Towards Abstract Categorial Grammars
Philippe de Groote
LORIA UMR no 7503 – INRIA Campus Scientifique, B.P. 239 54506 Vandœuvre les` Nancy Cedex – France [email protected]
Abstract
1. if a ∈ A, then a ∈ T (A); −Σ c : τ(c) (cons)
2. if α, β ∈ T (A), then (α −◦ β) ∈ T (A). x : α −Σ x : α (var) We now introduce the notion of a higher-order Σ = linear signature. It consists of a triple Γ, x : α −Σ t : β hA, C, τi, where: (abs) Γ −Σ (λx. t):(α −◦ β) 1. A is a finite set of atomic types;
Γ −Σ t :(α −◦ β) ∆ −Σ u : α 2. C is a finite set of constants; (app) Γ, ∆ −Σ (t u): β 3. τ : C → T (A) is a function that assigns to each constant in C a linear implicative type 2.2 Vocabulary, lexicon, grammar, and in T (A). language We now introduce the abstract notions of a vocab- Let X be a infinite countable set of λ-variables. ulary and a lexicon, on which the central notion of The set Λ(Σ) of linear λ-terms built upon a an abstract categorial grammar is based. higher-order linear signature Σ = hA, C, τi is in- A vocabulary is simply defined to be a higher- ductively defined as follows: order linear signature. 1. if c ∈ C, then c ∈ Λ(Σ); Given two vocabularies Σ1 = hA1,C1, τ1i and Σ2 = hA2,C2, τ2i, a lexicon L from Σ1 to Σ2 2. if x ∈ X, then x ∈ Λ(Σ); (in notation, L :Σ1 → Σ2) is defined to be a pair L = hF,Gi such that: 3. if x ∈ X, t ∈ Λ(Σ), and x occurs free in t exactly once, then (λx. t) ∈ Λ(Σ); 1. F : A1 → T (A2) is a function that inter- prets the atomic types of Σ1 as linear im- 4. if t, u ∈ Λ(Σ), and the sets of free variables plicative types built upon A2; of t and u are disjoint, then (t u) ∈ Λ(Σ). 2. G : C1 → Λ(Σ2) is a function that interprets Λ(Σ) is provided with the usual notion of cap- the constants of Σ1 as linear λ-terms built ture avoiding substitution, α-conversion, and β- upon Σ2; reduction (Barendregt, 1984). Given a higher-order linear signature Σ = 3. the interpretation functions are compatible hA, C, τi, each linear λ-term in Λ(Σ) may be as- with the typing relation, i.e., for any c ∈ C1, signed a linear implicative type in T (A). This the following typing judgement is derivable: type assignment obeys an inference system whose − G(c): Fˆ(τ (c)), judgements are sequents of the following form: Σ2 1 where Fˆ is the unique homomorphic exten- Γ −Σ t : α sion of F . where: As stated in Clause 3 of the above defini- 1. Γ is a finite set of λ-variable typing declara- tion, there exists a unique type homomorphism ˆ tions of the form ‘x : β’ (with x ∈ X and F : T (A1) → T (A2) that extends F . Simi- β ∈ T (A)), such that any λ-variable is de- larly, there exists a unique λ-term homomorphism ˆ clared at most once; G : Λ(Σ1) → Λ(Σ2) that extends G. In the se- quel, when ‘L ’ will denote a lexicon, it will also 2. t ∈ Λ(Σ); denote the homorphisms Fˆ and Gˆ induced by this lexicon. In any case, the intended meaning will It may be useful of thinking of the abstract lan- be clear from the context. guage as a set of abstract grammatical structures, Condition 3, in the above definition of a lexi- and of the object language as the set of concrete con, is necessary and sufficient to ensure that the forms generated from these abstract structures. homomorphisms induced by a lexicon commute Section 4 provides examples of ACGs that illus- with the typing relations. In other terms, for any trate this interpretation. lexicon L :Σ1 → Σ2 and any derivable judge- ment 2.3 Example In order to exemplify the concepts introduced so
x0 : α0, . . . , xn : αn −Σ1 t : α far, we demonstrate how to accomodate the PTQ fragment of Montague (1973). We concentrate on the following judgement Montague’s famous sentence: John seeks a unicorn (1) x0 : L (α0), . . . , xn : L (αn) −Σ2 L (t): L (α) For the purpose of the example, we make the two is derivable. This property, which is reminis- following assumptions: cent of Montague’s homomorphism requirement (Montague, 1970b), may be seen as an abstract 1. the formalism provides an atomic type realization of the compositionality principle. ‘string’ together with a binary associative We are now in a position of giving the defini- operator ‘+’ (that we write as an infix op- tion of an abstract categorial grammar. erator for the sake of readability); An abstract categorial grammar (ACG) is a 2. we have the usual logical connectives and quadruple = hΣ , Σ , , si where: G 1 2 L quantifiers at our disposal. 1. Σ1 = hA1,C1, τ1i and Σ2 = hA2,C2, τ2i We will see in Section 4 and 5 that these two as- are two higher-order linear signatures; Σ1 sumptions, in fact, are not needed. is called the abstract vovabulary and Σ2 is In order to handle the syntactic part of the ex- called the object vovabulary; ample, we define an ACG (G12). The first step consists in defining the two following vocabular- 2. L :Σ1 → Σ2 is a lexicon from the abstract vovabulary to the object vovabulary; ies:
