Exact Learning of Primitive Formal Systems Defining Labeled Ordered Tree Languages via Queries

Abstract

A formal graph system (FGS) is a logic programming system that directly manipulates graphs by dealing with graph patterns instead of terms of first-order predicate logic. In this paper, based on an FGS, we introduce a primitive formal ordered tree system (pFOTS) as a formal system defining labeled ordered tree languages. A pFOTS program is a finite set of graph rewriting rules. A logic program is well-known to be suitable to represent background knowledge. The query learning model is an established mathematical model of learning via queries in computational learning theory. In this learning model, we show the exact learnability of a pFOTS program consisting of one graph rewriting rule and background knowledge defined by a pFOTS program using a polynomial number of queries.

An Efficient Learning Algorithm for Regular Pattern Languages Using One Positive Example and a Linear Number of Membership Queries

“…We introduced a term tree pattern, which is a rooted ordered tree pattern that consists of ordered tree structures with edge labels and structured variables with labels, in [28]. Moreover, we introduced a primitive formal ordered tree system (pFOTS) as a formal system defining ordered tree languages [12]. For a pFOTS program Γ as background knowledge, we showed in [12] that the class of tree languages derived using Γ and one primitive graph rewriting rule is exactly learnable from one positive example using a polynomial number of membership queries.…”

“…Moreover, we introduced a primitive formal ordered tree system (pFOTS) as a formal system defining ordered tree languages [12]. For a pFOTS program Γ as background knowledge, we showed in [12] that the class of tree languages derived using Γ and one primitive graph rewriting rule is exactly learnable from one positive example using a polynomial number of membership queries. As future work, we will consider query learning algorithms for exactly identifying the class of tree languages generated by term tree patterns and the class of tree languages generated by from pFOTSs as background knowledge and primitive graph rewriting rules, from one positive example using a linear number of membership queries with respect to the number of edges of the positive example.…”

“…As other related work, there are query learning algorithms for identifying the classes of other languages, including the regular languages [7], the erasing pattern languages [8], the languages derived from elementary formal systems [9], [10], the tree languages derived from tree patterns [11], the tree languages derived from primitive formal ordered tree systems [12], and the sets of binary decision diagrams [13], [14].…”