The Wheel of Rational Numbers as an Abstract Data Type

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element such as infinity ∞ or error element ⊥. A wheel is an algebra in which division is totalised by setting 1/0 = ∞ but which also contains an error element ⊥ to help control its use. We construct the wheel of rational numbers as an abstract data type Qw and give it an equational specification without auxiliary operators under initial algebra semantics

The Transrational Numbers as an Abstract Data Type

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to the negative infinite value , and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics

PIE: pp-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption

A large part of current research in homomorphic encryption (HE) aims towards making HE practical for real-world applications. In any practical HE, an important issue is to convert the application data (type) to the data type suitable for the HE. The main purpose of this work is to investigate an efficient HE-compatible encoding method that is generic, and can be easily adapted to apply to the HE schemes over integers or polynomials. $p$-adic number theory provides a way to transform rationals to integers, which makes it a natural candidate for encoding rationals. Although one may use naive number-theoretic techniques to perform rational-to-integer transformations without reference to $p$-adic numbers, we contend that the theory of $p$-adic numbers is the proper lens to view such transformations. In this work we identify mathematical techniques (supported by $p$-adic number theory) as appropriate tools to construct a generic rational encoder which is compatible with HE. Based on these techniques, we propose a new encoding scheme PIE, that can be easily combined with both AGCD-based and RLWE-based HE to perform high precision arithmetic. After presenting an abstract version of PIE, we show how it can be attached to two well-known HE schemes: the AGCD-based IDGHV scheme and the RLWE-based (modified) Fan-Vercauteren scheme. We also discuss the advantages of our encoding scheme in comparison with previous works

Computational Visualistics: Dealing with Pictures in Computer Science

Building blocks from many disciplines have to be integrated into a general science of images. Computational visualistics has been formed as a contributing field embracing all aspects of dealing with images computationally. Two basic concepts of computer science are introduced. Applied to the concept "image", they determine the methodological core of computational visualistics. As the contribution of computer science to the subject of image theory, interactive pictures are examined. Finally, relations to other "image sciences" are sketched. * * * Bausteine vieler Disziplinen müssen in eine allgemeine Bildwissenschaft integriert werden. Als Beitrag aus der Informatik versteht sich die Computervisualistik, die alle Aspekte rechnergestützten Umgangs mit Bildern umfaßt. Zwei Grundbegriffe der Informatik werden vorgestellt und bestimmen, auf den Begriff "Bild" angewendet, den methodologischen Kern der Computervisualistik. Als Beitrag der Informatik zum Gegenstand der Bildtheorie werden interaktive Bilder betrachtet. Schließlich werden die Beziehungen zu anderen "Bildwissenschaften" kurz umrissen

From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps

Verification of Java Bytecode using Analysis and Transformation of Logic Programs

State of the art analyzers in the Logic Programming (LP) paradigm are nowadays mature and sophisticated. They allow inferring a wide variety of global properties including termination, bounds on resource consumption, etc. The aim of this work is to automatically transfer the power of such analysis tools for LP to the analysis and verification of Java bytecode (JVML). In order to achieve our goal, we rely on well-known techniques for meta-programming and program specialization. More precisely, we propose to partially evaluate a JVML interpreter implemented in LP together with (an LP representation of) a JVML program and then analyze the residual program. Interestingly, at least for the examples we have studied, our approach produces very simple LP representations of the original JVML programs. This can be seen as a decompilation from JVML to high-level LP source. By reasoning about such residual programs, we can automatically prove in the CiaoPP system some non-trivial properties of JVML programs such as termination, run-time error freeness and infer bounds on its resource consumption. We are not aware of any other system which is able to verify such advanced properties of Java bytecode

Square root meadows

Let Q_0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0^{-1}=0. Q_0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q_0(s ,\sqrt) of the signed rationals in which every number has a unique square root.Comment: 9 page

Inversive Meadows and Divisive Meadows

Inversive meadows are commutative rings with a multiplicative identity element and a total multiplicative inverse operation whose value at 0 is 0. Divisive meadows are inversive meadows with the multiplicative inverse operation replaced by a division operation. We give finite equational specifications of the class of all inversive meadows and the class of all divisive meadows. It depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. We show that inversive and divisive meadows of rational numbers can be obtained as initial algebras of finite equational specifications. In the spirit of Peacock's arithmetical algebra, we study variants of inversive and divisive meadows without an additive identity element and/or an additive inverse operation. We propose simple constructions of variants of inversive and divisive meadows with a partial multiplicative inverse or division operation from inversive and divisive meadows. Divisive meadows are more basic if these variants are considered as well. We give a simple account of how mathematicians deal with 1 / 0, in which meadows and a customary convention among mathematicians play prominent parts, and we make plausible that a convincing account, starting from the popular computer science viewpoint that 1 / 0 is undefined, by means of some logic of partial functions is not attainable.Comment: 18 pages; error corrected; 29 pages, combined with arXiv:0909.2088 [math.RA] and arXiv:0909.5271 [math.RA