A Calculational Theory of Pers as Types

In the calculational approach to programming, programs are derived from specifications by algebraic reasoning. This report presents a calculational programming framework based upon the notion of binary relations as programs, and partial equivalence relations (pers) as types. Working with relations as programs generalises the functional paradigm, admiting non-determinism and the use of relation converse. Working with pers as types permits a natural treatment of types that are subject to laws and restrictions

Making Functionality More General

The definition for the notion of a "function" is not cast in stone, but depends upon what we adopt as types in our language. With partial equivalence relations (pers) as types in a relational language, we show that the functional relations are precisely those satisfying the simple equation f = f o fu o f, where "o" and "u" are respectively the composition and converse operators for relations. This article forms part of "A calculational theory of pers as types"

The Index and Core of a Relation. With Applications to the Axiomatics of Relation Algebra

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -- specifically that all partial equivalence relations have an index -- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice

The Thins Ordering on Relations

Earlier papers \cite{VB2022,VB2023a,VB2023b} introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated -- specifically that all partial equivalence relations (pers) have an index -- and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the \textsf{thins} ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the \textsf{thins} ordering. We also characterise the relations that are maximal with respect to the \textsf{thins} ordering. Apart from our axiom of choice, the axiom system we employ is paired to a bare minimum and admits many models other than concrete relations -- we do not assume, for example, the existence of complements; in the case of concrete relations, the theorem is that the maximal elements of the \textsf{thins} ordering are the empty relation and the equivalence relations. This and other properties of \textsf{thins} provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations.Comment: The open problem posed in the first-submitted version of this paper has been successfully resolved. As a consequence, the additional axiom is no longer require

comparative study of the legislative processes in Finland, Slovenia and the United Kingdom as a source of inspiration for enhancing the efficiency of the Dutch legislative process

The main research question of the current study is when whether the efficiency of the Dutch legislative procedure for parliamentary acts indeed constitutes a problem, in particular if compared to the achievements of legislative processes in several other European countries and, if that turns out to be the case, whether lessons can be learned from those legislative processes and practices abroad with respect to pace and duration of the legislative process, phases and actors, transparency and the role of ICT. CONTENT: 1. Introduction 2. The selection of countries 3. Finland 4. Slovenia 5. United Kingdom 6. In comparison 7. Conclusio

A Calculational Theory of Pers as Types

In the calculational approach to programming, programs are derived from specifications by algebraic reasoning. This report presents a calculational programming framework based upon the notion of binary relations as programs, and partial equivalence relations (pers) as types. Working with relations as programs generalises the functional paradigm, admiting non-determinism and the use of relation converse. Working with pers as types permits a natural treatment of types that are subject to laws and restrictions