Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

International audienceThe verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq's logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples

A New Approach to Fuzzy Arithmetic

This work shows an application of a generalized approach for constructing dilation-erosion adjunctions on fuzzy sets. More precisely, operations on fuzzy quantities and fuzzy numbers are considered. By the generalized approach an analogy with the well known interval computations could be drawn and thus we can define outer and inner operations on fuzzy objects. These operations are found to be useful in the control of bioprocesses, ecology and other domains where data uncertainties exist.* This work is partly supported by the Ministry of Education, Youth and Science under contract DO 02-359/2008 "Computer Simulation and Innovative Model-Based Study of Bio-processes"

Verified computations for hyperbolic 3-manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method are an implementation of interval arithmetic and Krawczyk's Test. These techniques represent an improvement over existing algorithms as they are faster, while accounting for error accumulation in a more direct and user friendly way.Mathematic

Is Semantic Query Optimization Worthwhile?

The term quote semantic query optimization quote (SQO) denotes a methodology whereby queries against databases are optimized using semantic information about the database objects being queried. The result of semantically optimizing a query is another query which is syntactically different to the original, but semantically equivalent and which may be answered more efficiently than the original. SQO is distinctly different from the work performed by the conventional SQL optimizer. The SQL optimizer generates a set of logically equivalent alternative execution paths based ultimately on the rules of relational algebra. However, only a small proportion of the readily available semantic information is utilised by current SQL optimizers. Researchers in SQO agree that SQO can be very effective. However, after some twenty years of research into SQO, there is still no commercial implementation. In this thesis we argue that we need to quantify the conditions for which SQO is worthwhile. We investigate what these conditions are and apply this knowledge to relational database management systems (RDBMS) with static schemas and infrequently updated data. Any semantic query optimizer requires the ability to reason using the semantic information available, in order to draw conclusions which ultimately facilitate the recasting of the original query into a form which can be answered more efficiently. This reasoning engine is currently not part of any commercial RDBMS implementation. We show how a practical semantic query optimizer may be built utilising readily available semantic information, much of it already captured by meta-data typically stored in commercial RDBMS. We develop cost models which predict an upper bound to the amount of optimization one can expect when queries are pre-processed by a semantic optimizer. We present a series of empirical results to confirm the effectiveness or otherwise of various types of SQO and demonstrate the circumstances under which SQO can be effective

Introduction to the Maple Power Tool Intpakx

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006The Maple Power Tool intpakX [24] de nes Maple types for real intervals and complex disc intervals. On the level of basic operations, intpakX includes the four basic arithmetic operators, including extended interval division as an extra function. Furthermore, there are power, square, square root, logarithm and exponential functions, a set of standard functions, union, and intersection. Reimplementations of the Maple construction, conversion, and unapplication functions are available. Additionally, there is a range of operators for complex disc arithmetic. As applications, verified computation of zeroes (Interval Newton Me- thod) with the possibility to find all zeroes of a function on a specified interval, and range enclosure for real-valued functions of one or two variables are implemented, the latter using either interval evaluation or evaluation via the mean value form and adaptive subdivision of intervals. The user can choose between a non-graphical and a graphical version of the above algorithms displaying the resulting intervals of each iteration step. The source code (about 2000 lines of Maple{code) of the extension intpakX is freely available [23]

Interval linear systems as a necessary step in fuzzy linear systems

International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

Computing exact solutions of consensus halving and the Borsuk-Ulam theorem

We study the problem of finding an exact solution to the consensus halving problem. While recent work has shown that the approximate version of this problem is PPA-complete, we show that the exact version is much harder. Specifically, finding a solution with $n$ cuts is FIXP-hard, and deciding whether there exists a solution with fewer than $n$ cuts is ETR-complete. We also give a QPTAS for the case where each agent's valuation is a polynomial. Along the way, we define a new complexity class BU, which captures all problems that can be reduced to solving an instance of the Borsuk-Ulam problem exactly. We show that FIXP $\subseteq$ BU $\subseteq$ TFETR and that LinearBU $=$ PPA, where LinearBU is the subclass of BU in which the Borsuk-Ulam instance is specified by a linear arithmetic circuit