Papers by Roland Backhouse
Journal of Functional Programming, 2024
An equivalence relation can be constructed from a given (homogeneous, binary) relation in two ste... more An equivalence relation can be constructed from a given (homogeneous, binary) relation in two steps: first, construct the smallest reflexive and transitive relation containing the given relation (the "star" of the relation) and, second, construct the largest symmetric relation that is included in the result of the first step. The fact that the final result is also reflexive and transitive (as well as symmetric), and thus an equivalence relation, is not immediately obvious, although straightforward to prove. Rather than prove that the defining properties of reflexivity and transitivity are satisfied, we establish reflexivity and transitivity constructively by exhibiting a starth root-in a way that emphasises the creative process in its construction. The resulting construction is fundamental to algorithms that determine the strongly connected components of a graph as well as the decomposition of a graph into its strongly connected components together with an acyclic graph connecting such components.
The Capacity-CTorch Problem
Mathematics of Program Construction, 2008
Page 1. The Capacity-C Torch Problem Roland Backhouse School of Computer Science University of No... more Page 1. The Capacity-C Torch Problem Roland Backhouse School of Computer Science University of Nottingham, Nottingham NG8 1BB, England rcb@cs.nott.ac. uk Abstract. The torch problem (also known as the bridge problem ...
Theory of Computing Systems, 2007
Datatype-generic programs are programs that are parameterised by a datatype. We review the allego... more Datatype-generic programs are programs that are parameterised by a datatype. We review the allegorical foundations of a methodology of designing datatype-generic programs. The notion of F-reductivity, where F parametrises a datatype, is reviewed and a number of its properties are presented. The properties are used to give concise, effective proofs of termination of a number of datatype-generic programming schemas. The paper concludes with a concise proof of the well-foundedness of a datatype-generic occurs-in relation.
arXiv (Cornell University), Jan 30, 2024
More than 70 years ago, Jaques Riguet suggested the existence of an "analogie frappante" (strikin... more More than 70 years ago, Jaques Riguet suggested the existence of an "analogie frappante" (striking analogy) between so-called "relations de Ferrers" and a class of difunctional relations, members of which we call "diagonals". Inspired by his suggestion, we formulate an "analogie frappante" linking the notion of a block-ordered relation and the notion of the diagonal of a relation. We formulate several novel properties of the core/index of a diagonal, and use these properties to rephrase our "analogie frappante". Loosely speaking, we show that a block-ordered relation is a provisional ordering up to isomorphism and reduction to its core. (Our theorems make this informal statement precise.) Unlike Riguet (and others who follow his example), we avoid almost entirely the use of nested complements to express and reason about properties of these notions: we use factors (aka residuals) instead. The only (and inevitable) exception to this is to show that our definition of a "staircase" relation is equivalent to Riguet's definition of a "relation de Ferrers". Our "analogie frappante" also makes it obvious that a "staircase" relation is not necessarily block-ordered, in spite of the mental picture of such a relation presented by Riguet.
arXiv (Cornell University), Jan 30, 2024
Earlier papers introduced the notions of a core and an index of a relation (an index being a spec... more Earlier papers 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 thins ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the thins ordering. We also characterise the relations that are maximal with respect to the 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 thins ordering are the empty relation and the equivalence relations. This and other properties of thins provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations.
arXiv (Cornell University), Oct 12, 2023
An equivalence relation can be constructed from a given (homogeneous, binary) relation in two ste... more An equivalence relation can be constructed from a given (homogeneous, binary) relation in two steps: first, construct the smallest reflexive and transitive relation containing the given relation (the "star" of the relation) and, second, construct the largest symmetric relation that is included in the result of the first step. The fact that the final result is also reflexive and transitive (as well as symmetric), and thus an equivalence relation, is not immediately obvious, although straightforward to prove. Rather than prove that the defining properties of reflexivity and transitivity are satisfied, we establish reflexivity and transitivity constructively by exhibiting a particular starth root-in a way that emphasises the creative process in its construction. The constructed starth root is fundamental to algorithms that determine the strongly connected components of a graph as well as the decomposition of a graph into its strongly connected components together with an acyclic graph connecting such components.
Proceedings of the 2006 international conference on Datatype-generic programming
Generic Programming: An Introduction
Lecture Notes in Computer Science, 1999
Revised Lectures from the International Summer School and Workshop on Algebraic and Coalgebraic Methods in the Mathematics of Program Construction
MathSpad : ergonomic document preparation
Lecture Notes in Computer Science, Aug 12, 2008
The version presented here may differ from the published version or from the version of record. I... more The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.
On Difunctions
Journal of logical and algebraic methods in programming, Jun 1, 2023
Proceedings of the 5th International Conference on Mathematics of Program Construction
Springer eBooks, 2010
One-person solitaire-like games are explored with a view to using them in teaching algorithmic pr... more One-person solitaire-like games are explored with a view to using them in teaching algorithmic problem solving. The key to understanding solutions to such games is the identification of invariant properties of polynomial arithmetic. We demonstrate this via three case studies: solitaire itself, tiling problems and a novel class of one-person games. The known classification of states of the game of (peg) solitaire into 16 equivalence classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel algebraic formulation of the solution to a class of tiling problems. Finally, we introduce an infinite class of challenging one-person games, which we call "replacement-set games", inspired by earlier work by Chen and Backhouse on the relation between cyclotomic polynomials and generalisations of the seven-trees-in-one type isomorphism. We present an algorithm to solve arbitrary instances of replacement-set games and we show various ways of constructing infinite (solvable) classes of replacement-set games.
Springer eBooks, 1998
This volume contains the papers selected for presentation at the 4th International Conference on ... more This volume contains the papers selected for presentation at the 4th International Conference on Mathematics of Program Construction (MPC'98), which was held June 15-17, 1998 on the island Marstrand near GSteborg in Sweden. The general theme of this series of conferences is the use of crisp, clear mathematics in the discovery and design of algorithms and in the development of corresponding software or hardware. The conference theme reflects the growing interest in formal, mathematically based methods for the construction of software and hardware. The goal of the MPC conferences is to report on and significantly advance the state of the art in this area. Previous conferences were held in 1989 at Twente, The Netherlands, organized by the Rijksuniversiteit Groningen, in 1992 at Oxford, United Kingdom, and in 1995 at Kloster Irsee, Germany, organized by Augsburg University. The proceedings of these conferences were published as LNCS 375, 669, and 947, respectively. The program committee received 57 submissions, from which 17 were selected for presentation at the conference. Invited lectures were presented by David Harel, John Hughes, and Burghard von Karger.
Workshop on generic programming
The Schröder-Bernstein theorem
No abstract
We introduce the general notions of an index and a core of a relation. We postulate a limited for... more 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.
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Papers by Roland Backhouse