Minimalist Tree Languages Are Closed Under Intersection with Recognizable Tree Languages

Minimalist grammars are a mildly context-sensitive grammar framework within which analyses in mainstream chomskyian syntax can be faithfully represented. Here it is shown that both the derivation tree languages and derived tree languages of minimalist grammars are closed under intersection with regular tree languages. This allows us to conclude that taking into account the possibility of ‘semantic crashes’ in the standard approach to interpreting minimalist structures does not alter the strong generative capacity of the formalism. In addition, the addition to minimalist grammars of complexity filters is easily shown using a similar proof method to not change the class of derived tree languages. Minimalist grammars (in the sense of [1]) are a formalization of mainstream chomskyian syntax. In this paper I will show that both derived and derivation tree languages of minimalist grammars are closed under intersection with regular tree languages. The technique used in the proofs of this fact is similar to that of [2], where non-terminals of context-free derivation trees were paired with states of an automaton. While the closure of the derived tree languages under regular intersection can be seen to follow from that of the derivation tree languages (by virtue of the monadic second order relation between the two), the proof method extends immediately to cases of linguistic interest where the connection is not as obvious, as in the case of ‘complexity filters’ in the sense of [3]. The closure of derived tree languages under regular intersection guarantees that the kind of semantic interpretation performed in the minimalist literature [4], which makes use of only a finite domain of types, cannot in virtue of partiality (semantic ‘crashes’) lead to sets of semantically well-formed trees which could not be directly derived by some minimalist grammar. The remainder of the paper is organized as follows. The next section introduces minimalist grammars, as well as some relevant notation. Section 2 contains the proofs of closure under intersection with regular tree languages of both derivation and derived tree languages of minimalist grammars. Consequences and extensions of linguistic relevance are discussed in section 3. Finally, section 4 concludes. 1 Formal Preliminaries Given a finite set A, A∗ denotes the set of all finite sequences of elements over A. The symbol denotes the empty sequence. A ranked alphabet is a finite set S. Pogodalla and J.-P. Prost (Eds.): LACL 2011, LNAI 6736, pp. 129–144, 2011. c © Springer-Verlag Berlin Heidelberg 2011