We prove upper bounds for combinatorial parameters of finite relational structures, related to the complexity of learning a definable set. We show that monadic second order (MSO) formulas with parameters have bounded VC-dimension over structures of bounded clique-width, and first-order formulas with parameters have bounded VC-dimension over structures of bounded local clique-width (this includes planar graphs). We also show that MSO formulas of a fixed size have bounded strong consistency dimension over MSO formulas of a fixed larger size, for colored trees. These bounds imply positive learnability results for the PAC and equivalence query learnability of a definable set over these structures. The proofs are based on bounds for related definability problems for tree automata.
CITATION STYLE
Grohe, M., & Turán, G. (2002). Learnability and definability in trees and similar structures. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2285, pp. 645–657). Springer Verlag. https://doi.org/10.1007/3-540-45841-7_53
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