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A Scheme for Integrating Concrete Domains into Concept Languages
, 1991
"... A drawback which concept languages based on klone have is that all the terminological knowledge has to be defined on an abstract logical level. In many applications, one would like to be able to refer to concrete domains and predicates on these domains when defining concepts. Examples for such conc ..."
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Cited by 262 (20 self)
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A drawback which concept languages based on klone have is that all the terminological knowledge has to be defined on an abstract logical level. In many applications, one would like to be able to refer to concrete domains and predicates on these domains when defining concepts. Examples for such concrete domains are the integers, the real numbers, or also nonarithmetic domains, and predicates could be equality, inequality, or more complex predicates. In the present paper we shall propose a scheme for integrating such concrete domains into concept languages rather than describing a particular extension by some specific concrete domain. We shall define a terminological and an assertional language, and consider the important inference problems such as subsumption, instantiation, and consistency. The formal semantics as well as the reasoning algorithms are given on the scheme level. In contrast to existing klone based systems, these algorithms will be not only sound but also complete. The...
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 firstorder formulas over some temporarily fixed language and theory. In contrast to theorem provers, the methods applied know a ..."
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Cited by 105 (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 firstorder 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 generalpurpose 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 firstorder 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 62 (10 self)
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The linear quantifier elimination algorithm for ordered fields in [15] is applicable to a wide range of...
QEPCAD B: A program for computing with semialgebraic sets using CADs
 SIGSAM BULLETIN
, 2003
"... This report introduces QEPCAD B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the QEPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 199 ..."
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Cited by 56 (1 self)
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This report introduces QEPCAD B, a program for computing with real algebraic sets using cylindrical algebraic decomposition (CAD). QEPCAD B both extends and improves upon the QEPCAD system for quantifier elimination by partial cylindrical algebraic decomposition written by Hoon Hong in the early 1990s. This paper briefly discusses some of the improvements in the implementation of CAD and quantifier elimination via CAD, and provides somewhat more detail on extensions to the system that go beyond quantifier elimination. The author is responsible for most of the extended features of QEPCAD B, but improvements to the basic CAD implementation and to the SACLIB library on which QEPCAD is based are the results of many people’s work, including: George E.
Universally Quantified Interval Constraints
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decompo ..."
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Cited by 46 (0 self)
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Nonlinear real constraint systems with universally and/or existentially quantified variables often need be solved in such contexts as control design or sensor planning. To date, these systems are mostly handled by computing a quantifierfree equivalent form by means of Cylindrical Algebraic Decomposition (CAD). However, CAD restricts its input to be conjunctions and disjunctions of polynomial constraints with rational coefficients, while some applications such as camera control involve systems with arbitrary forms where time is the only universally quantified variable. In this paper, the handling of universally quantified variables is first related to the computation of innerapproximation of real relations.
Quantifier Elimination for Real Algebra  the Quadratic Case and Beyond
 AAECC
, 1993
"... . We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the firstorder theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifierfree formulas restricted by a non ..."
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Cited by 42 (4 self)
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. We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the firstorder theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifierfree formulas restricted by a nontrivial quadratic equation in x (the case considered also in [7]), and more generally in front of arbitrary quantifierfree formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9]. It yields a quantifier elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurences of the quantified variables. Moreover, for existential formulas ' of this kind it yields sample answers to the query represented by '. The method is implemented in reduce as part of the redlog package (see [4, 5]). Experiments show that the method is appl...
Nonlinear Loop Invariant Generation using Gröbner Bases
, 2004
"... We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generati ..."
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Cited by 41 (4 self)
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We present a new technique for the generation of nonlinear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the nonlinear 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 nonlinear 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 nonlinear (algebraic) loop invariants of the program. We discuss some tradeoffs between the completeness of the technique and the tractability of the constraintsolving problem generated. The application of the technique is demonstrated on a few examples.
ModelTheoretic Methods in Combined Constraint Satisfiability
 Journal of Automated Reasoning
, 2004
"... We extend NelsonOppen combination procedure to the case of theories which are compatible with respect to a common subtheory in the shared signature. The notion of compatibility relies on model completions and related concepts from classical model theory. ..."
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Cited by 40 (10 self)
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We extend NelsonOppen combination procedure to the case of theories which are compatible with respect to a common subtheory in the shared signature. The notion of compatibility relies on model completions and related concepts from classical model theory.
New Results on Quantifier Elimination Over Real Closed Fields and Applications to Constraint Databases
 Journal of the ACM
, 1999
"... In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depen ..."
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Cited by 35 (4 self)
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In this paper we give a new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now. Unlike previously known algorithms [3, 28, 22] the combinatorial part of the complexity (the part depending on the number of polynomials in the input) of this new algorithm is independent of the number of free variables. Moreover, under the assumption that each polynomial in the input depend only on a constant number of the free variables, the algebraic part of the complexity (the part depending on the degrees of the input polynomials) can also be made independent of the number of free variables. This new feature of our algorithm allows us to obtain a new algorithm for a variant of the quantifier elimination problem. We give an almost optimal algorithm for this new problem, which we call the uniform quantifier elimination problem. Using the uniform quantifier elimination algorithm, we give a...
Simplification of Quantifierfree 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 34 (15 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 "=,"