Caprotti, Olga

Saludes Closa, Jordi

UPCommons

The GF Mathematical Grammar Library: from OpenMath to natural languages

Saludes, Jordi

Chalmers Research

The GF mathematical grammar library: From openmath to natural languages

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The GF Mathematical Grammar Library: fromOpenMath to natural languagesOlga Caprotti1 and Jordi Saludes2 ∗1 Chalmers and University of Gothenburgcaprotti@chalmers.se2 Universitat Polite`cnica de Catalunyajordi.saludes@upc.edu1 IntroductionSince 2005 we have been developing a software library for rendering, reading, and translatingmathematical expressions, either expressed using formal languages such as OpenMath or LATEX,or in a number of natural languages. The work begun with the WebALT [1] project as a way toserve mathematical exercises in the native language of the student: in fact the library can beused to generate natural language descriptions of formally encoded mathematical expressionswith no loss of meaning. The applications of this technology, coming from the area of grammar-based machine translation are related to the possibility of parsing and generating high qualityrepresentations of mathematics.In this short paper we concentrate on few technical details that made the work interestingfrom the linguistic point of view. Therefore we introduce the computational linguistic softwareused as backbone to the work, called Grammatical Framework, and proceed with the presenta-tion of the mathematical library, its organization and modular design. We then discuss someexamples that required careful thought.1.1 The Grammatical FrameworkThe Grammatical Framework (GF) is a type theoretic programming language for writing gram-mars for multiple languages at once [3]. Multilingual applications use an interlingua: the se-mantics of an expression in natural language that should be rendered or translated is capturedin an abstract tree, which is its language-independent representation. As it turns out, the ab-stract tree representation is also a natural representation of mathematical expressions, one thatis also akin to the OpenMath abstract objects.These trees are described by an abstract grammar defining what is possible to express inthe specific application, whilst the concrete grammars (one for each language) define how theabstract meaning is converted to the given language. Once an abstract grammar is given, to addyet another language to the application amounts to adding a new concrete grammar. Ideally,if a concrete grammar for a language in the same linguistic group is already available, thegrammar for the new language is almost an exact copy of the existing grammar, modulo somelexicon adaptations. GF hides all linguistic details of a specific language from the programmerin a low-level resource grammar library, so that in principle a domain expert is able to developnew languages for a given application. Details of the GF Grammar Library, including languagecoverage, are online1.∗The research leading to these results has received funding from the European Union’s Seventh FrameworkProgramme (FP7/2007-2013) under grant agreement n° FP7-ICT-247914.1http://www.grammaticalframework.org/lib/doc/synopsis.html1The Mathematical Grammar Library Caprotti and SaludesA GF abstract grammar defines how expressions in given categories are combined. Anexample tree in the Mathematical Grammar Library (MGL) looks as follows:mkProp(lt_num (abs (plus (BaseValNum (Var2Num x) (Var2Num y))))(plus (BaseValNum (abs (Var2Num x)) (abs (Var2Num y)))))When linearized with the English and Spanish concrete grammars, it yields the naturallanguage expressions2:the absolute value of the sum of x and y is less than the sum of the absolute valueof x and the absolute value of yel valor absoluto de la suma de x e y es menor que la suma del valor absoluto de xy el valor absoluto de yAs mentioned above, the abstract tree is not far from the OpenMath expression. Thelinguistic function mkProp wraps the wording produced by the subexpressions. In terms ofcomputational linguistic technology, this approach differs from the standard statistical basedapproaches, such as Google Translate, in that it can generate high quality translations forarbitrarly deep nesting of subexpressions, as opposed to being limited by n-grams distance.The number of categories on a GF application is a trade-off between how much ambiguityis tolerable and the expressiveness of the whole system. The defined categoriesin the MGLare Value X, and Variable X where X is a Number, a Function, a Set or a Tensor (namelyvectors or matrices). The actual version of the library implements these by defining a fixedcategory for each combination {Variable,Value} × {Number,Set,Function,Tensor}. Thus, forinstance, VarNum = Variable Number and ValSet = Value Set. Other categories stand forpropositions, geometric constructions and indexes.Each abstract category corresponds to a linguistic category in a concrete grammar of aspecific language. Usually a Value points to a noun phrase and a Variable to a string. Morecomplex expressions, those combining categories, correspond in a natural way to linguisticentities composed from these elements: propositions are mapped into clauses with grammaticalpolarity, operations to sentences and simple exercises to texts.2 The Mathematical Grammar LibraryThe library can be organized in a matrix, where the horizontal axis runs over the languageswhile the vertical axis covers layers of complexity of mathematical expressions.At present the languages are: Bulgarian, Catalan, English, Finnish, French, German, Hindi,Italian, Polish, Romanian, Russian, Spanish, Swedish and Urdu. As a proof of concept, it in-cludes also a couple of computer software languages which are relevant to mathematics, namelyLATEX and Sage [2].The vertical axis runs over three layers of increasing complexity:1. Ground: