Szymczak, Bartlomiej Antoni

English

Categorial Grammar is a well established tool for describing natural language semantics. In the current paper we discuss some of its drawbacks and how it could be extended to overcome them. We use the extended version for deriving ontological semantics from text. A proof-of-concept implementation is also presented

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Proceedings of the International Multiconference onComputer Science and Information Technology pp. 295–298ISBN 978-83-60810-22-4ISSN 1896-7094Ontological Semantics in ModiﬁedCategorial GrammarBartłomiej Antoni SzymczakDTU Informatics, Technical University of DenmarkInternational Language Studies and Computational LinguisticsCopenhagen Business SchoolEmail: bas@imm.dtu.dkAbstract—Categorial Grammar is a well established tool fordescribing natural language semantics [1]. In the current paperwe discuss some of its drawbacks and how it could be extendedto overcome them. We use the extended version for deriving on-tological semantics from text. A proof-of-concept implementationis also presented.I. INTRODUCTIONIN THIS paper we extend the usual Categorial Grammarframework in order to achieve more ﬂexibility. We alsopresent how it can be used with an ontological component,which imposes well-formedness restrictions on sentences.The objective is to integrate formal ontologies with semanticdomains. Such an ontology-oriented semantics may be usefule.g. in content-based text search.Throughout the paper we use classical Church’s type theory,C, as presented in [2]. We use the convention that functionaltypes associate to the left, i.e. type γβα is the same as (γβ)α,which is also sometimes denoted as α → (β → γ).The work presented here takes place within the SIABOproject [3].II. ONTOLOGICAL SEMANTICSWe wish to construct ontological semantics for a fragmentof English, following the approach in [4] and [5].We can represent concepts appearing in a skeleton ontologyas constants of type α: Childα Tallα Runningα    . Thoseconcepts are to be thought of as sets, e.g. Childα is a setof all imaginable children. Tallα is the set of all tall objects,thus properties such as “tall” are introduced in the ontologyon the par with classes. Runningα is a set of all imaginableactions of running, etc. Introducing nominalized verb formsas concepts in the ontology is in line with the adoption ofthe Davidsonian view. We also disregard meaning of deter-miners in the present fragment, as the resulting semanticsis intended to be used for content–based text search. Wedon’t want to be speciﬁc about α, but in the current exampleit could be replaced by oι, which is traditionally used forrepresenting sets. We will represent intersection of conceptsusing constant ∩ααα, e.g. child ∩ tall will be represented as∩αααChildαTallα.A simple skeleton ontology can be represented using a setof factual clauses of the form:SuboααChildαPersonαSuboααPersonαPhysicalαSuboααPersonαAnimateαSuboααRunningαActionαSuboαα is a direct descendant relation, representing the linesthat are drawn in the Hasse diagrams. Isaoαα can be speciﬁedas the (reﬂexive) transitive closure of Suboαα in the followingway:∀cα[Isaoααcαcα]∀cα∀aα∀pα[Isaoααcαaα ⊃ Suboααcαpα ∧ Isaoααpαaα]We use constants of type ρ to represent roles (binary rela-tions), e.g. Agtρ, Locρ. We can use Peirce product [6] to createcompound concepts. We use constant :ααρ for that purpose,e.g. agt : child will be represented as :ααρ AgtρChildα. Sucha concept formation is well-known from Description Logics,where it would be represented as ∀agt child.For a sample sentence “The tall kid runs”, we would liketo derive the following ontologico–algebraic meaning:running ∩ agt : (child ∩ tall)This can be represented in type theory as a wﬀα:∩αααRunningα[:ααρ Agtρ[∩αααChildαTallα]]We shall extend the speciﬁcation of subsumption relation toaccomodate for the intersection:∀xα∀yα∀zα[[Isaoαα[∩αααxαyα]zα] ⊃ [Isaoααxαzα]]∀xα∀yα∀zα[[Isaoαα[∩αααxαyα]zα] ⊃ [Isaoααyαzα]]III. MODIFIED CATEGORIAL GRAMMARWe introduce type ω. Constants of this type represent wordsin English, e.g. W kidω, W tallω, W runsω. Notice that weuse different names for constants denoting words and thoserepresenting concepts, e.g. W tallω and Tallα. In this waythey are not confused.Let us deﬁne a new type, say, κ, which we will use forlexical entries. We also deﬁne three constants, which act astype constructors:978-83-60810-22-4/09/$25.00 c   2009 IEEE 295Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 30,2010 at 09:55:46 UTC from IEEE Xplore.  