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Abstract. Tree structured data such as HTML/XML files are represented by rooted trees with ordered children and edge labels. As a representation of a tree structured pattern in such tree structured data, we propose an ordered tree pattern, called a term tree, which is a rooted tree pattern consisting of ordered children and internal structured variables. A term tree is a generalization of standard tree patterns representing first order terms in formal logic. For a set of edge labels Λ and a term tree t, the term tree language of t, denoted by LΛ(t), is the set of all labeled trees which are obtained from a term tree t by substituting arbitrary labeled trees for all variables in t. In this paper, we propose polynomial time algorithms for the following two problems for two fundamental classes of term trees. The membership problem is, given a term tree t and a tree T , to decide whether or not LΛ(t) includes T . The minimal language problem is, given a set of labeled trees S, to find a term tree t such that LΛ(t) is minimal among all term tree languages which contain all trees in S. Then, by using these two algorithms, we show that the two classes of term trees are polynomial time inductively inferable from positive data.

SUMMARYTwo-Terminal Series Parallel (TTSP, for short) graphs are used as data models in applications for electric networks and scheduling problems. We propose a TTSP term graph which is a TTSP graph having structured variables, that is, a graph pattern over a TTSP graph. Let T G T T SP be the set of all TTSP term graphs whose variable labels are mutually distinct. For a TTSP term graph g in T G T T SP , the TTSP graph language of g, denoted by L(g), is the set of all TTSP graphs obtained from g by substituting arbitrary TTSP graphs for all variables in g. Firstly, when a TTSP graph G and a TTSP term graph g are given as inputs, we present a polynomial time matching algorithm which decides whether or not L(g) contains G. The minimal language problem for the class L T T SP = {L(g) | g ∈ T G T T SP } is, given a set S of TTSP graphs, to find a TTSP term graph g in T G T T SP such that L(g) is minimal among all TTSP graph languages which contain all TTSP graphs in S . Secondly, we give a polynomial time algorithm for solving the minimal language problem for L T T SP . Finally, we show that L T T SP is polynomial time inductively inferable from positive data. key words: inductive inference,

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