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Classic formulae for entropy and cross-entropy contain operations x0 and log2x that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as 0log20 and uncertainties in large scale calculations; partiality also introduces complications in logical analysis. Instead of adding conventions or splitting formulae into cases, we create a new algebra of real numbers with two symbols ±∞ for signed infinite values and a symbol named ⊥ for the undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback–Leibler divergence, and Shannon divergence can be expressed without concerning any further conventions. The algebra may form a basis for probability theory more generally.

Using the conceptual analysis of instruction sequence faults, failures, and defects as developed by the author in [10] and [12], a survey of testing is developed as an extension of a theory of instruction sequences. An attempt is made to develop a consistent terminology regarding instruction sequence testing while taking into account the literature on software testing at large.

The formal theory of division in arithmetical algebras reconstructs fractions as syntactic objects called fracterms. Basic to calculation, is the simplification of fracterms to fracterms with one division operator, a process called fracterm attening. We consider the equational axioms of a calculus for calculating with fracterms to determine what is necessary and sufficient for the fracterm calculus to allow fracterm flattening. For computation, arithmetical algebras require operators to be total for which there are several semantical methods. It is shown under what constraints up to isomorphism, the unique total and minimal enlargement of a field Q(\div) of rational numbers equipped with a partial division operator \div has fracterm attening.

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.

The notion of an instruction sequence fault is considered as a theoretical concept, for which the justification of the qualification of a fragment as faulty is mathematical instead of pragmatic, the latter approach being much more common. Starting from so-called Laski faults a range of patterns of faults and changes thereof for instruction sequences is developed.

Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to an instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.

The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced. Three examples of such specifications are given. A preference is formulated for a specification by means of infinitely many equations which can be presented via a finite number of so-called schematic equations phrased in terms of an infinite signature.On the basis of the latter specification three topics are discussed: (i) fracterm decomposition operators and the numerator paradox, (ii) foundational specifications of arithmetical datatypes, and (iii) poly-infix operations.

This paper presents an algebraic theory of instruction sequences with instructions for Turing tapes as basic instructions, the behaviours produced by the instruction sequences concerned under execution, and the interaction between such behaviours and Turing tapes provided by an execution environment. This theory provides a setting for the development of theory in areas such as computability and computational complexity that distinguishes itself by offering the possibility of equational reasoning and being more general than the setting provided by a known version of the Turing-machine model of computation. The theory is essentially an instantiation of a parameterized algebraic theory which is the basis of a line of research in which issues relating to a wide variety of subjects from computer science have been rigorously investigated thinking in terms of instruction sequences.

The number of instructions of an instruction sequence is taken for its logical SLOC, and is abbreviated with LLOC. A notion of quantitative expressiveness is based on LLOC and in the special case of operation over a family of single bit registers a collection of elementary properties is established. A dedicated notion of interface is developed and used for stating relevant properties of classes of instruction sequences.

Bayesian inference as applied in a legal setting is about belief transfer and involves a plurality of agents and communication protocols. A forensic expert (FE) may communicate to a trier of fact (TOF) first its value of a certain likelihood ratio with respect to FE's belief state as represented by a probability function on FE's proposition space. Subsequently FE communicates its recently acquired confirmation that a certain evidence proposition is true. Then TOF performs likelihood ratio transfer mediated reasoning thereby revising their own belief~state. The logical principles involved in likelihood transfer mediated reasoning are discussed in a setting where probabilistic arithmetic is done within a meadow, and with Adams conditioning placed in a central~role.

A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven.

A parameterized algebraic theory of instruction sequences, objects that represent the behaviours produced by instruction sequences under execution, and objects that represent the behaviours exhibited by the components of the execution environment of instruction sequences is the basis of a line of research in which issues relating to a wide variety of subjects from computer science have been rigorously investigated thinking in terms of instruction sequences. In various papers that belong to this line of research, use is made of an instantiation of this theory in which the basic instructions are instructions to read out and alter the content of Boolean registers and the components of the execution environment are Boolean registers. In this paper, we give a simplified presentation of the most general such instantiated theory.