Exact Learning of Tree Patterns

Tree patterns are natural candidates for representing rules and hypotheses in many tasks such as information extraction and symbolic mathematics. A tree pattern is a tree with labeled nodes where some of the leaves may be labeled with variables, whereas a tree instance has no variables. A tree pattern matches an instance if there is a consistent substitution for the variables that allows a mapping of subtrees to matching subtrees of the instance. A nite union of tree patterns is called a forest. In this thesis, we study the learnability of tree patterns from queries when the subtrees are ordered or unordered. The learnability is determined by the semantics of matching as de ned by the types of mappings from the pattern subtrees to the instance subtrees. Exact supervised learning is used. We rst show that ordered tree patterns and forests, with an in nite label alphabet (or equivalent condition), are learnable from equivalence (and membership) queries. Ordered forests and similar classes are shown to be as hard to learn as DNF without an in nite label alphabet or equivalent. We next show that unordered tree patterns and forests are not exactly learnable from equivalence and subset queries when the mapping between subtrees is one-to-one onto, regardless of the computational power of the learner. Tree and forest patterns are learnable from equivalence and membership queries for the one-to-one into mapping. Finally, we connect the problem of learning tree patterns to inductive logic programming by describing a class of tree patterns called Clausal trees that includes nonrecursive ingle-predicate Horn clauses and show that this class is learnable from equivalence and membership queries. Integrate divide / \ ---> / \ ^ x / \ / \ / + x n ^ / \ / \ n 1 x + / \ n 1 Figure 0.1: Simple Integration Rule Integrate Integrate Integrate / \ / \ / \ ^ x ^ x ^ x / \ / \ / \ x 3 x 5 x n (a) (b) (c) Figure 0.2: Simple Learning Illustration (Integration/Ordered Trees) Acknowledgments This research was partially supported by the NSF under grant number IRI-9520243. We thank Dana Angluin, Lisa Hellerstein, Roni Khardon, Stephen Kwek, David Page, Vijay Raghavan, and Chandra Reddy for interesting discussions on the topic of this paper. We thank the reviewers for many excellent suggestions. 1 A A A / \ / \ / \ B C B C z y / \ / \ | D E F G H (a) (b) Examples Pattern Figure 0.3: Learning with Abstract Trees 2 Chapter