Topic
Set-builder notation
About: Set-builder notation is a research topic. Over the lifetime, 161 publications have been published within this topic receiving 8302 citations. The topic is also known as: set comprehension.
Papers published on a yearly basis
Papers
Book•
1 Jan 1987
TL;DR: In this article, the authors present a largely balanced approach, which combines many elements of the different traditions of the subject, and includes a wide variety of examples, exercises, and applications, in order to illustrate the general concepts and results of the theory.
Abstract: Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.
4,212 citations
Book•
1 Jun 1986
TL;DR: Logic of propositions reasoning about predicates functions and operations set notation composite objects and invariants map notation sequence notation data rectification more on data types operation decomposition.
Abstract: Logic of propositions reasoning about predicates functions and operations set notation composite objects and invariants map notation sequence notation data rectification more on data types operation decomposition a small case study.
2,197 citations
Book•
1 Dec 1993
TL;DR: A logical framework for proofs in VDM and an example of a well-formedness proof obligation are presented.
Abstract: 1 Introduction.- 1.1 Background.- 1.2 How proofs arise in practice: an introductory example.- 1.3 A logical framework for proofs.- 1.4 Summary.- I A Logical Basis for Proof in VDM.- 2 Propositional LPF.- 2.1 Introduction.- 2.2 Basic axiomatisation.- 2.3 Derived rules reasoning by cases reasoning using contradiction.- 2.4 Using definitions: conjunction.- 2.5 Implication definedness further defined constructs.- 2.6 Summary.- 2.7 Exercises.- 3 Predicate LPF with Equality.- 3.1 Predicates.- 3.2 Types in predicates.- 3.3 Predicate calculus for LPF: proof strategies for quantifiers.- 3.4 Reasoning about equality: substitution and chains of equality.- 3.5 Extensions to typed predicate LPF with equality.- 3.6 Summary.- 3.7 Exercises.- 4 Basic Type Constructors.- 4.1 Introduction.- 4.2 Union types.- 4.3 Cartesian product types.- 4.4 Optional types.- 4.5 Subtypes.- 4.6 A note on composite types.- 4.7 Summary.- 4.8 Exercises.- 5 Numbers.- 5.1 Introduction.- 5.2 Axiomatising the natural numbers.- 5.3 Axiomatisation of addition and proof by induction.- 5.4 More on proof by induction.- 5.5 Using direct definitions.- 5.6 Summary.- 5.7 Exercises.- 6 Finite Sets.- 6.1 Introduction.- 6.2 Generators for sets set membership set induction.- 6.3 Proof using set induction.- 6.4 Quantification over sets.- 6.5 Subsets set equality cardinality.- 6.6 Other set constructors.- 6.7 Set comprehension.- 6.8 Reasoning about set comprehension.- 6.9 Summary.- 6.10 Exercises.- 7 Finite Maps.- 7.1 Introduction.- 7.2 Basic axiomatisation.- 7.3 Axiomatisation using generators.- 7.4 Extraction and abstraction of lemmas.- 7.5 Using subsidiary definitions.- 7.6 Polymorphic subtypes and associated induction rules.- 7.7 Map comprehension.- 7.8 Summary.- 7.9 Exercises.- 8 Finite Sequences.- 8.1 Introduction.- 8.2 Basic axiomatisation.- 8.3 Destructors.- 8.4 Equality between lists.- 8.5 Operators on lists.- 8.6 An alternative generator set.- 8.7 Summary.- 8.8 Exercises.- 9 Booleans.- 9.1 Introduction.- 9.2 Basic axiomatisation.- 9.3 Formation rules for boolean-valued operators.- 9.4 An example of a well-formedness proof obligation.- 9.5 Summary.- 9.6 Exercises.- II Proof in Practice.- 10 Proofs From Specifications.- 10.1 Introduction.- 10.2 Type definitions.- 10.3 The state.- 10.4 Functions and values.- 10.5 Operations.- 10.6 Validation proofs.- 10.7 Summary.- 10.8 Exercises.- 11 Verifying Reifications.- 11.1 Introduction.- 11.2 Data reification.- 11.3 Operation modelling.- 11.4 An example reification proof.- 11.5 Implementing functions.- 11.6 Implementation bias and unreachable states.- 11.7 Summary.- 11.8 Exercises.- 12 A Case Study in Air-Traffic Control.- 12.1 Introduction.- 12.2 The air-traffic control system.- 12.3 Formalisation of the state model.- 12.4 Top-level operations.- 12.5 First refinement step.- 12.6 Second refinement step.- 12.7 Concluding remarks.- 13 Advanced Topics.- 13.1 Introduction.- 13.2 Functions as a data type.- 13.3 Comparing elements of disjoint types.- 13.4 Recursive type definitions.- 13.5 Enumerated sets, maps and sequences.- 13.6 Patterns.- 13.7 Other expressions.- 13.8 Other types.- III Directory of Theorems.- 14 Directory of Theorems.- 14.1 Propositonal LPF.- 14.2 Predicate LPF with equality.- 14.3 Basic type constructors.- 14.4 Natural numbers.- 14.5 Finite sets.- 14.6 Finite maps.- 14.7 Finite sequences.- 14.8 Booleans.- 14.9 Specifications.- 14.10 Reifications.- 14.11 Case study I: abstract specification.- 14.12 Case study II: refinement.- Index of Symbols.- Index of Rules.
131 citations
23 Aug 1993
TL;DR: This paper documents the semantics of the design language by providing a mapping to the pi-calculus and describes the implications of this mapping for graph-oriented programming languages.
Abstract: Companion papers give examples of the development of concurrent programs using a design notation which employs a number of concepts from object-oriented programming languages. This paper documents the semantics of the design language by providing a mapping to the pi-calculus.
107 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 1 |
| 2018 | 1 |
| 2017 | 3 |
| 2015 | 7 |
| 2014 | 5 |