Restrictions apply. 296 PROCEEDINGS OF THE IMCSIT. VOLUME 4, 2009Bκ(oκκ)Fκ(oκκ)EκαLexical entries are represented using the predicate constantLexoκω. The lexicon consists of the set of factual clauses ofthe form:LexoκωW kidω[EκαChildα]The type constructor Eκα is used in the lexical entry in casethe meaning of a word is some “ﬁxed” ontological concept.The other constructors, Fκ(oκκ) and Bκ(oκκ) are reminiscentof the Categorial Grammar’s backward and forward slashes,respectively. A lexical entry for word “tall” might look asfollows:LexoκωW tallω[Fκ(oκκ)P talloκκ]In the above, P talloκκ is a predicate, which might bedeﬁned in the following way:∀cα[[P talloκκ[Eκαcα][Eκα[∩αααtallαcα]]]⊃ [IsaoααcαPhysicalα]]The novelty in our approach is the use of predicates inmeanings of words. In standard Categorial Grammar, lambdaterms are used for that purpose. They are combined usingβ–reduction, with the only provision of categorial agree-ment. Consider the sentence “vitamin smiles.” In traditionalCG, “vitamin” has category np and meaning vitamin. Theword “smiles” has category np\s and meaning λx smile(x).Since the categories ﬁt together, the meanings get com-bined using β–reduction, and the sentence gets the meaningsmile(vitamin). While the sentence is correct syntactically,it’s incorrect from an ontological point of view. Unfortunately,usual CG does not allow us to introduce any ontologicalrestrictions on the semantics. Our approach can reject this sen-tence, as we can use the following lexical entry for “smiles”:LexoκωW smilesω[Bκ(oκκ)P smilesoκκ]∀cα[[P smilesoκκ[Eκαcα][Eκα[∩αααSmilingα[:ααρ Agtρcα]]]]⊃ [IsaoααcαAnimateα]]The inability of β–reduction to fail has been already rec-ognized as a problem by G. Ben–Avi and N. Francez in [7].They have introduced a new formalism, which includes “β–reduction for ontologically–well typed λ–terms”, among 12deﬁnitions that constitute “the Ontological Lambek Calculus”.Our proposal, however, has a few further advantages:• It’s very formal – it’s formalized fully within C.• It’s very simple – it consists of a few formulas only.• It’s very ﬂexible – it allows adding arbitrary restrictions,e.g. one might like to use restrictions based on parthoodrelation rather than subsumption.• It can be implemented in a straight-forward way, aspresented in Section VFor those reasons, rather than using β-reduction, we proposeusing a general proof system, which is a machinery havinginherently the notion of failure at disposal. Now, desiredsemantic restriction can be handled by non-provability of acertain statement, e.g.:K 0 ∃xα[P smilesoκκ[EκαV itaminα]xα]In the above K denotes a set of formulas consisting oflexical assertions and rules introduced throughout this paper.So a vitamin cannot smile, but a child certainly can:K ⊢ ∃xα[P smilesoκκ[EκαChildα]xα]Not only are we interested in the provability of such a goal,but we would also like to know what is the resulting semantics.In this case it’s: Eκα[∩αααSmilingα[:ααρ AgtρChildα]]In other words:K ⊢ [P smilesoκκ[EκαChildα][Eκα[∩αααSmilingα[:ααρ AgtρChildα]]]]IV. ELIMINATION RULESRecall that the forward sequent rule of the natural–deductionLambek Calculus takes the form:a ⇒ Φ1 : A B b ⇒ Φ2 : Ba b ⇒ Φ1(Φ2) : Aand the backward rule:a ⇒ Φ1 : B b ⇒ Φ2 : B\Aa b ⇒ Φ2(Φ1) : AIn the above a b is the concatenation of a and b. Noticethat the only provision for the elimination to take place is theagreement of syntactic categories. This is because Φ1(Φ2) isalways a well-formed λ-term, which can be β-reduced.For the ease of reading, let us present the forward elimina-tion rule of the generalized grammar in a similar, though quiteinformal way:aθ ⇒ Fκ(oκκ)poκκ bθ ⇒ jκ K ⊢ poκκjκkκaθ bθ ⇒ kκLet us formalize our generalized elimination rules in C. Thederivation relation (⇒) can be represented using a constantRoκθ. The sequences of meanings will be represented using awell–known logic representation for lists. For the empty listwe use the Nθ constant, and the list constructor is representedby Lθθκ.The list concatenation can be represented using the listappending well–known from logic programming, though heredeﬁned for non–empty lists only:∀xκ∀yκ∀tθ[Aoθθθ[LθθκxκNθ][Lθθκyκtθ][Lθθκxκ[Lθθκyκtθ]]]∀hκ∀lθ∀mθ∀tθ[[Aoθθθ[Lθθκhκlθ]mθ[Lθθκhκtθ]]⊃ [Aoθθθlθmθtθ]]Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 30,2010 at 09:55:46 UTC from IEEE Xplore.  