Planar Languages and Learnability

Strings can be mapped into Hilbert spaces using feature maps such as the Parikh map. Languages can then be defined as the preimage of hyperplanes in the feature space, rather than using grammars or automata. These are the planar languages. In this paper we show that using techniques from kernel-based learning, we can represent and efficiently learn, from positive data alone, various linguistically interesting context-sensitive languages. In particular we show that the cross-serial dependencies in Swiss German, that established the non-context-freeness of natural language, are learnable using a standard kernel. We demonstrate the polynomial-time identifiability in the limit of these classes, and discuss some language theoretic properties of these classes, and their relationship to the choice of kernel/feature map.