Σ1 = h {n, np, s}, {J, Sre ,Sdicto, A, U}, 3. s ∈ T (A1) is a type of the abstract vocabu- {J 7→ np,S 7→ (np −◦ (np −◦ s)), lary; it is called the distinguished type of the re S 7→ (np −◦ (np −◦ s)), grammar. dicto A 7→ (n −◦ np),U 7→ n} i Any ACG generates two languages, an abstract language and an object language. The abstract Σ2 = h {string}, {John, seeks, a, unicorn}, {John 7→ string, seeks 7→ string, language generated by G (A(G )) is defined as follows: a 7→ string, unicorn 7→ string} i Then, we define a lexicon L12 from the abstract A( ) = {t ∈ Λ(Σ ) | − t: s is derivable} G 1 Σ1 vocabulary Σ1 to the object vocabulary Σ2:
In words, the abstract language generated by G L12 = h {n 7→ string, np 7→ string, is the set of closed linear λ-terms, built upon the s 7→ string}, abstract vocabulary Σ1, whose type is the distin- {J 7→ John, guished type s. On the other hand, the object lan- Sre 7→ λx. λy. x + seeks + y, guage generated by G (O(G )) is defined to be the Sdicto 7→ λx. λy. x + seeks + y, image of the abstract language by the term homo- A 7→ λx. a + x, morphism induced by the lexicon L : U 7→ unicorn} i
O(G ) = {t ∈ Λ(Σ2) | ∃u ∈ A(G ). t = L (u)} Finally we have G12 = hΣ1, Σ2, L12, si. The semantic part of the example is handled by difference is that our abstract vocabulary con- another ACG (G13), which shares with G12 the tains two constants corresponding to seek. Con- same abstract language. The object language of sequently, we have two distinct entries in the se- this second ACG is defined as follows: mantic lexicon, one for each possible reading. This is only a matter of choice. We could have Σ = h {e, t}, 3 adopt Morrill’s solution (which is closer to Mon- {JOHN, TRY-TO, FIND, UNICORN}, tague original analysis) by having only one ab- {JOHN 7→ e, stract constant S together with the following type TRY-TO 7→ (e −◦ ((e −◦ t) −◦ t)), assignment: FIND 7→ (e −◦ (e −◦ t)), UNICORN 7→ (e −◦ t)} i S 7→ (np −◦ (((np −◦ s) −◦ s) −◦ s))
Then, a lexicon from Σ1 to Σ3 is defined: Then the types of J and A, and the two lexicons should be changed accordingly. The semantic lex- L13 = h {n 7→ (e −◦ t), np 7→ ((e −◦ t) −◦ t), icon of this alternative solution would be simpler. s 7→ t}, The syntactic lexicon, however, would be more {J 7→ λP. P JOHN, involved, with entries such as: Sre 7→ λP. λQ. Q (λx. P S 7→ λx. λy. x + seeks + y (λz. z) A 7→ λx. λy. y (a + x) (λy. TRY-TO y (λz. FIND z x))), Sdicto 7→ 3 Three computational paradigms λP. λQ. P Compositional semantics associates meanings to (λx. TRY-TO x (λy. Q (λz. FIND y z))), utterances by assigning meanings to atomic items, A 7→ λP. λQ. ∃x. P x ∧ Q x, and by giving rules that allows to compute the U 7→ λx. UNICORN x} i meaning of a compound unit from the meanings of its parts. In the type logical approach, follow- This allows the ACG G13 to be defined as ing the Montagovian tradition, meanings are ex- hΣ1, Σ3, L13, si. pressed as typed λ-terms and combine via func- The abstract language shared by G12 and G13 tional application. contains the two following terms: Dalrymple et al. (1995) offer an alternative to this applicative paradigm. They present a deduc- Sre J (AU) (2) Sdicto J (AU) (3) tive approach in which linear logic is used as a glue language for assembling meanings. Their The syntactic lexicon L12 applied to each of these approach is more in the tradition of logic pro- terms yields the same image. It β-reduces to the gramming. following object term: The grammatical framework introduced in the John + seeks + a + unicorn previous section realizes the compositionality principle in a abstract way. Indeed, it provides On the other hand, the semantic lexicon L13 compositional means to associate the terms of yields the de re reading when applied to (2): a given language to the terms of some other language. Both the applicative and deductive ∃x. UNICORN x ∧ TRY-TOJOHN (λz. FIND z x) paradigms are available. and it yields the de dicto reading when applied to 3.1 Applicative paradigm (3): In our framework, the applicative paradigm con- TRY-TOJOHN (λy. ∃x. UNICORN x ∧ FIND y x) sists simply in computing, according to the lex- icon of a given grammar, the object image of Our handling of the two possible readings an abstract term. From a computational point of of (1) differs from the type-logical account of view it amounts to performing substitution and β- Morrill (1994) and Carpenter (1996). The main reduction. 3.2 Deductive paradigm 4.1 Context-free grammars The deductive paradigm, in our setting, answers Let G = hT,N,P,Si be a context-free grammar, the following problem: does a given term, built where T is the set of terminal symbols, N is the upon the object vocabulary of an ACG, belong set of non-terminal symbol, P is the set of rules, to the object language of this ACG. It amounts and S is the start symbol. We write L(G) for the to a kind of proof-search that has been de- language generated by G. We show how to con- scribed by Merenciano and Morrill (1997) and by struct an ACG GG = hΣ1, Σ2, L ,Si correspond- Pogodalla (2000). This proof-search relies on lin- ing to G. ear higher-order matching, which is a decidable The abstract vocabulary Σ1 = hA1,C1, τ1i is problem (de Groote, 2000). defined as follows:
3.3 Transductive paradigm 1. The set of atomic types A1 is defined to be The example developped in Section 2.3 suggests the set of non-terminal symbols N. a third paradigm, which is obtained as the com- 2. The set of constants C is a set of symbols in position of the applicative paradigm with the de- 1 1-1-correspondence with the set of rules P . ductive paradigm. We call it the transductive paradigm because it is reminiscent of the math- 3. Let c ∈ C1 and let ‘X → ω’ be the rule cor- ematical notion of transduction (see Section 4.2). responding to c. τ1 is defined to be the func- This paradigm amounts to the transfer from one tion that assigns the type [[ω]]X to c, where object language to another object language, using [[·]]X obeys the following inductive defini- a common abstract language as a pivot. tion:
4 Relating ACGs to other grammatical (a) [[]]X = X; formalisms (b) [[Y ω]]X = (Y −◦ [[ω]]X ), for Y ∈ N; In this section, we illustrate the expressive power (c) [[aω]]X = [[ω]]X , for a ∈ T . of ACGs by showing how some other families of The definition of the object vocabulary Σ2 = formal grammars may be subsumed. It must be hA2,C2, τ2i is as follows: stressed that we are not only interested in a weak form of correspondence, where only the gener- 1. A2 is defined to be {∗}. ated languages are equivalent, but in a strong form of correspondence, where the grammatical struc- 2. The set of constants C2 is defined to be the tures are preserved. set of terminal symbols T . First of all, we must explain how ACGs may 3. τ2 is defined to be the function that assigns manipulate strings of symbols. In other words, the type ‘string’ to each c ∈ C2. we must show how to encode strings as linear λ- terms. The solution is well known: it suffices It remains to define the lexicon L = hF,Gi: to represent strings of symbols as compositions of functions. Consider an arbitrary atomic type 1. F is defined to be the function that interprets ∗, and define the type ‘string’ to be (∗ −◦ ∗). each atomic type a ∈ A1 as the type ‘string’. Then, a string such as ‘abbac’ may be repre- 2. Let c ∈ C1 and let ‘X → ω’ be sented by the linear λ-term λx. a (b (b (a (c x)))), the rule corresponding to c. G is de- where the atomic strings ‘a’, ‘b’, and ‘c’ are fined to be the function that interprets c as declared to be constants of type (∗ −◦ ∗). In λx1 . . . . λxn. |ω|, where x1 . . . xn is the se- this setting, the empty word () is represented quence of λ-variables occurring in |ω|, and by the identity function (λx. x) and concatena- | · | is inductively defined as follows: tion (+) is defined to be functional composition (λf. λg. λx. f (g x)), which is indeed an associa- (a) || = λx. x; tive operator that admits the identity function as a (b) |Y ω| = y + |ω|, for Y ∈ N, and where unit. y is a fresh λ-variable; (c) |aω| = a + |ω|, for a ∈ T . 4.3 Tree adjoining grammars The construction that allows to handle the tree It is then easy to prove that GG is such that: adjoining grammars of Joshi (Joshi and Schabes, 1997) may be seen as a generalization of the con- 1. the abstract language A(GG) is isomorphic to the set of parse-trees of G. struction that we have described for the context- free grammars. Nevertheless, it is a little bit more 2. the language generated by G coincides with involved. For instance, it is necessary to triplicate the non-terminal symbols in order to distinguish the object language of GG, i.e., O(GG) = L(G). the initial trees from the auxiliary trees. We do not have enough room in this paper for For instance consider the CFG whose produc- giving the details of the construction. We will tion rules are the following: rather give an example. Consider the TAG with the following initial tree and auxiliary tree: S → , S S → aSb, NA { CC S {{ CC {{ CC which generates the language anbn. The cor- a { S d | BB responding ACG has the following abstract lan- || BB || BB guage, object language, and lexicon: || ∗ B b SNA c
Σ1 = h {S}, {A, B}, It generates the non context-free language n n n n {A 7→ S, B 7→ ((S −◦ S)} i a b c d . This TAG may be represented by the ACG, G = hΣ1, Σ2, L ,Si, where: Σ = h {∗}, {a, b}, 2 Σ = h {S, S0,S00}, {A, B, C}, {a 7→ string, b 7→ string} i 1 {A 7→ ((S00 −◦ S0) −◦ S), B 7→ (S00 −◦ ((S00 −◦ S0) −◦ S0)), L = h {S 7→ string}, C 7→ (S00 −◦ S0)} i {A 7→ λx. x, B 7→ λx. a + x + b} i
4.2 Regular grammars and rational Σ2 = h {∗}, {a, b, c, d}, transducers {a 7→ string, b 7→ string, c 7→ string, d 7→ string} i Regular grammars being particular cases of context-free grammars, they may be handled by L = h {S 7→ string,S0 7→ string, the same construction. The resulting ACGs S00 7→ string}, (which we will call “regular ACGs” for the pur- {A 7→ λf. f (λx. x), pose of the discussion) may be seen as finite state B 7→ λx. λg. a + g (b + x + c) + d, automata. The abstract language of a regular C 7→ λx. x} i ACG correspond then to the set of accepting se- quences of transitions of the corresponding au- One of the keystones in the above translation is to represent an adjunction node A as a functional tomaton, and its object language to the accepted 00 0 language. parameter of type A −◦ A . Abrusci et al. (1999) use a similar idea in their translation of the TAGs More interestingly, rational transducers may into non-commutative linear logic. also be accomodated. Indeed, two regular ACGs that shares the same abstract language correspond 5 Beyond the multiplicative fragment to a regular language homomorphism composed with a regular language inverse homomorphism. The linear λ-calculus on which we have based Now, after Nivat’s theorem (Nivat, 1968), any ra- our definition of an ACG may be seen as a rudi- tional transducer may be represented as such a bi- mentary functional programming language. The morphism. results in Section 4 indicate that, in theory, this rudimentary language is powerful enough. Never- 5.2 Exponentials theless, in practice, it would be useful to increase The exponentials of linear logic are modal oper- the expressive power of the multiplicative kernel ators that may be used to go beyond linearity. In defined in Section 2 by providing features such particular, the exponential ‘!’ allows the intuition- as records, enumerated types, conditional expres- istic implication ‘→’ to be defined, which cor- sions, etc. responds to the possibility of dealing with non- From a methodological point of view, there is linear λ-terms. A need for such non-linear λ- a systematic way of considering such extensions. terms is already present in the example of Sec- It consists of enriching the type system of the tion 2.3. Indeed, the way of getting rid of the formalism with new logical connectives. Indeed, second assumption we made at the beginning of each new logical connective may be interpreted, section 2.3 is to declare the logical symbols (i.e., through the Curry-Howard isomorphism, as a new the existential quantifier and the conjunction that type constructor. Nonetheless, the possible addi- occurs in the interpretation of A in Lexicon L13) tional connectives must satisfy the following re- as constants of the object vocabulary Σ3. Then, quirements: the interpretation of A would be something like:
1. they must be provided with introduction and λP. λQ. EXISTS (λx. AND (P x)(Q x)). elimination rules that satisfy Prawitz’s inver- sion principle (Prawitz, 1965) and the result- Now, this expression must be typable, which is ing system must be strongly normalizable; not possible in a purely linear framework. Indeed, the λ-term to which EXISTS is applied is not linear 2. the resulting term language (or at least an in- (there are two occurrences of the bound variable teresting fragment of it) must have a decid- x). Consequently, EXISTS must be given ((e → able matching problem. t) −◦ t) as a type.
The first requirement ensures that the new types 5.3 Quantifiers come with appropriate data constructors and dis- Quantifiers may also play a part. Uses of first- criminators, and that the associated evaluation order quantification, in a type logical setting, are rule terminates. This is mandatory for the applica- exemplified by Morrill (1994), Moortgat (1997), tive paradigm of Section 3. The second require- and Ranta (1994). As for second-order quantifi- ment ensures that the deductive paradigm (and cation, it allows for polymorphism. consequently the transductive paradigm) may be fully automated. 6 Grammars as first-class citizen The other connectives of linear logic are natural The difference we make between an abstract vo- candidates for extending the formalism. In partic- cabulary and an object vocabulary is purely con- ular, they all satisfy the first requirement. On the ceptual. In fact, it only makes sense relatively to other hand, the satisfaction of the second require- a given lexicon. Indeed, from a technical point ment is, in most of the cases, an open problem. of view, any vocabulary is simply a higher-order linear signature. Consequently, one may think of 5.1 Additives a lexicon L12 :Σ1 → Σ2 whose object lan- The additive connectives of linear logic ‘&’ and guage serves as abstract language of another lex- ‘⊕’ corresponds respectively to the cartesian icon L23 :Σ2 → Σ3. This allows lexicons to be product and the disjoint union. The cartesian sequentially composed. Moreover, one may eas- product allows records to be defined. The dis- ily construct a third lexicon L13 :Σ1 → Σ3 that joint union, together with the unit type ‘1’, al- corresponds to the sequential composition of L23 lows enumerated types and case analysis to be with L12. From a practical point of view, this defined. Consequently, the additive connectives means that the sequential composition of two lex- offer a good theoretical ground to provide ACG icons may be compiled. From a theoretical point with feature structures. of view, it means that the ACGs form a category whose objects are vocabularies and whose arrows R. Montague. 1970a. English as a formal language. are lexicons. This opens the door to a theory In B. Visentini et al., editor, Linguaggi nella So- where operations for constructing new grammars cieta` e nella Tecnica, Milan. Edizioni di Commu- nita.` Reprinted: (Montague, 1974, pages 188–221). from other grammars could be defined. R. Montague. 1970b. Universal grammar. Theoria, 7 Conclusion 36:373–398. Reprinted: (Montague, 1974, pages 222–246). This paper presents the first steps towards the de- sign of a powerful grammatical framework based R. Montague. 1973. The proper treatment of quan- tification in ordinary english. In J. Hintikka, on a small set of computational primitives. The J. Moravcsik, and P. Suppes, editors, Approaches to fact that these primitives are well known from natural language: proceedings of the 1970 Stanford programming theory renders the framework suit- workshop on Grammar and Semantics, Dordrecht. able for an implementation. A first prototype is Reidel. Reprinted: (Montague, 1974, pages 247– 270). currently under development. R. Montague. 1974. Formal Philosophy: selected pa- pers of Richard Montague, edited and with an intro- References duction by Richmond Thomason. Yale University Press. M. Abrusci, C. Fouquere,´ and J. Vauzeilles. 1999. Tree-adjoining grammars in a fragment of the M. Moortgat. 1997. Categorial type logic. In J. van Lambek calculus. Computational Linguistics, Benthem and A. ter Meulen, editors, Handbook of 25(2):209–236. Logic and Language, chapter 2. Elsevier. H.P. Barendregt. 1984. The lambda calculus, its syn- G. Morrill. 1994. Type Logical Grammar: Catego- tax and semantics. North-Holland, revised edition. rial Logic of Signs. Kluwer Academic Publishers, Dordrecht. J. van Benthem. 1986. Essays in Logical Semantics. Reidel, Dordrecht. M. Nivat. 1968. Transduction des langages de Chom- sky. Annales de l’Institut Fourier, 18:339–455. B. Carpenter. 1996. Type-Logical Semantics. MIT Press, Cambridge, Massachussetts and London R. T. Oehrle. 1994. Term-labeled categorial type sys- England. tems. Linguistic & Philosophy, 17:633–678.
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