Restrictions apply. BARTŁOMIEJ ANTONI SZYMCZAK: ONTOLOGICAL SEMANTICS IN MODIFIED CATEGORIAL GRAMMAR 297The above use of Aoθθθ asserts that the concatenation of aand b yields g, or g is split into a and b. Now the forwardelimination rule can be formalized in C:∀aθ∀bθ∀gθ∀jκ∀kκ∀poκκ[Roκθgθkκ]⊃ [Aoθθθaθbθgθ]∧ [Roκθaθ[Fκ(oκκ)poκκ]]∧ [Roκθbθjκ]∧ [poκκjκkκ]Similarly, the backward elimination rule:∀aθ∀bθ∀gθ∀jκ∀kκ∀poκκ[Roκθgθkκ]⊃ [Aoθθθaθbθgθ]∧ [Roκθaθjκ]∧ [Roκθbθ[Bκ(oκκ)poκκ]]∧ [poκκjκkκ]We also need the following grammatical axiom, whichexpresses that if the list of meanings contains only oneelement, that element is the resulting meaning:∀kκ[Roκθ[LθθκkκNθ]kκ]The English text can also be represented formally in C. Weintroduce a new type ζ for that purpose. We will representthe text as a list of words. An empty list of words will berepresented by a constant Tnilξ and a list constructor by aconstant Tξξω.V. IMPLEMENTATION OUTLINEUsing C instead of some kind of untyped logic as the under-lying formalism has important advantages regarding the imple-mentation. It forces us to think of the type of every formula,subformula and symbol. Thanks to that, the resulting formal-ization of the grammar is well-suited for implementation in astrongly-typed programming language. Using such a language(e.g. Mercury instead of Prolog) helps avoid very hard-to-ﬁndbugs and allows the compiler to generate much faster code.We are interested in translating the formal speciﬁcationgiven so far to a Prolog–like logic programming language,so that we can execute speciﬁc queries. For that purpose, wehave formalized the grammar in C using only deﬁnite clauses.Furthermore, we perform the following steps:• We write all the constants in lower–case• We write all variables in upper–case• We use i for intersection constant and p for the Peirceproduct constant• We drop the subscripts indicating types in all formulas• We replace ‘⊃’ and ‘∧’ with ‘:-’ and ‘,’, respectively.• We add a period at the end of each formula.• We replace curried notation of argument application witha non–curried one.• We simply remove the universal quantiﬁcation over allvariables, as it’s implicit.Following all the given steps results in the program pre-sented below. The result is a perfectly valid program in theMercury logic programming language. Mercury is howeverstrongly typed, so we need to add a few auxiliary deﬁnitionsin order to compile it.The type α can be modeled as follows::-type alpha ---> child; tall; physical;action; person; running;smiling; animate;...i(alpha,alpha);p(role,alpha).We need to add the following type, mode and determinismspeciﬁcation for the subsumption predicate::-pred sub(alpha::in,alpha::out) is nondet.This tells Mercury that predicate sub takes an entityof type α as input, and computes an entity of type α asoutput. Furthermore, we specify that sub is a nondeterministicpredicate, meaning that it can compute multiple outputs forone input. We need to provide similar speciﬁcations for allpredicates in our program.The factual subsumption database, translated directly fromour previous deﬁnition in C:sub(child,person).sub(person,physical).sub(person,animate).sub(running,action).sub(smiling,action)....The type ρ of roles can be deﬁned in Mercury as::-type role --->tmp ; loc ; prp ; wrt; chr ; cum ; bmo ; cby; cau ; cmp ; pof ; agt; pnt ; src ; rst ; dst; via ; ...Clauses deﬁning the isa relation:isa(C,C).isa(C,A):-sub(C,P),isa(P,A).isa(i(X,_Y),Z):-isa(X,Z).isa(i(_X,Y),Z):-isa(Y,Z).The predicates included in lexical entries take the followingform in Mercury:p_tall(e(C),e(i(tall,C))):-isa(C,physical).p_runs(e(C),e(i(running,p(agt,C)))):-isa(C,animate).p_smiles(e(C),e(i(smiling,p(agt,C)))):-isa(C,animate).p_the(e(I),e(I))....Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on July 30,2010 at 09:55:46 UTC from IEEE Xplore.  Restrictions apply. 298 PROCEEDINGS OF THE IMCSIT. VOLUME 4, 2009The lexicon, translated to Mercury:lex(w_kid,e(child)).lex(w_tall,f(p_tall)).lex(w_runs,b(p_runs)).lex(w_smiles,b(p_smiles)).lex(w_the,f(p_the)).lex(w_a,f(p_the))....The deﬁnition of non–empty list appending:a(l(X,n),l(Y,T),l(X,l(Y,T))).a(l(H,L),M,l(H,T)):-a(L,M,T).The rules formalized in C take the following form inMercury:r(l(X,n),X).r(G,E1):-a(G1,G2,G),r(G1,f(P)),r(G2,E2),P(E2,E1).r(G,E1):-a(G1,G2,G),r(G1,E2),r(G2,b(P)),P(E2,E1).The rules are the only place in the program, where we usehigher–order predicates.The following predicate assigns a meaning to English text:m(T,C):-map_lex(T,L),r(L,e(C)).Let us add an auxiliary predicate work for testing purposes:work(C):-m(t(w_the,t(w_tall,t(w_kid,t(w_smiles,tnil)))),C).The map_lex predicate is used for lexicon look–up of alist of words.map_lex(tnil,n).map_lex(t(W,T),l(I,MT)):-lex(W,I),map_lex(T,MT).Notice that the program does not require higher orderuniﬁcation, except for the simplest case where a variable isbound to a predeﬁned predicate and such a predicate is called.Most logic programming languages provide a facility for suchbehaviour. We could change the syntax slightly and turn ourcode into a valid Prolog program by utilizing the call librarypredicate.Notice that the program is a direct implementation of thetheory, hence quite inefﬁcient. Instead of the presented top–down approach, we could derive the semantics in the bottom–up manner in order to avoid unneeded search.For the sample query:? work(C).We get the following (exactly one) result:C = i(smiling, p(agt, i(tall, child)))VI. CONCLUSIONWe have presented an extension to Categorial Grammar,which allows constructing arbitrary semantics and enforcingarbitrary semantic restrictions in a very ﬂexible manner. Thefunctional composition and β-reduction are replaced withproof rule application. We have utilized it for constructingontologico-algebraic meaning using ontological restrictions,dropping at the same time syntactic categories and syntacticrestrictions. Finally, a proof-of-concept implementation hasbeen given.ACKNOWLEDGMENTI am very grateful to my supervisor, prof. Jørgen FischerNilsson, for his most kind help.REFERENCES[1] B. Carpenter, Type-logical semantics. Cambridge, MA, USA: MIT Press,1998.[2] P. Andrews, “Classical type theory,” 2001, peter Andrews. Classical typetheory. In Alan Robinson and Andrei Voronkov, editors, Handbook ofAutomated Reasoning,volume 2, chapter 15, pages 965-1007. North-Holland, 2001. 43.[3] SIABO.dk, “Siabo project website,” http://www.siabo.dk, 2008.[4] P. A. Jensen and J. F. Nilsson, “Ontology-based semantics for preposi-tions,” in Syntax and Semantics of Prepositions, ser. Text, Speech andLanguage Technology, Vol. 29. Springer, 2006.[5] B. A. Szymczak, “Formal ontologies for semantic text processing,”Master’s thesis, Informatics and Mathematical Modelling, Technical Uni-versity of Denmark, DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby, 2007, supervised by Prof. Jørgen Fischer Nilsson,IMM, DTU. http://www2.imm.dtu.dk/pubdb/p.php?5399.[6] C. Brink, K. Britz, and R. A. Schmidt, “Peirce algebras,” Formal Aspectsof Computing, vol. 6, no. 3, pp. 339–358, April 1994, also availableas Research Report MPI-I-92-229, Max-Planck-Institut f¨ ur Informatik,Saarbr¨ ucken, Germany (July 1992), and as Research Report RR 140,Department of Mathematics, University of Cape Town, Cape Town, SouthAfrica (August 1992). An extended abstract appears in Nivat, M., Rattray,C., Rus, T. and Scollo, G. (eds), Algebraic Methodology and SoftwareTechnology (AMAST’93): Proceedings of the 3rd International Confer-ence on Algebraic Methodology and Software Technology. Workshops inComputing Series, Springer-Verlag, London, 165-168 (1994).[7] G. Ben-Avi and N. Francez, “Categorial grammar with ontology-reﬁnedtypes,” in Proceedings of CG04, 2004.Authorized licensed use limited to: Danmarks Tekniske Informationscenter. 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Ontological semantics in modified categorial grammar

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Ontological semantics in modified categorial grammar

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