Showing papers by "Jan A. Bergstra published in 2019"

Using Hoare logic in a process algebra setting

Abstract: This paper concerns the relation between process algebra and Hoare logic. We investigate the question whether and how a Hoare logic can be used for reasoning about how data change in the course of a process when reasoning equationally about that process. We introduce an extension of ACP (Algebra of Communicating Processes) with features that are relevant to processes in which data are involved, present a Hoare logic for the processes considered in this process algebra, and discuss the use of this Hoare logic as a complement to pure equational reasoning with the equational axioms of the process algebra.

Division by Zero: a Survey of Options

Abstract: The idea that, as opposed to the conventional viewpoint, division by zero may produce a meaningful result, is long standing and has attracted inter- est from many sides. We provide a survey of some options for defining an outcome for the application of division in case the second argument equals zero. The survey is limited by a combination of simplifying assumptions which are grouped together in the idea of a premeadow, which generalises the notion of an associative transfield.

A Promise Theoretic Account of the Boeing 737 Max MCAS Algorithm Affair

Abstract: Many public controversies involve the assessment of statements about which we have imperfect information. Without a structured approach, it is quite difficult to develop an approach to reasoning which is not based on ad hoc choices. Forms of logic have been used in the past to try to bring such clarity, but these fail for a variety of reasons. We demonstrate a simple approach to bringing a standardized approach to semantics, in certain discourse, using Promise Theory. As a case, we use Promise Theory (PT) to collect and structure publicly available information about the case of the MCAS software component for the Boeing 737 Max flight control system.

Dual Number Meadows

Abstract: The class of dual number meadows is introduced. By definition this class is a quasivariety. Dual number meadows contain a non-zero element the square of which is zero. These structures are non-involutive and coregular. Some properties of the equational theory of dual number meadows are discussed and an initial algebra specification is given for the minimal dual number meadow of characteristic zero which contains the dual rational numbers. Several open problems are stated.

Process Algebra with Strategic Interleaving

Abstract: In process algebras such as ACP (Algebra of Communicating Processes), parallel processes are considered to be interleaved in an arbitrary way. In the case of multi-threading as found in contemporary programming languages, parallel processes are actually interleaved according to some interleaving strategy. An interleaving strategy is what is called a process-scheduling policy in the field of operating systems. In many systems, for instance hardware/software systems, we have to do with both parallel processes that may best be considered to be interleaved in an arbitrary way and parallel processes that may best be considered to be interleaved according to some interleaving strategy. Therefore, we extend ACP in this paper with the latter form of interleaving. The established properties of the extension concerned include an elimination property, a conservative extension property, and a unique expansion property.

Quantitative Expressiveness of Instruction Sequence Classes for Computation on Single Bit Registers

Abstract: 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 are established. A dedicated notion of interface is developed and is used for stating relevant properties of classes of instruction sequences

Adams Conditioning and Likelihood Ratio Transfer Mediated Inference

Abstract: Forensic science advocates the use of inference mechanisms which may be viewed as simple multi-agent protocols. An important protocol of this kind involves an agent FE (forensic expert) who communicates to a second agent TOF (trier of fact) first its value of a certain likelihood ratio with respect to its own belief state which is supposed to be captured by a probability function on FE's proposition space. Subsequently FE communicates its recently acquired confirmation that a certain evidence proposition is true. The inference part of this sort of reasoning, here referred to as likelihood ratio transfer mediated reasoning, involves TOF's revision of its own belief state, and in particular an evaluation of the resulting belief in the hypothesis proposition. Different realizations of likelihood ratio transfer mediated reasoning are distinguished: if the evidence hypothesis is included in the prior proposition space of TOF then a comparison is made between understanding the TOF side of a belief revision step as a composition of two successive steps of single likelihood Adams conditioning followed by a Bayes conditioning step, and as a single step of double likelihood Adams conditioning followed by Bayes conditioning; if, however the evidence hypothesis is initially outside the proposition space of TOF an application of proposition kinetics for the introduction of the evidence proposition precedes Bayesian conditioning, which is followed by Jeffrey conditioning on the hypothesis proposition.

Program Algebra for Turing-Machine Programs

Abstract: 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 computa- tional 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.

Program algebra for Turing-machine programs

Abstract: 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.