arXiv subjects
Publications and source records attributed to 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.
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.
An attempt is made to define the concept of execution of an instruction sequence. It is found to be a special case of directly putting into effect of an instruction sequence. Directly putting into effect of an instruction sequences comprises interpretation as well as execution. Directly putting into effect is a special case of putting into effect with other special cases classified as indirectly putting into effect.
We discuss an algebraic approach to propositional logic with side effects. To this end, we use Hoare's conditional [1985], which is a ternary connective comparable to if-then-else. Starting from McCarthy's notion of sequential evaluation [1963] we discuss a number of valuation congruences and we introduce Hoare-McCarthy algebras as the structures that characterize these congruences.
A steering fragment of an instruction sequence consists of a sequence of steering instructions. These are decision points involving the check of a propositional statement in sequential logic. The question is addressed why composed propositional statements occur in steering fragments given the fact that a straightforward transformation allows their elimination. A survey is provided of constraints that may be implicitly assumed when composed propositional statements occur in a meaningful instruction sequence.
General definitions as well as rules of reasoning regarding control code production, distribution, deployment, and usage are described. The role of testing, trust, confidence and risk analysis is considered. A rationale for control code testing is sought and found for the case of safety critical embedded control code.
We investigate the expressiveness of backward jumps in a framework of formalized sequential programming called program algebra. We show that - if expressiveness is measured in terms of the computability of partial Boolean functions - then backward jumps are superfluous. If we, however, want to prevent explosion of the length of programs, then backward jumps are essential.
We introduce an algebra of instruction sequences by presenting a semigroup C in which programs can be represented without directional bias: in terms of the next instruction to be executed, C has both forward and backward instructions and a C-expression can be interpreted starting from any instruction. We provide equations for thread extraction, i.e., C's program semantics. Then we consider thread extraction compatible (anti-)homomorphisms and (anti-)automorphisms. Finally we discuss some expressiveness results.
We define focus-method interfaces and some connections between such interfaces and instruction sequences, giving rise to instruction sequence components. We provide a flexible and practical notation for interfaces using an abstract datatype specification comparable to that of basic process algebra with deadlock. The structures thus defined are called progression rings. We also define thread and service components. Two types of composition of instruction sequences or threads and services (called `use' and `apply') are lifted to the level of components.
A combination of program algebra with the theory of meadows is designed leading to a theory of computation in algebraic structures which use in addition to a zero test and copying instructions the instruction set $\{x \Leftarrow 0, x \Leftarrow 1, x\Leftarrow -x, x\Leftarrow x^{-1}, x\Leftarrow x+y, x\Leftarrow x\cdot y\}$. It is proven that total functions on cancellation meadows can be computed by straight-line programs using at most 5 auxiliary variables. A similar result is obtained for signed 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.
Earlier work on program and thread algebra detailed the functional, observable behavior of programs under execution. In this article we add the modeling of unobservable, mechanistic processing, in particular processing due to jump instructions. We model mechanistic processing preceding some further behavior as a delay of that behavior; we borrow a unary delay operator from discrete time process algebra. We define a mechanistic improvement ordering on threads and observe that some threads do not have an optimal implementation.
A meadow is a zero totalised field (0^{-1}=0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomatization of these operators and thus obtain a finite basis for differential cancellation meadows. Using the Zariski topology we prove the existence of a differential cancellation meadow.
We go into the need for, and the requirements on, a formal theory of budgets. We present a simple algebraic theory of rational budgets, i.e., budgets in which amounts of money are specified by functions on the rational numbers. This theory is based on the tuplix calculus. We go into the importance of using totalized models for the rational numbers. We present a case study on the educational budget of a university department offering master programs.
Analytic execution architectures have been proposed by the same authors as a means to conceptualize the cooperation between heterogeneous collectives of components such as programs, threads, states and services. Interface groups have been proposed as a means to formalize interface information concerning analytic execution architectures. These concepts are adapted to organization architectures with a focus on financial transfers. Interface groups (and monoids) now provide a technique to combine interface elements into interfaces with the flexibility to distinguish between directions of flow dependent on entity naming. The main principle exploiting interface groups is that when composing a closed system of a collection of interacting components, the sum of their interfaces must vanish in the interface group modulo reflection. This certainly matters for financial transfer interfaces. As an example of this, we specify an interface group and within it some specific interfaces concerning the financial transfer architecture for a part of our local academic organization. Financial transfer interface groups arise as a special case of more general service architecture interfaces.
A rigid loop is a for-loop with a counter not accessible to the loop body or any other part of a program. Special instructions for rigid loops are introduced on top of the syntax of the program algebra PGA. Two different semantic projections are provided and proven equivalent. One of these is taken to have definitional status on the basis of two criteria: `normative semantic adequacy' and `indicative algorithmic adequacy'.