Closure properties of linear context-free tree languages with an application to optimality theory

Abstract

Context-free tree grammars, originally introduced by Rounds [Math. Systems Theory 4(3) (1970) 257–287], are powerful grammar devices for the definition of tree languages. The properties of the class of context-free tree languages have been studied for more than three decades now. Particularly important here is the work by Engelfriet and Schmidt [J. Comput. System Sci. 15(3) (1977) 328–353, 16(1) (1978) 67–99]. In the present paper, we consider a subclass of the class of context-free tree languages, namely the class of linear context-free tree languages. A context-free tree grammar is linear, if no rule permits the copying of subtrees. For this class of linear context-free tree languages we show that the grammar derivation mode, which is very important for the general class of context-free tree languages, is immaterial. The main result we present is the closure of the class of linear context-free tree languages under linear frontier-to-root tree transduction mappings. Two further results are the closure of this class under linear root-to-frontier tree transduction mappings and under intersection with regular tree languages. The results of the first part of the paper are applied to the formalisation of optimality theory. Optimality theory (OT), introduced by Prince and Smolensky [Tech. Report 1993], is a linguistic framework in which the mapping of one level of linguistic representation to another is based on rules and filters. The rules generate candidate expressions in the target representation, which are subsequently checked against the filters, so that only those candidates remain that survive this filtering process. A proposal to formalise the description of OT using formal language theory and in particular automata theory was presented by Karttunen [Proceedings of International Workshop on Finite-State Methods in Natural Language Processing, 1998, pp. 1–12] and Frank and Satta [Comput. Linguistics 24 (1998) 307–315]. The main result of these papers is that if the generator is defined as a finite-state string transducer and the filters are defined by finite-state string automata, then the whole OT-system can be defined by means of a finite-state string transducer. Considering the fact that most parts of linguistics have trees as their underlying data structures instead of strings, we show here that generators can be extended to linear frontier-to-root tree transducers on linear context-free tree languages—with constraints being regular tree languages—while the computation of optimal candidates can still be performed using finite-state techniques (over trees).