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37
Efficient solving of quantified inequality constraints over the real numbers
 ACM Transactions on Computational Logic
"... Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In t ..."
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Cited by 33 (8 self)
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Let a quantified inequality constraint over the reals be a formula in the firstorder predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. 1
Beyond The Universal Turing Machine
, 1998
"... We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a phi ..."
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Cited by 31 (1 self)
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We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of welldefined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.
Continuous FirstOrder Constraint Satisfaction
 ARTIFICIAL INTELLIGENCE, AUTOMATED REASONING, AND SYMBOLIC COMPUTATION, NUMBER 2385 IN LNCS
, 2002
"... This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of con ..."
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Cited by 28 (12 self)
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This paper shows how to use constraint programming techniques for solving firstorder constraints over the reals (i.e., formulas in the firstorder predicate language over the structure of the real numbers). More specifically, based on a narrowing operator that implements an arbitrary notion of consistency for atomic constraints over the reals (e.g., boxconsistency), the paper provides a narrowing operator for firstorder constraints that implements a corresponding notion of firstorder consistency, and a solver based on such a narrowing operator. As a consequence, this solver can take over various favorable properties from the field of constraint programming.
Approximate quantified constraint solving by cylindrical box decomposition
 RESEARCH INSTITUTE FOR SYMBOLIC COMPUTATION (RISC
, 2000
"... This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a firstorder formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the pro ..."
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Cited by 20 (7 self)
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This paper applies interval methods to a classical problem in computer algebra. Let a quantified constraint be a firstorder formula over the real numbers. As shown by A. Tarski in the 1930's, such constraints, when restricted to the predicate symbols , are in general solvable. However, the problem becomes undecidable, when we add function symbols like sin. Furthermore, all exact algorithms known up to now are too slow for big examples, do not provide partial information before computing the total result, cannot satisfactorily deal with interval constants in the input, and often generate huge output. As a remedy we propose an approximation method based on interval arithmetic. It uses a generalization of the notion of cylindrical decomposition  as introduced by G. Collins. We describe an implementation of the method and demonstrate that, for quantied constraints without equalities, it can efficiently give approximate information on problems that are too hard for current exact methods.
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Cited by 15 (4 self)
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
A New Number Core for Robust Numerical and Geometric Libraries (Extended Abstract)
, 1998
"... We describe a new numerical core that can serve as the basis for robust numerical and geometric libraries. A novel feature of core is its hierarchy of numerical accuracies which can be accessed simultaneously by a conventional C/C++ program. We propose to build a core library (Core) of critical geom ..."
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Cited by 13 (7 self)
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We describe a new numerical core that can serve as the basis for robust numerical and geometric libraries. A novel feature of core is its hierarchy of numerical accuracies which can be accessed simultaneously by a conventional C/C++ program. We propose to build a core library (Core) of critical geometric and numerical primitives around this numerical core. This library will be portable, efficient, robust and easytouse. The portability and efficiency are based on a stateoftheart compiler technology (Trimaran) that can produce optimized code for a wide range of hardware architectures, particularly the new EPIC (explicitly parallel instruction computing) technology heralded by the Intel Merced chip. Numerical robustness is based on a scientifically sound approach called exact geometric computation. Easeofuse is based on the possibility of using Core with minimal change in programmer's behavior. Our library will, for the first time, make very powerful robustness techniques widely accessi...
Towards Automatic Proofs of Inequalities Involving Elementary Functions
 In Pragmatics of Decision Procedures in Automated Reasoning (PDPAR
, 2006
"... Inequalities involving functions such as sines, exponentials and logarithms lie outside the scope of decision procedures, and can only be solved using heuristic methods. Preliminary investigations suggest that many such problems can be solved by reduction to algebraic inequalities, which can then be ..."
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Cited by 10 (5 self)
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Inequalities involving functions such as sines, exponentials and logarithms lie outside the scope of decision procedures, and can only be solved using heuristic methods. Preliminary investigations suggest that many such problems can be solved by reduction to algebraic inequalities, which can then be decided by a decision procedure for the theory of real closed fields (RCF). The reduction involves replacing each occurrence of a function by a lower or upper bound (as appropriate) typically derived from a power series expansion. Typically this requires splitting the domain of the function being replaced, since most bounds are only valid for specific intervals. 1
The Automatic Solution of Recurrence Relations  I. Linear Recurrences of Finite Order with Constant Coefficients
"... We describe algorithmic techniques for the efficient solution of a wide class of linear recurrences of finite order with constant coefficients. We give ..."
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Cited by 8 (1 self)
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We describe algorithmic techniques for the efficient solution of a wide class of linear recurrences of finite order with constant coefficients. We give
Nonholonomicity of sequences defined via elementary functions
 Annals of Combinatorics
"... We present a new method for proving nonholonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generali ..."
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Cited by 6 (3 self)
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We present a new method for proving nonholonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results; e.g., nonholonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at odd integers is not holonomic.
Safety Verification of Nonlinear Hybrid Systems is QuasiDecidable∗
, 2013
"... Safety verification of hybrid systems is undecidable, except for very special cases. In this paper, we circumvent undecidability by providing a verification algorithm that provably terminates for all robust problem instances, but need not necessarily terminate for nonrobust problem instances. A pro ..."
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Cited by 6 (1 self)
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Safety verification of hybrid systems is undecidable, except for very special cases. In this paper, we circumvent undecidability by providing a verification algorithm that provably terminates for all robust problem instances, but need not necessarily terminate for nonrobust problem instances. A problem instance x is robust iff the given property holds not only for x itself, but also when x is perturbed a little bit. Since, in practice, welldesigned hybrid systems are usually robust, this implies that the algorithm terminates for the cases occurring in practice. In contrast to earlier work, our result holds for a very general class of hybrid systems, and it uses a continuous time model. 1