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Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases
"... We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integrodifferential algebras. The algebraic treatment of boundary problems brings up ..."
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We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integrodifferential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integrodifferential operators, is used for both stating and solving linear boundary problems. The other structure, called integrodifferential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integrodifferential polynomials for generating an automated proof establishing a canonical simplifier for integrodifferential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.
Knowledge Archives in Theorema: A LogicInternal Approach
"... Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching la ..."
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Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logicinternal in the sense that they have a natural translation to higherorder logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.
Mathematical Knowledge Archives in Theorema
"... Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambigu ..."
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Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces, addressing the domains of categories and functors as namespaces with variable opera− tions. All constructs are logic–internal in the sense that they have a natural translation to higher–order logic so that �mathematical knowledge management � can be treated by the object logic itself. 1
ISSAC 2008 Poster Abstracts General Polynomial Reduction with TH∃OREM ∀ Functors: Applications to IntegroDifferential Operators and Polynomials
"... General Polynomial Reduction. We outline a prototype implementation of the algorithms for integrodifferential operators/polynomials in [12]. Our approach based on a generic implementation of noncommutative monoid rings with reduction, programmed in the functors language of the TH∃OREM ∀ system. The ..."
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General Polynomial Reduction. We outline a prototype implementation of the algorithms for integrodifferential operators/polynomials in [12]. Our approach based on a generic implementation of noncommutative monoid rings with reduction, programmed in the functors language of the TH∃OREM ∀ system. The integrodifferential operators—realized by a suitable quotient of noncommutative polynomials over a given integrodifferential algebra—can be used for solving and manipulating boundary problems for linear ordinary differential equations. For describing extensions of integrodifferential algebras algorithmically, we use integrodifferential polynomials. We use a fixed Gröbner basis for normalizing integrodifferential operators. Gröbner bases were invented by Buchberger [2, 3] for commutative polynomials and reinvented in [1] for noncommutative ones. While [9] analyzes the computational aspects of the latter, it does not support two features that are important for our present setting: the usage of infinitely many indeterminates and reduction modulo an (algorithmic) infinite system of polynomials. Among the systems implementing noncommutative Gröbner bases, most address certain special classes (e.g. algebras of solvable type or homogeneous polynomials) which do not include our present case. To our best knowledge, none of these allow polynomials with infinitely many indeterminates and reduction modulo an infinite system of polynomials. For details, see the website