this paper we introduce a notion of induction over an arbitrary datatype and go on to show how the notion is used to establish unicity of a certain (broad) class of equations. Our overall goal is to develop a calculational theory of mathematical induction. That is we want to be able to calculate relations on which inductive arguments may be based using laws that relate admitting induction to the mechanisms for constructing datatypes. We also want to incorporate such calculations into a methodology for calculating inductive hypotheses rather than leaving their creation to inspired guesswork. This is a bold aim, in view of the vast amount of knowledge and experience that already exists on proof by induction, but recent advances in the role played by Galois connections in the calculus of relations have led us to speculate that significant progress can be made in the short term. The theory developed in this paper is general and not specific to any particular datatype. We define a notion of F -reductivity (so called in order to avoid confusion with existing notions of inductivity), where F stands for a "relator", and show that F -reductive relations always exist, whatever the value of F . We also give laws for constructing reductive relations from existing reductive relations. We conclude the paper by introducing the dual notion of F -inductivity and briefly contrast it with F -reductivity. The organisation of this note is as follows. In section 2 we give a very brief introduction to the relational calculus. In section 3 the notion of reductivity is defined. This notion is a generalisation of well-foundedness, or inductivity. Then in section 4 we define a class of equations and prove that an equation from that class has a unique solution if one of its components enjoys a red...

Several concise formulations of mathematical induction are presented and proven 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 lends itself very well for use in calculational proofs. As a non-trivial example a proof of a generalisation of Newman's lemma is given. The paper begins with an introduction to relation algebra and is reasonably selfcontained. (Some knowledge of lattice theory, in particular the Knaster-Tarski fixed point theorem for complete lattices, is assumed.) The style is expository and suggestions for exercises are included. The basic concept underlying many of the calculations is the notion of a Galois connection, and the paper could be seen as an introductory tutorial in the use of this concept. The idea of formal reasoning --- by which we mean the manipulation of uninterpreted formulae accordi...

Abstract not found

We have recast the problem of truth maintenance in a setting of algebraic equations over Boolean lattices. If a method of labeling propositions to justify them according to some reasoning agent's constraints of belief happens to conform to the postulates of Boolean lattices, the labeling system can be reformulated as an algebraic equation solving system. All truth maintenance systems known to us can be so reformulated. This note summarizes our investigations into the existence and structure of solutions of these algebraic systems. Our central result is a unique factorization theorem for lattice equational systems and their solutions. Our theoretical results are interpreted to compare various styles of truth maintenance and to reveal certain computational difficulties implicit in the algebraic structure of truth maintenance. I.

Several concise formulations of mathematical induction are presented and proven 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 a Newman's lemma. The second is a calculation of a condition under which the union of two well-founded relations is wellfounded. 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.

ion Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 6.3 The Beautiful Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 6.4 The Rolling Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 6.5 The Square Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 6.6 The Exchange Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94 6.7 The Diagonal Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100 6.7.1 One Half : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100 6.7.2 The Other Half : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 103 6.8 Mutual Recursion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106 7 Monads 111 7.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 111 7.2 Monads and Adjunctions : : : : : : : : : : : : : : : : : : : : : : : : : : : : 112 7.3 Basic Adjunction : : : : : : : : : : : : ...

New properties and implications of inclusion systems are investigated in the present paper. Many properties of lattices, factorization systems and special practical cases can be abstracted and adapted to our framework, making the various versions of inclusion systems useful tools for computer scientists and mathematicians.

Dijkstra and Scholten have formulated a theorem stating that all disjunctivity properties of a predicate transformer are preserved by the construction of least prefix points. An alternative proof of their theorem is presented based on two fundamental fixed point theorems, the abstraction theorem and the fusion theorem, and the fact that suprema in a lattice are defined by a Galois connection. The abstraction theorem seems to be new; the fusion theorem is known but its importance does not seem to be fully recognised. The abstraction theorem, the fusion theorem, and Dijkstra and Scholten's theorem are then generalised to the context of category theory and shown to be valid. None of the theorems in this context seems to be known, although specific instances of Dijkstra and Scholten's theorem are known. The main point of the paper is to discuss the process of drawing inspiration from lattice theory to formulate theorems in category theory (first advocated by Lambek in 1968). W...

Abstract. A bimonotone linear inequality is a linear inequality with at most two nonzero coefficients that are of opposite signs (if both different from zero). A linear inequality defines a halfspace that is a sublattice of Rn (a subset closed with respect to componentwise maximum and minimum) if and only if it is bimonotone. Veinott has shown that a polyhedron is a sublattice if and only if it can be defined by a finite system of bimonotone linear inequalities, whereas Topkis has shown that every sublattice of Rn (and of more general product lattices) is the solution set of a system of nonlinear bimonotone inequalities. In this paper we prove that a subset of Rn is the solution set of a countable system of bimonotone linear inequalities if and only if it is a closed convex sublattice. We also present necessary and/or sufficient conditions for a sublattice to be the intersection of the cartesian product of its projections on the coordinate axes with the solution set of a (possibly infinite) system of bimonotone linear inequalities. We provide explicit constructions of such systems of bimonotone linear inequalities under certain assumptions on the sublattice. We obtain Veinott’s polyhedral representation theorem and a 0-1 version of Birkhoff’s Representation Theorem as corollaries. We also point out a few potential pitfalls regarding properties of sublattices of Rn. 1.