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188
REDLOG Computer Algebra Meets Computer Logic
- ACM SIGSAM Bulletin
, 1996
"... . redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of first-order formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know a ..."
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Cited by 130 (30 self)
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. redlog is a package that extends the computer algebra system reduce to a computer logic system, i.e., a system that provides algorithms for the symbolic manipulation of first-order formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know about the underlying algebraic theory and make use of it. Though the focus is on simplification, parametric linear optimization, and quantifier elimination, redlog is designed as a general-purpose system. We describe the functionality of redlog as it appears to the user, and explain the design issues and implementation techniques. ? The second author was supported by the dfg (Schwerpunktprogramm: Algorithmische Zahlentheorie und Algebra) 1 Introduction redlog stands for reduce logic system. It provides an extension of the computer algebra system (cas) reduce to a computer logic system (cls) implementing symbolic algorithms on first-order formulas w.r.t. temporarily fixed firstorder languag...
Applying Linear Quantifier Elimination
, 1993
"... The linear quantifier elimination algorithm for ordered fields in [15] is applicable to a wide range of... ..."
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Cited by 87 (11 self)
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The linear quantifier elimination algorithm for ordered fields in [15] is applicable to a wide range of...
Non-linear Loop Invariant Generation using Gröbner Bases
, 2004
"... We present a new technique for the generation of non-linear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the non-linear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generati ..."
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Cited by 60 (4 self)
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We present a new technique for the generation of non-linear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the non-linear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generation has been focussed on the construction of linear invariants for linear programs. Consequently, there has been little progress toward non-linear invariant generation. In this paper, we demonstrate a technique that encodes the conditions for a given template assertion being an invariant into a set of constraints, such that all the solutions to these constraints correspond to non-linear (algebraic) loop invariants of the program. We discuss some trade-offs between the completeness of the technique and the tractability of the constraint-solving problem generated. The application of the technique is demonstrated on a few examples.
Real Quantifier Elimination in Practice
- Algorithmic Algebra and Number Theory
, 1998
"... We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applicat ..."
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Cited by 42 (6 self)
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We give a survey of three implemented real quantifier elimination methods: partial cylindrical algebraic decomposition, virtual substitution of test terms, and a combination of Gröbner basis computations with multivariate real root counting. We examine the scope of these implementations for applications in various fields of science, engineering, and economics.
Simplification of Quantifier-free Formulas over Ordered Fields
- Journal of Symbolic Computation
, 1995
"... this article is to provide a collection of practicable methods that have been implemented and extensively tested for their relevance. We further show how to combine different ideas for simplification in such a way that a formula is obtained which cannot be further simplified with any of the describe ..."
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Cited by 35 (16 self)
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this article is to provide a collection of practicable methods that have been implemented and extensively tested for their relevance. We further show how to combine different ideas for simplification in such a way that a formula is obtained which cannot be further simplified with any of the described methods. In other words, our simplifiers viewed as a function are idempotent. Achieving this is by no means trivial. On the algorithmic side, we introduce the concept of a background theory that is implicitly enlarged when entering a formula for simplification. Originally developed for detecting interactions between atomic formulas on different Boolean levels, it has turned out that this concept captures also other simplifiers that we had developed some time ago. These simplifiers, namely the Grobner simplifier and the Tableau simplifiers, could even be generalized due to this new viewpoint. 1.1. definitions Our formulas combine atomic formulas using the Boolean connectives "," "," "\Gamma!," "/\Gamma," "/!," and ":." Conjunction and disjunction are not binary but allow an arbitrary number of arguments. The atomic formulas are equations constructed with "=,"
Calculating Invariant Rings of Finite Groups over Arbitrary Fields
- J. Symbolic Computation
, 1995
"... An algorithm is presented which calculates invariant rings of finite linear groups over an arbitrary field K. Up to now, such algorithms have been available only for the case that the characteristic of K does not divide the group order. Some applications to the question whether a modular invariant r ..."
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Cited by 34 (9 self)
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An algorithm is presented which calculates invariant rings of finite linear groups over an arbitrary field K. Up to now, such algorithms have been available only for the case that the characteristic of K does not divide the group order. Some applications to the question whether a modular invariant ring is CohenMacaulay or isomorphic to a polynomial ring are discussed. Contents Introduction 2 1 Calculating Primary Invariants 2 2 Calculating Secondary Invariants 5 3 Checking Properties of Invariant Rings 9 3.1 The Cohen-Macaulay Property : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.2 Polynomial Rings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 References 12 This article appeared as IWR Preprint 95-12, Heidelberg 1995. 1 Introduction If G is a finite linear group over a field K such that char(K) - jGj, there are various effective methods to calculate the invariant ring I of G, i.e., to find a finite system of generators of I as an algebra over K ...
Using Galois Ideals for Computing Relative Resolvents
, 2000
"... In this paper we show that some ideals which occur in Galois theory are generated by triangular sets of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may and should be exploited in order to obtain more efficient algorithms, and it e ..."
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Cited by 33 (7 self)
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In this paper we show that some ideals which occur in Galois theory are generated by triangular sets of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may and should be exploited in order to obtain more efficient algorithms, and it enables us to present a new algebraic method for computing relative resolvents which works with any polynomial invariant.
New Algorithm for Discussing Gröbner Bases with Parameters
- J. Symb. Comput
, 2002
"... Let F be a set of polynomials in the variables x = x1,..., xn with coefficients in R[a], where R is a UFD and a = a1,..., am a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to ..."
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Cited by 30 (9 self)
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Let F be a set of polynomials in the variables x = x1,..., xn with coefficients in R[a], where R is a UFD and a = a1,..., am a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Gröbner basis, when the parameters in F are substituted in an extension field K of R. This new algorithm improves Weispfenning’s comprehensive Gröbner basis CGB algorithm, obtaining a reduced complete set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Gröbner basis of the generic case specializes properly. These constructive methods provide a very satisfactory discussion and rich geometrical interpretation in the applications. c ○ 2002 Academic Press 1.
Computing Parametric Geometric Resolutions
, 2001
"... Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we f ..."
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Cited by 29 (8 self)
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Given a polynomial system of n equations in n unknowns that depends on some parameters, we de ne the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d n , where d is a bound on the degrees of the input system. We then present a probabilistic algorithm to compute such a resolution; in short, its complexity is polynomial in the size of the output and the probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.
