Jan A. Bergstra

Jan A. Bergstra, Alban Ponse

Proposition algebra is based on Hoare's conditional connective, which is a ternary connective comparable to if-then-else and used in the setting of propositional logic. Conditional statements are provided with a simple semantics that is based on evaluation trees and that characterizes so-called free valuation congruence: two conditional statements are free valuation congruent if, and only if, they have equal evaluation trees. Free valuation congruence is axiomatized by the four basic equational axioms of proposition algebra that define the conditional connective. Valuation congruences that identify more conditional statements than free valuation congruence are repetition-proof, contractive, memorizing, and static valuation congruence. Each of these valuation congruences is characterized using a transformation on evaluation trees: two conditional statements are C-valuation congruent if, and only if, their C-transformed evaluation trees are equal. These transformations are simple and natural, and only for static valuation congruence a slightly more complex transformation is used. Also, each of these valuation congruences is axiomatized in proposition algebra. A spin-off of our approach can be called "normalization functions for proposition algebra": for each valuation congruence C considered, two conditional statements are C-valuation congruent if, and only if, the C-normalization function returns equal images.

Jan A. Bergstra, Alban Ponse, Daan J. C. Staudt

Short-circuit evaluation denotes the semantics of propositional connectives in which the second argument is evaluated only if the first argument does not suffice to determine the value of the expression. Short-circuit evaluation is widely used in programming, with sequential conjunction and disjunction as primitive connectives. We study the question which logical laws axiomatize short-circuit evaluation under the following assumptions: compound statements are evaluated from left to right, each atom (propositional variable) evaluates to either true or false, and atomic evaluations can cause a side effect. The answer to this question depends on the kind of atomic side effects that can occur and leads to different "short-circuit logics". The basic case is FSCL (free short-circuit logic), which characterizes the setting in which each atomic evaluation can cause a side effect. We recall some main results and then relate FSCL to MSCL (memorizing short-circuit logic), where in the evaluation of a compound statement, the first evaluation result of each atom is memorized. MSCL can be seen as a sequential variant of propositional logic: atomic evaluations cannot cause a side effect and the sequential connectives are not commutative. Then we relate MSCL to SSCL (static short-circuit logic), the variant of propositional logic that prescribes short-circuit evaluation with commutative sequential connectives. We present evaluation trees as an intuitive semantics for short-circuit evaluation, and simple equational axiomatizations for the short-circuit logics mentioned that use negation and the sequential connectives only.

Jan A. Bergstra, Alban Ponse

Partial algebras and datatypes are discussed with the use of signatures that allow partial functions, and a three-valued short-circuit (sequential) first order logic with a Tarski semantics. The propositional part of this logic is also known as McCarthy calculus and has been studied extensively. Axioms for the fracterm calculus of partial meadows are given. The case is made that in this way a rather natural formalisation of fields with division operator is obtained. It is noticed that the logic thus obtained cannot express that division by zero must be undefined. An interpretation of the three-valued sequential logic into $\bot$-enlargements of partial algebras is given, for which it is concluded that the consequence relation of the former logic is semi-computable, and that the $\bot$-enlargement of a partial meadow is a common meadow.

Jan A. Bergstra

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.

Jan A. Bergstra

An alignment is developed between the terminology of outcome oriented decision taking and a terminology for promise issuing. Differences and correspondences are investigated between the concepts of decision and promise. For decision taking, two forms are distinguished: the external outcome delivering form and internalized decision taking. Internalized decision taking is brought in connection with Marc Slors' theory of self-programming. Examples are produced for decisions and promises in four different several settings each connected with software technology: instruction sequence effectuation, informational money transfer, budget announcement, and division by zero.

Jan A. Bergstra, Karl de Leeuw

The famous new money Bitcoin is classified as a technical informational money (TIM). Besides introducing the idea of a TIM, a more extreme notion of informational money will be developed: exclusively informational money (EXIM). The informational coins (INCOs) of an EXIM can be in control of an agent but are not owned by any agent. INCOs of an EXIM cannot be stolen, but they can be lost, or thrown away. The difference between an EXIM and a TIM shows up when considering a user perspective on security matters. Security for an EXIM user is discussed in substantial detail, with the remarkable conclusion that computer security (security models, access control, user names, passwords, firewalls etc.) is not always essential for an EXIM, while the application of cryptography based information security is unavoidable for the use of an EXIM. Bitcoin seems to meet the criteria of an EXIM, but the assertion that "Bitcoin is an EXIM", might also be considered problematic. As a thought experiment we will contemplate Bitguilder, a hypothetical copy of Bitcoin that qualifies as an EXIM. A business ethics assessment of Bitcoin is made which reveals a number of worries. By combining Bitguilder with a so-called technical informational near-money (TINM) a dual money system, having two units with a fluctuating rate, may be obtained. It seems that a dual money can remedy some, but not all, of the ethical worries that arise when contemplating Bitcoin after hypothetically having become a dominant form of money. The contributions that Bitcoin's designers can potentially make to the evolution of EXIMs and TIMs is analyzed in terms of the update of the portfolio of money related natural kinds that comes with Bitcoin.

Jan A. Bergstra

Motivated by an intention to remedy current complications with Dutch terminology concerning informatics, the term informaticology is positioned to denote an academic counterpart of informatics where informatics is conceived of as a container for a coherent family of practical disciplines ranging from computer engineering and software engineering to network technology, data center management, information technology, and information management in a broad sense. Informaticology escapes from the limitations of instrumental objectives and the perspective of usage that both restrict the scope of informatics. That is achieved by including fiction science in informaticology and by ranking fiction science on equal terms with computer science and data science, and framing (the study of) game design, evelopment, assessment and distribution, ranging from serious gaming to entertainment gaming, as a chapter of fiction science. A suggestion for the scope of fiction science is specified in some detail. In order to illustrate the coherence of informaticology thus conceived, a potential application of fiction to the ontology of instruction sequences and to software quality assessment is sketched, thereby highlighting a possible role of fiction (science) within informaticology but outside gaming.

Jan A. Bergstra

Decision Taking is discussed in the context of the role it may play for a selling agent in a search market, in particular for agents involved in the sale of valuable and relatively unique items, such as a dwelling, a second hand car, or a second hand recreational vessel. Detailed connections are made between the architecture of decision making processes and a sample of software technology based concepts including instruction sequences, multi-threading, and thread algebra. Ample attention is paid to the initialization or startup of a thread dedicated to achieving a given objective, and to corresponding decision taking. As an application, the selling of an item is taken as an objective to be achieved by running a thread that was designed for that purpose.

Jan A. Bergstra

Decision taking is discussed in the context of the role it may play for various types of agents, and it is contrasted with action determination. Some remarks are made about the role of decision taking and action determination in the ongoing debate concerning the reverse polder development of the hertogin Hedwige polder.

Jan A. Bergstra

Decision taking can be performed as a service to other parties and it is amenable to outtasking rather than to outsourcing. Outtasking decision taking is compatible with selfsourcing of decision making activities carried out in preparation of decision taking. Decision taking as a service (DTaaS) is viewed as an instance of so-called decision casting. Preconditions for service casting are examined, and compliance of decision taking with these preconditions is confirmed. Potential advantages and disadvantages of using decision taking as a service are considered.

Jan A. Bergstra

The notion of an instruction sequence fault is considered from various perspectives. Four different viewpoints on what constitutes a fault, or how to use the notion of a fault, are formulated. An integration of these views is proposed.