A DEFINITE CLAUSE VERSION OF CATEGORIAL GRAMMAR Remo Pareschi," Department of Computer and Information Science, University of Pennsylvania, 200 S. 33 rd St., Philadelphia, PA 19104,t and Department of Artificial Intelligence and Centre for Cognitive Science, University of Edinburgh, 2 Buccleuch Place, Edinburgh EH8 9LW, Scotland remo(~linc.cis.upenn.edu ABSTRACT problem by adopting an intuitionistic treatment of implication, which has already been proposed We introduce a first-order version of Catego- elsewhere as an extension of Prolog for implement- rial Grammar, based on the idea of encoding syn- ing hypothetical reasoning and modular logic pro- tactic types as definite clauses. Thus, we drop gramming. all explicit requirements of adjacency between combinable constituents, and we capture word- order constraints simply by allowing subformu- 1 Introduction lae of complex types to share variables ranging over string positions. We are in this way able Classical Categorial Grammar (CG) [1] is an ap- to account for constructiods involving discontin- proach to natural language syntax where all lin- uous constituents. Such constructions axe difficult guistic information is encoded in the lexicon, via to handle in the more traditional version of Cate- the assignment of syntactic types to lexical items. gorial Grammar, which is based on propositional Such syntactic types can be viewed as expressions types and on the requirement of strict string ad- of an implicational calculus of propositions, where jacency between combinable constituents. atomic propositions correspond to atomic types, We show then how, for this formalism, parsing and implicational propositions account for com- can be efficiently implemented as theorem proving. plex types. A string is grammatical if and only Our approach to encoding types:as definite clauses if its syntactic type can be logically derived from presupposes a modification of standard Horn logic the types of its words, assuming certain inference syntax to allow internal implications in definite rules. clauses. This modification is needed to account for In classical CG, a common way of encoding the types of higher-order functions and, as a con- word-order constraints is by having two symmet- sequence, standard Prolog-like Horn logic theorem ric forms of "directional" implication, usually in- proving is not powerful enough. We tackle this dicated with the forward slash / and the backward * I am indebted to Dale Miller for help and advice. I slash \, constraining the antecedent of a complex am also grateful to Aravind Joshi, Mark Steedman, David type to be, respectively, right- or left-adjacent. A x,Veir, Bob Frank, Mitch Marcus and Yves Schabes for com- word, or a string of words, associated with a right- ments and discussions. Thanks are due to Elsa Grunter and Amy Feh.y for advice on typesetting. Parts of this research (left-) oriented type can then be thought of as a were supported by: a Sloan foundation grant to the Cog- right- (left-) oriented function looking for an ar- nitive Science Program, Univ. of Pennsylvania; and NSF gument of the type specified in the antecedent. A grants MCS-8219196-GER, IRI-10413 AO2, ARO grants DAA29-84-K-0061, DAA29-84-9-0027 and DARPA grant convention more or less generally followed by lin- NOOO14-85-K0018 to CIS, Univ. of Pezmsylvani& guists working in CG is to have the antecedent and t Address for correspondence the consequent of an implication respectively on 270 the right and on tile left of the connective. Thus, sal and existential quantifiers. Let A be a syntactic tile type-assignment (1) says that the ditransitive variable ranging over the set of atoms, i. e. the set verb put is a function taking a right-adjacent ar- of atomic first-order formulae, and let D and G be gulnent of type NP, to return a function taking a syntactic variables ranging, respectively, over the right-adjacent argument of type PP, to return a set of definite clauses and the set of goal clauses. function taking a left-adjacent argument of type We introduce the notions of definite clause and NP, to finally return an expression of the atomic of goal clause via the two following mutually re- type S. cursive definitions for the corresponding syntactic variables D and G: (1) put: ((b~xNP)/PP)/NP The Definite Clause Grammar (DCG) framework • D:=AIG--AIVzDID1AD2 [14] (see also [13]), where phrase-structure gram- mars can be encoded as sets of definite clauses • G:=AIG1AG=I3~:GID~G (which are themselves a subset of Horn clauses), We call ground a clause not containing variables. and the formalization of some aspects of it in [15], We refer to the part of a non-atomic definite clause suggests a more expressive alternative to encode coming on the left of the implication connective word-order constraints in CG. Such an alterna- as to the