tree Patterns , 1999, ALT.

[16]  E. Mark Gold,et al.  Language Identification in the Limit , 1967, Inf. Control..

[17]  Hiroki Arimura,et al.  Inductive inference of unbounded unions of pattern languages from positive data , 2000, Theor. Comput. Sci..

[18]  Carl H. Smith,et al.  On the Intrinsic Complexity of Learning , 1995, Inf. Comput..

[19]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .