On the Complexity of Consistent Identification of Some Classes of Structure Languages Christophe Costa Florˆ encio 1⋆ UiL OTS (Utrecht University) Trans 10, 3512 JK Utrecht, Netherlands costa@let.uu.nl Abstract. In [5,7] ‘discovery procedures’ for CCGs were defined that accept a sequence of structures as input and yield a set of grammars. In [11] it was shown that some of the classes based on these procedures are learnable. The complexity of learning them was still left open. In this paper it is shown that learning some of these classes is NP-hard under certain restrictions. Keywords: identification in the limit, inductive inference, consistent learning, complexity of learning, classical categorial grammar. 1 Identification in the Limit In the seminal paper [9] the concept of identification in the limit was introdu- ced. In this model of learning a learning function receives an endless stream of sentences from the target language, called a text, and hypothesizes a grammar for the target language at each time-step. A class of languages is called learnable if and only if there exists a learning function such that after a finite number of presented sentences it guesses the right language on every text for every language from that class and does not deviate from this hypothesis. Research within this framework is known as formal learnability theory. In this paper only those aspects of formal learnability theory that are rel- evant to the proof of NP-hardness will be discussed. See [15] and [10] for a comprehensive overview of the field. In formal learnability theory the set Ω denotes the hypothesis space, which can be any class of finitary objects. Members of Ω are called grammars. The set S denotes the sample space, a recursive subset of Σ ∗ for some fi- xed finite alphabet Σ. Elements of S are called sentences, subsets of S (which obviously are sets of sentences) are called languages. The function L maps elements of Ω to subsets of S. If G is a grammar in Ω, then L(G) is called the language generated by (associated with) G. L is also called the naming function. The question whether a sentence belongs to a language generated by a grammar is called the universal membership problem. ⋆ I would like to thank Dick de Jongh and Peter van Emde-Boas for their valuable comments. A.L. Oliveira (Ed.): ICGI 2000, LNAI 1891, pp. 89–102, 2000. c  Springer-Verlag Berlin Heidelberg 2000