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

A thread calculus with molecular dynamics

We present a theory of threads, interleaving of threads, and interaction between threads and services with features of molecular dynamics, a model of computation that bears on computations in which dynamic data structures are involved. Threads can interact with services of which the states consist of structured data objects and computations take place by means of actions which may change the structure of the data objects. The features introduced include restriction of the scope of names used in threads to refer to data objects. Because that feature makes it troublesome to provide a model based on structural operational semantics and bisimulation, we construct a projective limit model for the theory.Comment: 47 pages; examples and results added, phrasing improved, references replace

Bitcoin and Islamic Finance

It is argued that a Bitcoin-style money-like informational commodity may constitute an effective instrument for the further development of Islamic Finance. The argument involves the following elements: (i) an application of circulation theory to Bitcoin with the objective to establish the implausibility of interest payment in connection with Bitcoin, (ii) viewing a Bitcoin-like system as a money-like exclusively informational commodity with the implication that such a system need not support debt, (iii) the idea that Islamic Finance imposes different requirements compared to conventional financial policies on a money concerning its use as a tool for achieving social and economic objectives, and (iv) identification of two aspects of mining, gambling and lack of trust, that may both be considered problematic from the perspective of compliance with the rules of Islamic Finance and a corresponding proposal to modify the architecture of mining in order to improve compliance with these rules

Qualifications of Instruction Sequence Failures, Faults and Defects: Dormant, Effective, Detected, Temporary, and Permanent

Starting out from the survey of instruction sequence faults from [6] program faults are classified according to the conventional criteria of being dormant, effective, detected, temporary, and permanent. Being retrospectively approved is introduced as an additional qualification. For this theoretical investigation the context is simplified by contemplating instruction sequences as a theoretical model for programs, and by assuming that instruction sequences are supposed to compute total transformations on finite bit sequences of a fixed length only. The main conclusion which can be drawn from this work concerns the notion of dormancy. First of all it is noticed that the unconventional notion of a dormant failure is both plausible and amenable to a straightforward and convincing definition. The conventional notion of a dormant fault, however, is much harder to grasp and the definition of a dormant fault which is provided in the paper may be disputed. The notion of a dormant fault seems to admit no convincing intuition. All faults are defects but not the other way around. The idea of a fault exclusively depends on an instruction sequence and a specification of which it is considered to be a candidate implementation. In the presence of a design, however, in addition to faults, the notion of a deviation from design (DFD) defect arises, which constitutes a class of defects many of which are not faults. For DFD defects the notion of dormancy admits a straightforward and convincing definition