Uniform Representation of Basic Algebraic Structures in Computer Algebra

This work presents a methodological framework suitable for the design of innovative sym bolic computation systems A signi cant set of basic algebraic structures is de ned and implemented by a uniform representation based on the algebraic structure of truncated power series By following this approach we succeed in avoiding the need for explicit coercion in computing with the de ned structures We provide a Clos prototype imple mentation Introduction This work aims to propose a neat methodological framework suitable for the design of innovative symbolic computation systems We describe the speci cation and the prototype implementation of some basic alge braic structures such as algebraic numbers multivariate polynomials and square matrices represented in a uniform way By uniform we intend that the mathematical entities at is sue are represented by a set of inheritance related classes in an object oriented language such that their data structure representation is structurally compatible This compat ibility allows for an important e ect of uniform representation the prototype software system we present can execute several legal computations over instances of our structures without having to perform coercions In particular we can perform an operation between a number and a polynomial as well as a polynomial and a matrix without any need for explicit coercions We give a speci cation of the three basic algebraic structures cited above by expressing their common algebraic nature of Ring and Truncated Power Series TPS structures This speci cation is translated in the de nition of a hierarchy of data structures in which numbers polynomials and matrices are uniformly represented as specializations of the TPS The idea of a uniform representation for numbers and polynomials has been already presented in Here we develop further the notion of uniform representation and its application In particular we provide an interpretation of matrices over ring elements as TPS In order to obtain a sound speci cation of all these structures we have to solve some problems that mainly depend on the nature of matrices and of their operations In particular from our speci cation it comes out that external multiplication cannot be safely expressed in our framework In order to accomplish this requirement we have to model explicitly our matrix structure as a specialization of the algebraic structure of R module On the other hand the ring operations between matrices matrix plus a matrix a matrix times a matrix are simply expressed by their de nition in the matrix structure After the speci cation of our structures is given by following the formalism de ned in we present their one to one implementation in suitable classes In this implementa tion the derivation of a structure from another is expressed by specialization inheritance Clos is the programming language used It has been chosen for a number of rea sons because of its object orientation because it has been already used within a project in which we have been involved aiming to the development of a language for symbolic computation systems and for our involvement in another research activity aiming to provide an object oriented language with class reasoning features In presenting the prototype implementation we show some software details and some signi cant computa tional examples In the following section we explain the formalism used for the speci cation of our algebraic structures Sec describes the organization of implementation In Sec we Given a matrix M with elements in R we mean the multiplication of a R by M discuss some computational examples and Sec previews future work The details about speci cation of our structures and some signi cant parts of the Clos code are shown in appendix Classi cation of formal structures The classi cation of formal structures can be given in an algebraic context as proposed in by de ning three planes for their static de nition abstract parametric and ground plane In the abstract plane classical algebraic structures are described by inherited properties Sorts are not speci ed and only symbolic computations are performed In the parametric plane parametric structures are de ned they enrich the de nition of abstract structures by partial parameterized sorts and by additional operations and properties In the ground plane the ground structures take place they are completely speci ed and both symbolic and numeric computations can be performed in this plane This classi cation points out the di erent speci cation requirements of structures lay ing on di erent planes leading to a higher level of correctness in their treatment The whole discussion about this approach is given in In the following we provide an example of speci cation of the abstract structure Ring and of its derived structure Unit Ring Let us note that the speci cation includes also properties of operators even if they cannot be exploited at the moment This choice is due to our work being developed in a general research framework and doesn t a ect the coherence of this exposition Ring from Abelian Group from Semiring properties a b c a b c a b a c a b c a b c a c b c s s s s s inv s Unit Ring from Ring operations properties a a a a Usually an abstract class is a generalization of the properties that are common to a collection of more specialized classes It can also be a particular specialization of other ab stract classes the main feature is the fact that methods either are not implemented or they are partially implemented A parametric class inherits methods and properties from one or more abstract classes but methods and sorts are parametrically speci ed In a ground class all the sorts and methods are completely speci ed Direct object instantiations of mathematical objects are possible only for ground classes actually such an instantiation would be meaningless for abstract classes and simply not possible for a parametric class without providing further type arguments for the actualization parameters Algebraic numbers polynomials and matrices by TPS Let N P and M symbols standing for the representation of respectively algebraic num bers multivariate polynomials and matrices In the following we will deal with structures of type T by the following notation T N jPT jMT We consider for instance polynomials over N we will denote their set by PN polynomials over polynomials over N PPN or matrices whose elements are polynomials over N MPN