This paper presents a set of tools for mechanical reasoning of numerical bounds using interval arithmetic. The tools implement two techniques for reducing decorrelation: interval splitting and Taylor’s series expansions. Although the tools are designed for the proof assistant system PVS, expertise on PVS is not required. The ultimate goal of the tools is to provide guaranteed proofs of numerical properties with a minimal human-theorem prover interaction. 1

The development of methodologies for the automatic implementation of floating-point algorithms in fixed-point architectures is required for the minimization of cost, power consumption and time to market of digital signal processing applications. In this paper, a new methodology of implementation in Digital Signal Processors (DSP) under accuracy constraint is presented. In comparison with the existing methodologies, the DSP architecture is completely taken into account for optimizing the execution time under accuracy constraint. The justification and the different stages of our methodology are presented. 1.

A. NEUMAIER [1] has given the fundamentals of interval analysis on directed acyclic graphs (DAGs) for global optimization and constraint propagation. We show in this paper how constraint propagation on DAGs can be made efficient and practical by: (i) working on partial DAG representations; and (ii) enabling the flexible choice of the interval inclusion functions during propagation. We then propose a new simple algorithm which coordinates constraint propagation and exhaustive search for solving numerical constraint satisfaction problems. The experiments carried out on different problems show that the new approach outperforms previously available propagation techniques by an order of magnitude or more in speed, while being roughly the same quality w.r.t. enclosure properties. I.

Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 1

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Abstract Various extensions of the CSP framework exist to address ill-defined, real-world optimisation problems. One extension, the uncertain CSP (UCSP) tackles the aspect of data errors and incompleteness by ensuring that the problem is faithfully represented with what is known for sure about the data, and by seeking reliable solutions that do not approximate such uncertainties. The extended model has a great impact on the solving complexity. For instance, by introducing bounded interval coefficients, the default representation of an arithmetic linear constraint is of degree 2. A challenge lies in determining constraint classes that allow one to reformulate the UCSP model such that polynomial algorithms exist. In this paper we present two novel sufficient conditions, built on algebraic properties of constraints, that ensure a tractable reformulation exists. We give an algorithm to test for the conditions for binary constraints, and demonstrate as instances some previously identified practical UCSP reformulations. 1

Interval analysis provides techniques that make it possible to determine all solutions to a nonlinear algebraic equation system and to do so with mathemat-ical and computational certainty. Such methods are based on the processing of granules in the form of intervals and thus can be regarded as one facet of granular computing. We review here some of the key concepts used in these methods and then focus on some specific application areas, namely ecological modeling, transition state analysis, and the modeling of phase equilibrium. 1

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This paper proposes a novel generic scheme enabling the combination of multiple inclusion representations to propagate numerical constraints. The scheme allows bringing into the constraint propagation framework the strength of inclusion techniques coming from di#erent areas such as interval arithmetic, a#ne arithmetic or mathematical programming.

To satisfy energy and complexity constraints, embedded wireless systems require fixed-point arithmetic implementation. To optimize the fixed-point specification, existing approaches are based on fixed-point simulations to evaluate the performances. In this paper, the approach used to optimize the fixed-point specification for a WCDMA receiver is presented. The dynamic range and the fixed-point accuracy are evaluated analytically in our approach. The analytical accuracy constraint expression according to the bit error rate (BER) is proposed. The results show that the optimized fixedpoint specification depends on the input receiver signal-to-noise ratio (SNR). 1.