Papers by jaap van der woude
as Coherently Constructive Lattice Theory
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Calculations with Relations, an Example
Beauty Is Our Business, 1990
Predicate transformer semantics (PTS) has proved to be very fruitful, in particular for program d... more Predicate transformer semantics (PTS) has proved to be very fruitful, in particular for program derivation. It is well known, although not very well documented, that the version of PTS with universal conjunctivity for wlp and without the law of the excluded miracle is equivalent to some relation semantics (RS).
Siam Journal on Computing - SIAMCOMP, 1984
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library .Jan de Vries and Jaap van der Woude
Journal of Logical and Algebraic Methods in Programming, 2021
Central to algorithmic graph theory are the concepts of acyclicity and strongly connected compone... more Central to algorithmic graph theory are the concepts of acyclicity and strongly connected components of a graph, and the related search algorithms. This article is about combining mathematical precision and concision in the presentation of these concepts. Concise formulations are given for, for example, the reflexive-transitive reduction of an acyclic graph, reachability properties of acyclic graphs and their relation to the fundamental concept of "definiteness", and the decomposition of paths in a graph via the identification of its strongly connected components and a pathwise homomorphic acyclic subgraph. The relevant properties are established by precise algebraic calculation. The combination of concision and precision is achieved by the use of point-free relation algebra capturing the algebraic properties of paths in graphs, as opposed to the use of pointwise reasoning about paths between nodes in graphs.
A Relational Perspective on Types With Laws
With relational transformational programming in mind, an extension of a "lawless &qu... more With relational transformational programming in mind, an extension of a "lawless " relational theory of datatypes is proposed in order to study and manipulate quotient types within a Tarski-like calculus of relations. The extended notion of type, pertype (from partial equivalence relation), is shown to admit a complete lattice structure by constructing the order via a Galois connection. A pertyping of relations is developed and inductive pertypes generated by equations are discussed. Pertypes do occur in model theory for -calculus but we are unaware of manipulations with inductive "lawful" types based on a simple relational calculus. 1 Introduction Program construction and its theory are developing steadily towards an algebraic discipline combining methods from category theory, algebraic logic and lattice theory. The driving forces are the needs for uninterpreted manipulation, problem structuring and generalisation, and polymorphy, with a major role for calculational type theo...
Polynomial Relators (Extended Abstract)
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Lecture Notes in Computer Science, 1989
Domain Operators and Domain Kinds
Model-based specification of design patterns
Acta Informatica, 1999
Relational Heterogeneity Relaxed by Subtyping
Lecture Notes in Computer Science, 2011
ABSTRACT Homogeneous relation algebra is an elegant calculational framework with many application... more ABSTRACT Homogeneous relation algebra is an elegant calculational framework with many applications in computing science. In one application of relation algebra, called Ampersand, heterogeneous relation algebra is used as a specification language for business processes and information systems. For this purpose a typed version of relation algebra is needed together with subtyping. This requires heterogeneous relational algebra. However, the partiality of the composition and union operators in heterogeneous relational algebra are detrimental to its manipulative power. This paper proposes a practical solution to this problem. The authors suggest to relax the partiality of the heterogeneous operators. By suitable choices this homogenisation allows for a type-based specification language, which has sufficient manipulative power.
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Theoretical Computer Science, 1997
Several concise formulations of mathematical induction are presented and proved equivalent. The f... more Several concise formulations of mathematical induction are presented and proved equivalent. The formulations are expressed in variable-free relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when combined with the explicit use of Galois connections, lends itself very well for use in calculational proofs. Two non-trivial examples are presented. The first is a proof of Newman's lemma. The second is a calculation of a condition under which the union of two well-founded relations is well-founded. In both cases the calculations lead to generalisations of the known results. In the case of the latter example, one lemma generalises three different conditions.
Science of Computer Programming, 1989
DOI to the publisher's website. • The final author version and the galley proof are versions of t... more DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:
Pacific Journal of Mathematics, 1987
Consider homomorphisms φ: X-* Y and ψ: Y-> Z, where φ is open and N-to-one, ψ is almost periodic.... more Consider homomorphisms φ: X-* Y and ψ: Y-> Z, where φ is open and N-to-one, ψ is almost periodic. In the paper by R. J Sacker and G. R. Sell, it was shown that, under a certain condition on the phase group, the composition ψ ° φ: X-» Z is almost periodic (provided that Z is trivial and X is minimal). In this paper almost periodicity of ψ ° φ is studied under connectedness conditions on the fibers of ψ. For instance it is shown that if ψ is almost periodic with connected fibers than ψ φ is almost periodic. If ψ is locally almost periodic with locally connected fibers then φ ° ψ is locally almost periodic.
Pacific Journal of Mathematics, 1985
CHARACTERIZATIONS OF (H)PI EXTENSIONS JAAP VAN DER WOUDE A generalization of I. U. Bronstein's ch... more CHARACTERIZATIONS OF (H)PI EXTENSIONS JAAP VAN DER WOUDE A generalization of I. U. Bronstein's characterization for PD extensions is given and, exploiting similar ideas, HPI extensions are characterized intrinsically.
Demonic operators and monotype factors
Mathematical Structures in Computer Science, 1993
This paper tackles the problem of constructing a compact, point-free proof of the associativity o... more This paper tackles the problem of constructing a compact, point-free proof of the associativity of demonic composition of binary relations and its distributivity through demonic choice. In order to achieve this goal, a definition of demonic composition is proposed in which angelic composition is restricted by means of a so-called ‘monotype factor’. Monotype factors are characterised by a Galois connection similar to the Galois connection between composition and factorisation of binary relations. The identification of such a connection is argued to be highly conducive to the desired compactness of calculation.
Fixed-point calculus
Information Processing Letters, 1995
The aim of this paper is to present a small calculus of extreme fixed points and to show it in ac... more The aim of this paper is to present a small calculus of extreme fixed points and to show it in action. The fixed-point theorem that was the main incentive for writing this paper is the fusion theorem presented in Section 3. It exploits the calculational properties of Galois connections.
Ergodic Theory and Dynamical Systems, 1981
We study minimal flows and their extensions by means of the associated maximally highly proximal ... more We study minimal flows and their extensions by means of the associated maximally highly proximal flows. These, in turn, can be represented by highly proximal generators, which are certain subsets of the universal minimal flow. From this point of view we obtain information on relative disjointness, coalescence, the Bronstein property, and RIC extensions.
Ergodic Theory and Dynamical Systems, 1984
We discuss weak disjointness of homomorphisms of minimal transformation groups and use the techni... more We discuss weak disjointness of homomorphisms of minimal transformation groups and use the techniques involved to deepen our knowledge of the equicontinuous structure relation.
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Papers by jaap van der woude