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28
InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 821 (23 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
The computation of Gröbner bases on a shared memory multiprocessor
 Proc. DISCO ‘90, Springer LNCS 429
, 1990
"... The computation of Gro╠êbner bases on a shared memory multiprocessor ..."
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Cited by 20 (0 self)
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The computation of Gro╠êbner bases on a shared memory multiprocessor
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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Cited by 17 (14 self)
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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.
Decomposition of Test Sets in Stochastic Integer Programming
 MATHEMATICAL PROGRAMMING
, 2000
"... Graver test sets for linear twostage stochastic integer programs are studied. It is shown that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program. A finite algorithm to compute the building blocks directly, ..."
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Cited by 12 (8 self)
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Graver test sets for linear twostage stochastic integer programs are studied. It is shown that test sets can be decomposed into finitely many building blocks whose number is independent on the number of scenarios of the stochastic program. A finite algorithm to compute the building blocks directly, without prior knowledge of test set vectors, is presented. Once computed, building blocks can be employed to solve the stochastic program by a simple augmentation scheme, again without explicit knowledge of test set vectors. Preliminary computational experience is reported.
Computing generating sets of lattice ideals,
 Journal of Symbolic Computation
, 2009
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Towards the Automated Synthesis of a Gröbner Bases Algorithm
, 2004
"... We discuss the question of whether the central result of algorithmic Gr obner bases theory, namely the notion of Spolynomials together with the algorithm for constructing Gr obner bases using Spolynomials, can be obtained by "artificial intelligence", i.e. a systematic (algorithmic) ..."
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Cited by 9 (6 self)
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We discuss the question of whether the central result of algorithmic Gr obner bases theory, namely the notion of Spolynomials together with the algorithm for constructing Gr obner bases using Spolynomials, can be obtained by "artificial intelligence", i.e. a systematic (algorithmic) algorithm synthesis method. We present the "lazy thinking" method for theorem and algorithm invention and apply it to the "critical pair / completion" algorithm scheme. We present a road map that demonstrates that, with this approach, the automated synthesis of the author's Gr obner bases algorithm is possible. Still, significant technical work will be necessary to improve the current theorem provers, in particular the ones in the Theorema system, so that the road map can be transformed into a completely computerized process.
On the Computation of Hilbert Bases and Extreme Rays of Cones
, 2008
"... In this paper we present a novel projectandlift approach to compute the set of minimal generators of the semigroup (Λ ∩ R n +,+) for lattices Λ ⊆ Z n. This problem class includes the computation of Hilbert bases of cones {z: Az = 0, z ∈ R n +} for integer matrices A. A similar approach can be used ..."
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Cited by 9 (3 self)
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In this paper we present a novel projectandlift approach to compute the set of minimal generators of the semigroup (Λ ∩ R n +,+) for lattices Λ ⊆ Z n. This problem class includes the computation of Hilbert bases of cones {z: Az = 0, z ∈ R n +} for integer matrices A. A similar approach can be used to compute only the extreme rays of such cones. Finally, some combinatorial applications and computational experience are presented.
Decomposition Methods for TwoStage Stochastic Integer Programs
 IN ‘ONLINE OPTIMIZATION OF LARGE SCALE SYSTEMS
, 2001
"... Stochastic programs are proper tools for realtime optimization if realtime features arise due to lack of data information at the moment of decision. The paper's focus is at twostage linear stochastic programs involving integer requirements. After a discussion of basic structural properties, t ..."
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Cited by 8 (0 self)
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Stochastic programs are proper tools for realtime optimization if realtime features arise due to lack of data information at the moment of decision. The paper's focus is at twostage linear stochastic programs involving integer requirements. After a discussion of basic structural properties, two decomposition approaches are developed. While the first approach is directed to decomposition of the stochastic program itself, the second deals with decomposition of the related Graver test set.
Automatic Generation of EpsilonDelta Proofs of Continuity
 ARTIFICIAL INTELLIGENCE AND SYMBOLIC COMPUTATION
, 1998
"... As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilondelta arguments. These proofs involve two or three quantifiers on the logical side, and on the computational side, they involve algebra, ..."
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Cited by 8 (1 self)
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As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilondelta arguments. These proofs involve two or three quantifiers on the logical side, and on the computational side, they involve algebra, trigonometry, and some calculus. At the border of logic and computation, they involve several types of arguments involving inequalities, including transitivity chaining and several types of bounding arguments, in which bounds are sought that do not depend on certain variables. Control mechanisms have been developed for intermixing logical deduction steps with computational steps and with inequality reasoning. Problems discussed here as examples involve the continuity and uniform continuity of various specific functions.