body of the clause, and to the one on tive eliminates all notions of directionality from the right as to the head. With respect to standard the logical connectives, and any explicit require- Horn logic syntax, the main novelty in the defini- ment of adjacency between functions and argu- tions above is that we permit implications in goals ments, and replaces propositions with first-order and in the bodies of definite clauses. Extended • formulae. Thus, atomic types are viewed as atomic Horn logic syntax of this kind has been proposed formulae obtained from two-place predicates over to implement hypothetical reasoning [3] and mod- string positions represented as integers, the first ules [7] in logic programming. We shall first make and the second argument corresponding, respec- clear the use of this extension for the purpose of tively, to the left and right end of a given string. linguistic description, and we shall then illustrate Therefore, the set of all sentences of length j its operational meaning. generated from a certain lexicon corresponds to the type S(0,j). Constraints over the order of constituents are enforced by sharing integer in- dices across subformulae inside complex (func- 3 First-order tional) types. Categorial Grammar This first-order version of CG can be viewed as a logical reconstruction of some of the ideas behind 3.1 Definite Clauses as Types the recent trend of Categorial Unification Gram- mars [5, 18, 20] 1. A strongly analogous develop- We take CONN (for "connects") to be a three- ment characterizes the systems of type-assignment place predicate defined over lexical items and pairs for the formal languages of Combinatory Logic and of integers, such that CONN(item, i,j) holds if Lambda Calculus, leading from propositional type and only if and only if i = j - 1, with the in- systems to the "formulae-as-types" slogan which is tuitive meaning that item lies between the two behind the current research in type theory [2]. In consecutive string positions i and j. Then, a this paper, we show how syntactic types can be en- most direct way to translate in first-order logic coded using an extended version of standard Horn the type-assignment (1) is by the type-assignment logic syntax. (2), where, in the formula corresponding to the as- signed type, the non-directional implication con- nective --, replaces the slashes. 2 Definite Clauses with In- ternal Implications (2) put : VzVyYzVw[CONN(put, y - 1, y) --* Let A and ---* be logical connectives for conjunc- (NP(y, z) -- tion and implication, and let V and 3 be the univer- (PP(z, w) -- 1Indeed, Uszkoreit [18] mentions the possibility of en- coding order constraints among constituents via variables (NP(z, y - 1) --* ranging over string positions in the DCG style. s(=, ~o))))] 271 A definite clause equivalent of tile formula in (2) 3.3 Arithmetic Predicates is given by the type-assignment (3) 2 . The fact that we quantify over integers allows us to use arithmetic predicates to determine sub- sets of indices over which certain variables must VzVyVzVw[CONN(put, y -- 1, y) A (3) put: range. This use of arithmetic predicates charac- NP(y, z) ^ terizes also Rounds' ILFP notation [15], which ap- PP(z, w) A pears in many ways interestingly related to the gP(z, y - 1) --* S(x, w)] framework proposed here. We show here below how this capability can be exploited to account for a case of extraction which is particularly prob- Observe that the predicate CONNwill need also lematic for bidirectional propositional CG. to be part of types assigned to "non-functional" lexical items. For example, we can have for the noun-phrase Mary the type-assignment (4). 3.3.1 Non-perlpheral Extraction Both the propositional type (5) and the first- order type (6) are good enough to describe the (4) Mary : Vy[OONN(Mary, y- 1,y) .-.-* kind of constituent needed by a relative pronoun NP(y - 1, y)] in the following right-oriented case of peripheral extraction, where the extraction site is located at one end of the sentence. (We indicate the extrac- 3.2 Higher-order Types and Inter- tion site with an upward-looking arrow.) nal Implications Propositional CG makes crucial use of func- which [Ishallput a book on T ] tions of higher-order type. For example, the type- assignment (5) makes the relative pronoun which However, a case of non.peripheral extraction, into a function taking a right-oriented function where the extraction site is in the middle, such from noun-phrases to sentences and returning a as relative clause 3. This kind of type-assignment has been used by several linguists to provide attractive which [ I shall put T on the table ] accounts of certain cases of extraction [16, 17, 10]. is difficult to describe in bidirectional proposi- tional CG, where all functions must take left- or (5) which: REL/(S/NP) right-adjacent arguments.