Yasuhito Mukouchi

Characteristic Sets for Unions of Regular Pattern Languages and Compactness

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.

Language Learning with a Neighbor System

We consider inductive language learning from positive examples, some of which may be incorrect. In the present paper, the error or incorrectness we consider is the one described uniformly in terms of a distance over strings. Firstly, we introduce a notion of a recursively generable distance over strings, and define a k-neighbor closure of a language L as the collection of strings each of which is at most k distant from some string in L. Then we define a k-neighbor system as the collection of original languages and their j-neighbor closures with j ≤ k, and adopt it as a hypothesis space. In ordinary learning paradigm, a target language, whose examples are fed to an inference machine, is assumed to belong to a hypothesis space without any guarantee. In this paper, we allow an inference machine to infer a neighbor closure instead of the original language as an admissible approximation. We formalize such kind of inference, and give some sufficient conditions for a hypothesis space.

Learning Languages Generated by Elementary Formal Systems and Its Application to SH Languages

The Elementary Formal Systems (EFSs, for short) are originally introduced by Smullyan to develop his recursion theory. In a word, EFSs are a kind of logic programs which use strings instead of terms in first order logic, and they are shown to be natural devices to define languages.

Refutable Language Learning with a Neighbor System

We consider inductive language learning and machine discovery from examples with some errors. In the present paper, the error or incorrectness we consider is the one described uniformly in terms of a distance over strings. Firstly, we introduce a notion of a recursively generable distance over strings, and for a language L, we define a k-neighbor language L? as a language obtained from L by (i) adding some strings not in L each of which is at most k distant from some string in L and by (ii) deleting some strings in L each of which is at most k distant from some string not in L. Then we define a k-neighbor system of a base language class as the collection of k-neighbor languages of languages in the class, and adopt it as a hypothesis space. We give formal definitions of k-neighbor (refutable) inferability, and discuss necessary and sufficient conditions on such kinds of inference.

Inferring a Rewriting System from Examples

In their previous paper, Mukouchi and Arikawa discussed both refutability and inferability of a hypothesis space from examples. If a target language is a member of the hypothesis space, then an inference machine should identify it in the limit, otherwise it should refute the hypothesis space itself in a finite time. They pointed out the necessity of refutability of a hypothesis space from a view point of machine discovery. Recently, Mukouchi focused sequences of examples successively generated by a certain kind of system. He call such a sequence an observation with time passage, and a sequence extended as long as possible a complete observation. Then the set of all possible complete observations is called a phenomenon of the system. In this paper, we introduce phenomena generated by rewriting systems known as 0L systems and pure grammars, and investigate their inferability in the limit from positive examples as well as refutable inferability from complete examples. First, we show that any phenomenon class generated by 0L systems is inferable in the limit from positive examples. We also show that the phenomenon class generated by pure grammars such that left hand side of each production is not longer than a fixed length is inferable in the limit from positive examples, while the phenomenon class of unrestricted pure grammars is shown not to be inferable. We also obtain the result that the phenomenon class of pure grammars such that the number of productions and that of axioms are not greater than a fixed number is inferable in the limit from positive examples as well as refutably inferable from complete examples.

Learning Decision Trees over Erasing Pattern Languages

In this paper, we consider a learning problem of decision trees over erasing patterns from positive examples in the framework of identification in the limit due to Gold and Angluin. An erasing pattern is a string pattern with constant symbols and erasable variables. A decision tree over erasing patterns can be applied to identify or express transmembrane domains of amino acid sequences, and gives intuitive knowledge expressions.We first show that the ordinary decision trees with height 1 over erasing regular patterns are learnable but those with height at most 2 are not learnable from positive examples. Then we introduce a co‐pattern pc for an erasing pattern p, and we redefine the language of a decision tree over erasing patterns as a language obtainable by finitely many applications of union operations and intersection operations to the languages of erasing patterns and co‐patterns. Under the new definition of decision trees, we show that these decision trees with height at most n are learnable from pos...

Refutable/Inductive Learning from Neighbor Examples and Its Application to Decision Trees over Patterns

The paper develops the theory of refutable/inductive learning as a foundation of discovery science from examples. We consider refutable/inductive language learning from positive examples, some of which may be incorrect. The error or incorrectness we consider is the one described uniformly in terms of a distance over strings. We define a k-neighbor closure of a language L as the collection of strings each of which is at most k distant from some string in L. In ordinary learning paradigm, a target language is assumed to belong to a hypothesis space without any guarantee. In this paper, we allow an inference machine to infer a neighbor closure instead of the original language as an admissible approximation. We formalize such kind of learning, and give some sufficient conditions for a hypothesis space.As its application to concrete problems, we deal with languages defined by decision trees over patterns. The problem of learning decision trees over patterns has been studied from a viewpoint of knowledge discovery for Genome information processing in the framework of PAC learning from both positive and negative examples. We investigate their learnability in the limit from neighbor examples as well as refutable learnability from complete examples, i.e., from both positive and negative examples. Furthermore, we present some procedures which plays an important role for designing efficient learning algorithms for decision trees over regular patterns.