arXiv subjects

J. A. Bergstra

Publications and source records attributed to J. A. Bergstra.

At least 19 recordsLinked to original sources

arXiv2025

Space-time process algebra with asynchronous communication

We introduce a process algebra that concerns the timed behaviour of distributed systems with a known spatial distribution. This process algebra provides a communication mechanism that deals with the fact that a datum sent at one point in space can only be received at another point in space at the point in time that the datum reaches that point in space. The variable-binding integration operator used in related process algebras to model such a communication mechanism is absent from this process algebra. This is considered an advantage because the variable-binding operator does not really fit in with an algebraic approach and a process algebra with this operator is not firmly founded in established metatheory.

cs.LO↗

arXiv2020

Using Hoare logic in a process algebra setting

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.

cs.LO↗

arXiv2020

Process algebra with strategic interleaving

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.

cs.LO↗

arXiv2019

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

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.

cs.OH↗

arXiv2019

Program algebra for Turing-machine programs

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.

cs.PL↗

arXiv2019

On the complexity of the correctness problem for non-zeroness test instruction sequences

This paper concerns the question to what extent it can be efficiently determined whether an arbitrary program correctly solves a given problem. This question is investigated with programs of a very simple form, namely instruction sequences, and a very simple problem, namely the non-zeroness test on natural numbers. The instruction sequences concerned are of a kind by which, for each $n > 0$, each function from $\{0,1\}^n$ to $\{0,1\}$ can be computed. The established results include the time complexities of the problem of determining whether an arbitrary instruction sequence correctly implements the restriction to $\{0,1\}^n$ of the function from $\{0,1\}^*$ to $\{0,1\}$ that models the non-zeroness test function, for $n > 0$, under several restrictions on the arbitrary instruction sequence.

cs.LO↗

arXiv2018

A short introduction to program algebra with instructions for Boolean registers

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.

cs.PL↗

arXiv2018

Instruction sequences expressing multiplication algorithms

For each function on bit strings, its restriction to bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We describe instruction sequences of this kind that compute the function on bit strings that models multiplication on natural numbers less than $2^N$ with respect to their binary representation by bit strings of length $N$, for a fixed but arbitrary $N > 0$, according to the long multiplication algorithm and the Karatsuba multiplication algorithm. We find among other things that the instruction sequence expressing the former algorithm is longer than the one expressing the latter algorithm only if the length of the bit strings involved is greater than $2^8$. We also go into the use of an instruction sequence with backward jump instructions for expressing the long multiplication algorithm. This leads to an instruction sequence that it is shorter than the other two if the length of the bit strings involved is greater than $2$.

cs.PL↗

arXiv2017

Axioms for behavioural congruence of single-pass instruction sequences

In program algebra, an algebraic theory of single-pass instruction sequences, three congruences on instruction sequences are paid attention to: instruction sequence congruence, structural congruence, and behavioural congruence. Sound and complete axiom systems for the first two congruences were already given in early papers on program algebra. The current paper is the first one that is concerned with an axiom system for the third congruence. The presented axiom system is especially notable for its axioms that have to do with forward jump instructions.

cs.PL↗

arXiv2017

Instruction sequence expressions for the secure hash algorithm SHA-256

The secure hash function SHA-256 is a function on bit strings. This means that its restriction to the bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We describe such instruction sequences for the restrictions to bit strings of the different possible lengths by means of uniform terms from an algebraic theory.

cs.PL↗

arXiv2017

A Hoare-like logic of asserted single-pass instruction sequences

We present a formal system for proving the partial correctness of a single-pass instruction sequence as considered in program algebra by decomposition into proofs of the partial correctness of segments of the single-pass instruction sequence concerned. The system is similar to Hoare logics, but takes into account that, by the presence of jump instructions, segments of single-pass instruction sequences may have multiple entry points and multiple exit points. It is intended to support a sound general understanding of the issues with Hoare-like logics for low-level programming languages.

cs.LO↗

arXiv2017

Contradiction-tolerant process algebra with propositional signals

In a previous paper, an ACP-style process algebra was proposed in which propositions are used as the visible part of the state of processes and as state conditions under which processes may proceed. This process algebra, called ACPps, is built on classical propositional logic. In this paper, we present a version of ACPps built on a paraconsistent propositional logic which is essentially the same as CLuNs. There are many systems that would have to deal with self-contradictory states if no special measures were taken. For a number of these systems, it is conceivable that accepting self-contradictory states and dealing with them in a way based on a paraconsistent logic is an alternative to taking special measures. The presented version of ACPps can be suited for the description and analysis of systems that deal with self-contradictory states in a way based on the above-mentioned paraconsistent logic.

cs.LO↗

arXiv2016

Instruction sequence size complexity of parity

Each Boolean function can be computed by a single-pass instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Auxiliary Boolean registers are not necessary for this. In the current paper, we show that, in the case of the parity functions, shorter instruction sequences are possible with the use of an auxiliary Boolean register in the presence of instructions to complement the content of auxiliary Boolean registers. This result supports, in a setting where programs are instruction sequences acting on Boolean registers, a basic intuition behind the storage of auxiliary data, namely the intuition that this makes possible a reduction of the size of a program.

cs.CC↗

arXiv2016

On instruction sets for Boolean registers in program algebra

In previous work carried out in the setting of program algebra, including work in the area of instruction sequence size complexity, we chose instruction sets for Boolean registers that contain only instructions of a few of the possible kinds. In the current paper, we study instruction sequence size bounded functional completeness of all possible instruction sets for Boolean registers. We expect that the results of this study will turn out to be useful to adequately assess results of work that is concerned with lower bounds of instruction sequence size complexity.

cs.PL↗

arXiv2016

Transformation of fractions into simple fractions in divisive meadows

Meadows are alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Divisive meadows are meadows with the multiplicative inverse operation replaced by a division operation. Viewing a fraction as a term over the signature of divisive meadows that is of the form p / q, we investigate which divisive meadows admit transformation of fractions into simple fractions, i.e. fractions without proper subterms that are fractions.

math.RA↗

arXiv2015

Probabilistic thread algebra

We add probabilistic features to basic thread algebra and its extensions with thread-service interaction and strategic interleaving. Here, threads represent the behaviours produced by instruction sequences under execution and services represent the behaviours exhibited by the components of execution environments of instruction sequences. In a paper concerned with probabilistic instruction sequences, we proposed several kinds of probabilistic instructions and gave an informal explanation for each of them. The probabilistic features added to the extension of basic thread algebra with thread-service interaction make it possible to give a formal explanation in terms of non-probabilistic instructions and probabilistic services. The probabilistic features added to the extensions of basic thread algebra with strategic interleaving make it possible to cover strategies corresponding to probabilistic scheduling algorithms.

cs.LO↗

arXiv2015

On algorithmic equivalence of instruction sequences for computing bit string functions

Every partial function from bit strings of a given length to bit strings of a possibly different given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We look for an equivalence relation on instruction sequences of this kind that captures to a reasonable degree the intuitive notion that two instruction sequences express the same algorithm.

cs.LO↗

arXiv2014

Instruction sequence notations with probabilistic instructions

This paper concerns instruction sequences that contain probabilistic instructions, i.e. instructions that are themselves probabilistic by nature. We propose several kinds of probabilistic instructions, provide an informal operational meaning for each of them, and discuss related work. On purpose, we refrain from providing an ad hoc formal meaning for the proposed kinds of instructions. We also discuss the approach of projection semantics, which was introduced in earlier work on instruction sequences, in the light of probabilistic instruction sequences.

cs.PL↗