Journal of Logic, Language, and Information 8: 83–110, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 83 The Complexity of Modellability in Finite and Computable Signatures of a Constraint Logic for Head-Driven Phrase Structure Grammar PAUL JOHN KING 1 , KIRIL IVANOV SIMOV 2 , and BJØRN ALDAG 1 1 Seminar für Sprachwissenschaft, Eberhard-Karls-Universität, Kl. Wilhelmstr. 113, 72074 Tübingen, Germany (E-mail: {king,aldag}@sfs.nphil.uni-tuebingen.de); 2 Laboratory for Linguistic Modelling, Bulgarian Academy of Sciences, Avenue Acad. G. Bonchev, Sofia, Bulgaria (E-mail: kivs@bgcict.acad.bg) (Received 9 April 1997; in final form 21 November 1997) Abstract. The SRL (speciate re-entrant logic) of King (1989) is a sound, complete and decid- able logic designed specifically to support formalisms for the HPSG (head-driven phrase structure grammar) of Pollard and Sag (1994). The SRL notion of modellability in a signature is particularly important for HPSG, and the present paper modifies an elegant method due to Blackburn and Spaan (1993) in order to prove that – modellability in each computable signature is  0 1 , – modellability in some finite signature is  0 1 -hard (hence not decidable), and – modellability in some finite signature is decidable. Since each finite signature is a computable signature, we conclude that  0 1 -completeness is the least upper bound on the complexity of modellability both in finite signatures and in computable signatures, though not a lower bound in either. Key words: Complexity theory, constraint logics, HPSG formalisms 1. Introduction HPSG (head-driven phrase structure grammar) is one of several grammars created during the eighties partly in protest at the increasingly vague and indeterminate mathematical formalisms underlying the various forms of Chomskyan grammar. Pollard and Sag (1994) envisage a formalism for HPSG in which formal-syntactic entities called descriptions are closed under negation, conjunction and disjunction, each description denotes a set of linguistic objects, and the negation, conjunction and disjunction of the formal syntax denote respectively the complement, intersec- tion and union of sets of linguistic objects. A theory is a set of descriptions, and the denotation of a theory is the intersection of the denotations of its members. The formalism expresses an account of a natural language as a theory that denotes the collection of linguistic objects that the mature users of the natural language