Developments from enquiries into the learnability of the pattern languages from positive data

The paper deals with the class RPk of sets of at most k regular patterns. A semantics of a set P of regular patterns is a union L(P) of languages defined by patterns in P. A set Q of regular patterns is said to be a more general than P, denoted by P ⊆ Q, if for any p ∈ P, there is a more general pattern q in Q than p. It is known that the syntactic containment P ⊆ Q for sets of regular patterns is efficiently computable. We prove that for any sets P and Q in RPk, (i) S2(P) ⊆ L(Q), (ii) the syntactic containment P ⊆ Q and (iii) the semantic containment L(P) ⊆ L(Q) are equivalent mutually, provided #Σ ≥ 2k - 1, where Sn(P) is the set of strings obtained from P by substituting strings with length at most n for each variable. The result means that S2 (P) is a characteristic set of L(P) within the language class for RPk under the condition above. Arimura et al. showed that the class RPk has compactness with respect to containment, if #Σ ≥ 2k+1. By the equivalency above, we prove that RPk has compactness if and only if #Σ ≥ 2k - 1.

The results obtained enable us to design efficient learning algorithms of unions of regular pattern languages such as already presented by Arimura et al. under the assumption of compactness.

Characteristic Sets for Unions of Regular Pattern Languages and Compactness

In this paper, we study two operations of taking subspaces and homomorphic images of identifiable concept classes from positive data. We give sufficient conditions for the identifiable classes to be identifiable from positive data after the applications of those two operations. As one of the examples to show the effectiveness of the obtained theorems, we will apply them to the class of zero-reversible languages, and obtain some interesting identifiable language classes related to reversible languages. Further, we will show a connection of those theories to the theory of approximate identification in the limit from positive data([Kob96]). Another important contribution of this paper is an algebraic extension of Angluin's theorem in [Ang80] based on an algebraic characterization of zero-reversible languages given by [Pin87]. This generalized theorem tells us the importance of Pin's characterization of zero-reversible languages using finitely generated groups in the context of identification in the limit from positive data.

Identifiability of Subspaces and Homomorphic Images of Zero-Reversible Languages

Mathematical Foundations of Learning Theory.

A regular pattern is a string of constant symbols and distinct variables. A semantics of a set P of regular patterns is a union L(P) of erasing pattern languages generated by patterns in P. The paper deals with the class RPk of sets of at most k regular patterns, and an efficient learning from positive examples of the language class defined by RPk. In efficient learning languages, the complexity for the MINL problem to find one of minimal languages containing a given sample is one of very important keys. Arimura et al.[5] introduced a notion of compactness w.r.t. containment for more general framework, called generalization systems, than RPk of language description which guarantees the equivalency between the semantic containment L(P) ? L(Q) and the syntactic containment P ? Q, where ? is a syntactic subsumption over the generalization systems.Under the compactness, the MINL problem reduces to finding one of minimal sets in RPk for a given sample under the subsumption ?. They gave an efficient algorithm to find such minimal sets under the assumption of compactness and some conditions.We first show that for each k ? 1, the class RPk has compactness if and only if the number of constant symbols is greater than k+1. Moreover, we prove that for each P ? RPk, a finite subset S2(P) is a characteristic set of L(P) within the class, where S2(P) consists of strings obtained from P by substituting strings with length two for each variable. Then our class RPk is shown to be polynomial time inferable from positive examples using the efficient algorithm of the MINL problem due to Arimura et al.[5], provided the number of constant symbols is greater than k + 1.

Compactness and Learning of Classes of Unions of Erasing Regular Pattern Languages

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Masako Sato

Papers

Inductive inference of formal languages