literals, indexes and variables2. OpenMath: modeled after the following Content Dictionaries, considered useful forexpressing the mathematical fragments at the time of the WebALT project:2Notice the special form of the conjunction “x e y”: The usual Spanish conjunction “y” must be changed foreuphony before a vowel that sounds alike. It is automatically taken care by the GF Spanish resource grammar.2The Mathematical Grammar Library Caprotti and Saludes• arith1, arith2, complex1, integer1, integer2, logic1, nums1, quant1, relation1,rounding1;• calculus1, fns1, fns2, interval1, limit1, transc1, veccalc1;• linalg1, linalg2;• minmax1, plangeo1, s data1, set1, setname1.3. Operations: takes care of simple mathematical exercises. These appear in drilling ex-ercises and usually begin with directives such as ‘Compute’, ‘Find’, ‘Prove’, ‘Give anexample of’, etc.Objects in the OpenMath standard [5] relate to GF types, namely each symbol in a ContentDictionary (CD) roughly corresponds to a production of the same name in a GF module namedafter that CD. Application of functions to numbers are expressed by the production At thattakes a Value Function and a Value Number and return a Value Number. More examples arein the table 1.Following the lines of the Small Type System [4, principle 4], we imposed that binary asso-ciative functions take a list of values and return a value of the same kind. For example, plusin arith1 has signature plus : [ValNum] → ValNum, while the category [ValNum] (meaninga list of numeric values) is declared to take at least two values. Therefore is impossible byconstruction to add a single number (i. e. “the sum of 3”).3 Linguistic peculiaritiesSome interesting points on the implementation are related to language specifics. For example,the simple exercise that asks for computing a numeric value 3:DoComputeN ComputeV (determinant (Var2Tensor M))gives in English:Compute the determinant of M .This pattern is shared in most of the languages, so it got abstracted into an incomplete concretegrammar file OperationsI. From this module, one can get OperationsL for language L simplyby specifiying the lexicon and paradigms modules for this L, in a similar way a function is appliedto its arguments. But in French is impolite to use an imperative in this case; Therefore themodule OperationsFre should re-implement this production in a specific way.Another point worth mentioning is function application. Notice the different forms:• “the cosine of 3”• “f at 3”• “the derivative of the sine at 3”• “x to the cosine of x where x is 3”They are all mathematically equivalent but differ in structure: in the first case, the functionbeing applied is a named symbol (the cosine) while in the last one is a λ-abstraction. In theother cases, it is a function variable or it comes from a functional operator.3determinant belongs to the OpenMath layer of the library and Var2Tensor makes a value out of a variable.DoComputeN denotes an exercise asking to compute a number, while ComputeV gives finer control on which verbto use to denote computation (‘to compute’ in this case).3The Mathematical Grammar Library Caprotti and SaludesOpenMath GFSymbol name in CD name in module CDInteger n n converted to Value from predefined type Int inmodule LiteralsVariable name name in category Variable XApplication of a on b a bBinding λz app lambda z app, where z is a Variable and app a Value.Attribution, Error, Bytearray Not supportedTable 1: Some equivalences between the OpenMath standard and grammar library4 Applications and future developmentThe library is publicly available at the MOLTO repository [7] and is documented at [8]. It isbeing used in the mathbar demo in the MOLTO project [6] accessible from [9]. An example ofnatural language interaction with a computer algebra system can be retrieved from the sagedirectory of the library distribution and has been recently presented at [12].For the future, the library needs to grow in breadth and shape: at this moment, it issystematically tested for three languages but depends on domain experts native speakers topolish the remaining ones.Integration of natural language productions and formulas is also prominent in the TODOlist. This is a variegated issue as [10] shows, but it is necessary for fluent mathematics inapplications. Also more natural renderings of logical propositions [11] will open the door tousage in automatic reasoners and theorem provers.References[1] http://webalt.math.helsinki.fi/content/index_eng.html Last viewed June 2012.[2] http://www.sagemath.org/ Last viewed May 2012.[3] A. Ranta, “Grammatical Framework: programming with Multilingual Grammars,” CSLIStudies in Computational Linguistics, Standford, 2011.[4] J. H. Davenport, “A small OpenMath type system,” ACM SIGSAM Bulletin, vol. 34,no. 2, pp. 16–21, Jun. 2000.[5] OpenMath Consortium, “The OpenMath Standard,” OpenMath Deliverable, vol. 1, 2000.[6] The MOLTO project. http://www.molto-project.eu/ Last viewed May 2012.[7] The mathematical library svn://molto-project.eu/mgl.[8] J. Saludes et al., “Simple drill grammar library,” http://www.molto-project.eu/sites/default/files/d61.pdf. Last viewed May 2012.[9] Grammatical framework demos. http://www.grammaticalframework.org/demos/index.html. Last viewed May 2012.[10] Mohan Ganesalingam, “The Language of Mathematics.” PhD thesis, Cambridge Univer-sity, 2009.[11] A. Ranta, “Translating between language and logic: what is easy and what is difficult.”Automated Deduction, CADE-23, 2011.[12] Dominique Archambault, Olga Caprotti, Aarne Ranta and Jordi Saludes, “Using GFin multimodal assistants for mathematics.” Digitization and E-Inclusion in Mathematicsand Science 2012, Tokyo, Japan.4

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The GF Mathematical Grammar Library: from OpenMath to natural languages

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