Since 2005 we have been developing a software library for rendering, reading, and translating mathematical 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 to serve mathematical exercises in the native language of the student: in fact the library can be used to generate natural language descriptions of formally encoded mathematical expressions with no loss of meaning. The applications of this technology, coming from the area of grammarbased machine translation are related to the possibility of parsing and generating high quality representations of mathematics. In this short paper we concentrate on few technical details that made the work interesting from the linguistic point of view. Therefore we introduce the computational linguistic software used as backbone to the work, called Grammatical Framework, and proceed with the presentation of the mathematical library, its organization and modular design. We then discuss some examples that required careful thought.
1.1
The Grammatical Framework
The Grammatical Framework (GF) is a type theoretic programming language for writing
grammars for multiple languages at once [
These trees are described by an abstract grammar de ning what is possible to express in the speci c application, whilst the concrete grammars (one for each language) de ne how the abstract meaning is converted to the given language. Once an abstract grammar is given, to add yet 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, the grammar for the new language is almost an exact copy of the existing grammar, modulo some lexicon adaptations. GF hides all linguistic details of a speci c language from the programmer in a low-level resource grammar library, so that in principle a domain expert is able to develop new languages for a given application. Details of the GF Grammar Library, including language coverage, are online1.
A GF abstract grammar de nes how expressions in given categories are combined. An example tree in the Mathematical Grammar Library (MGL) looks as follows: m k P r o p ( l t _ n u m ( abs ( plus ( B a s e V a l N u m ( V a r 2 N u m x ) ( V a r 2 N u m y ) ) ) ) ( plus ( B a s e V a l N u m ( abs ( V a r 2 N u m x ) ) ( abs ( V a r 2 N u m y ) ) ) ) ) When linearized with the English and Spanish concrete grammars, it yields the natural language expressions2: the absolute value of the sum of x and y is less than the sum of the absolute value of x and the absolute value of y el valor absoluto de la suma de x e y es menor que la suma del valor absoluto de x y el valor absoluto de y
As mentioned above, the abstract tree is not far from the OpenMath expression. The linguistic function mkProp wraps the wording produced by the subexpressions. In terms of computational linguistic technology, this approach di ers from the standard statistical based approaches, such as Google Translate, in that it can generate high quality translations for arbitrarly deep nesting of subexpressions, as opposed to being limited by n-grams distance.
The number of categories on a GF application is a trade-o between how much ambiguity is tolerable and the expressiveness of the whole system. The de ned categoriesin the MGL are Value X, and Variable X where X is a Number, a Function, a Set or a Tensor (namely vectors or matrices). The actual version of the library implements these by de ning a xed category for each combination fVariable; Valueg fNumber; Set; Function; Tensorg. Thus, for instance, VarNum = Variable Number and ValSet = Value Set. Other categories stand for propositions, geometric constructions and indexes.
Each abstract category corresponds to a linguistic category in a concrete grammar of a speci c language. Usually a Value points to a noun phrase and a Variable to a string. More complex expressions, those combining categories, correspond in a natural way to linguistic entities composed from these elements: propositions are mapped into clauses with grammatical polarity, operations to sentences and simple exercises to texts. 2
The library can be organized in a matrix, where the horizontal axis runs over the languages while 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
includes also a couple of computer software languages which are relevant to mathematics, namely
LATEX and Sage [
The vertical axis runs over three layers of increasing complexity:
2. OpenMath: modeled after the following Content Dictionaries, considered useful for expressing 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 for euphony before a vowel that sounds alike. It is automatically taken care by the GF Spanish resource grammar. • 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 exercises and usually begin with directives such as `Compute', `Find', `Prove', `Give an example of', etc.
Objects in the OpenMath standard [
Following the lines of the Small Type System [4, principle 4], we imposed that binary associative functions take a list of values and return a value of the same kind. For example, plus in arith1 has signature plus : [ValNum] ! ValNum, while the category [ValNum] (meaning a list of numeric values) is declared to take at least two values. Therefore is impossible by construction to add a single number (i. e. \the sum of 3"). 3
Some interesting points on the implementation are related to language speci cs. For example, the simple exercise that asks for computing a numeric value 3: D o C o m p u t e N C o m p u t e V
( d e t e r m i n a n t ( V a r 2 T e n s o r M ) ) gives in English:
This pattern is shared in most of the languages, so it got abstracted into an incomplete concrete grammar le OperationsI. From this module, one can get OperationsL for language L simply by speci ying the lexicon and paradigms modules for this L, in a similar way a function is applied to its arguments. But in French is impolite to use an imperative in this case; Therefore the module OperationsFre should re-implement this production in a speci c way.
Another point worth mentioning is function application. Notice the di erent 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 di er in structure: in the rst case, the function being applied is a named symbol (the cosine) while in the last one is a -abstraction. In the other 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 ner control on which verb to use to denote computation (`to compute' in this case).
GF name in module CD n converted to Value from prede ned type Int in module Literals name in category Variable X a b lambda z app, where z is a Variable and app a Value. Not supported
The library is publicly available at the MOLTO repository [
For the future, the library needs to grow in breadth and shape: at this moment, it is systematically tested for three languages but depends on domain experts native speakers to polish the remaining ones.
Integration of natural language productions and formulas is also prominent in the TODO
list. This is a variegated issue as [