Euler Diagrams 2004 Preliminary Version A Survey of Reasoning Systems Based on Euler Diagrams Abstract
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paper_title: Constraint diagrams: visualizing invariants in object-oriented models

paper_content:

A new visual notation is proposed for precisely expressing constraints on object-oriented models, as an alternative to mathematical logic notation used in methods such as Syntropy and Catalysis. The notation is potentially intuitive, expressive, integrates well with existing visual notations, and has a clear and unambiguous semantics. It is reminiscent of informal diagrams used by mathematicians for illustrating relations, and borrows much from Venn diagrams. It may be viewed as a generalization of instance diagrams.
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paper_title: The Logical Status of Diagrams

paper_content:

Acknowledgements 1. Introduction 2. Preliminaries 3. Venn-I 4. Venn-II 5. Venn-II and L0 6. Diagrammatic versus linguistic representation 7. Conclusion Appendix References Index.
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paper_title: SD2: a sound and complete diagrammatic reasoning system

paper_content:

SD2 is a system of Venn-type diagrams that can be used to reason diagrammatically about sets, their cardinalities and their relationships. They augment the systems of Venn-Peirce diagrams investigated by Shin (1994) to include lower and upper bounds for the cardinalities of the sets represented by regions of diagrams. We summarise their syntax and semantics and introduce inference rules for reasoning with the system. We discuss the soundness of the system and develop a proof strategy for completeness simpler than that adopted by Shin. We expect this strategy to extend to other, richer spider diagram systems and to constraint diagrams, the visual notation that has been used in conjunction with object-oriented modelling notations such as the Unified Modelling Language.
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paper_title: Logic and Visual Information

paper_content:

From the Publisher: ::: The importance of visual information is clear from its frequent presence in everyday reasoning and communication, and also in computation. This book examines the logical foundations of visual information presented in the form of diagrams, graphs, charts, tables and maps.
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paper_title: Implementing Euler/Venn Reasoning Systems

paper_content:

This paper proposes an implementation of a Euler/Venn reasoning system using directed acyclic graphs and shows that this implementation is correct with respect to a modified Shin/Hammer mathematical model of Euler/Venn Reasoning. In proving its correctness it will also be shown that the proposed implementation preserves or inherits the soundness and completeness properties of the mathematical model of the Euler/Venn system.
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paper_title: Logic and Visual Information

paper_content:

From the Publisher: ::: The importance of visual information is clear from its frequent presence in everyday reasoning and communication, and also in computation. This book examines the logical foundations of visual information presented in the form of diagrams, graphs, charts, tables and maps.
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paper_title: The Logical Status of Diagrams

paper_content:

Acknowledgements 1. Introduction 2. Preliminaries 3. Venn-I 4. Venn-II 5. Venn-II and L0 6. Diagrammatic versus linguistic representation 7. Conclusion Appendix References Index.
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paper_title: The information content of Euler/Venn diagrams

paper_content:

An ignition system for internal combustion engines. It generates a controlled-duration continuous-wave high-frequency spark, and employs an output transformer in an oscillator which includes a control winding for starting and stopping the oscillator. There is an electronic switch in series with the control winding; and the spark intervals, including duration thereof, are determined by photoelectric engine-timed means that employ a phototransistor. There is a control circuit for the electronic switch, which circuit includes means for minimizing the response time of the phototransistor.
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paper_title: Towards a model theory of Venn diagrams

paper_content:

In a regenerative type gas turbine application, a mounting apparatus for a recuperator to support the recuperator between the relatively stationary end walls of a gas turbine housing while simultaneously permitting movement of the recuperator in plural directions with respect to the walls caused by thermal expansion.
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paper_title: Modeling Heterogeneous Systems

paper_content:

Reasoning practices and decision making often require information from many different sources, which can be both sentential and diagrammatic. In such situations, there are many advantages to reasoning with the diagrams themselves, as opposed to re-expressing the information content of the diagram in sentential form and reasoning in an abstract sentential language. Thus for these practices, being able to extract and re-express pieces of information from one kind of representation into another is essential. The main goal of this paper is to propose a general framework for the modeling of heterogeneous reasoning systems and, most importantly, heterogeneous rules of inference in those systems. Unlike some other work in designing heterogeneous systems, our purpose will not be to define just one notion of heterogeneous inference, but rather to provide a framework in which many different kinds of heterogeneous rules of inference can be defined. After proposing this framework, we will then show how it can be applied to a sample heterogeneous system to define a number of different heterogeneous rules of inference. We will also discuss how the framework can be used to define rules of inference similar to the Observe Rule in Barwise and Etchemendy's Hyperproof system.
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paper_title: Using DAG Transformations to Verify Euler/Venn Homogeneous and Euler/Venn FOL Heterogeneous Rules of Inference

paper_content:

Abstract In this paper we will present a graph-transformation based method for the verification of heterogeneous first order logic (FOL) and Euler/Venn proofs. It has been shown that a special collection of directed acyclic graphs (DAGs) can be used interchangeably with Euler/Venn diagrams in reasoning processes [4]. Thus, proofs which include Euler/Venn diagrams can be thought of as proofs with DAGs where steps involving only Euler/Venn diagrams can be treated as particular DAG transformations. In the work reported here, we will show how the characterization of these manipulations can be used to verify Euler/Venn proofs. Also, a method for verifying the use of heterogeneous Euler/Venn and FOL reasoning rules will be presented that is also based upon DAG transformations.
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paper_title: The Logical Status of Diagrams

paper_content:

Acknowledgements 1. Introduction 2. Preliminaries 3. Venn-I 4. Venn-II 5. Venn-II and L0 6. Diagrammatic versus linguistic representation 7. Conclusion Appendix References Index.
---
paper_title: Logic and Visual Information

paper_content:

From the Publisher: ::: The importance of visual information is clear from its frequent presence in everyday reasoning and communication, and also in computation. This book examines the logical foundations of visual information presented in the form of diagrams, graphs, charts, tables and maps.
---
paper_title: Implementing Euler/Venn Reasoning Systems

paper_content:

This paper proposes an implementation of a Euler/Venn reasoning system using directed acyclic graphs and shows that this implementation is correct with respect to a modified Shin/Hammer mathematical model of Euler/Venn Reasoning. In proving its correctness it will also be shown that the proposed implementation preserves or inherits the soundness and completeness properties of the mathematical model of the Euler/Venn system.
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paper_title: A case study of the design and implementation of heterogeneous reasoning systems

paper_content:

In recent years we have witnessed a growing interest in heterogeneous reasoning systems. A heterogeneous reasoning system incorporates representations from a number of different representation systems, in our case a sentential and a diagrammatic system. The advantage of heterogeneous systems is that they allow a reasoner to bridge the gaps among various formalisms and construct threads of proof which cross the boundaries of the systems of representation. In doing this, these heterogeneous systems allow the reasoner to take advantage of each component system’s ability to express information in that component’s area of expertise. The purpose of this paper is twofold: to propose a general theoretical framework, inspired by Barwise and Seligman’s work in Information Theory [Barwise and Seligman, 1997], for the design of heterogeneous reasoning systems and to use this framework as the basis of an implementation of a First Order Logic and Euler/Venn reasoning system.
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paper_title: Reasoning with Extended Venn-Peirce Diagrammatic Systems

paper_content:

A method for making molten metal for casting directly from cold starting material. Cold starting material is placed in a movable vessel, and a layer of particulate material selected from the group consisting of coke and flux is spread on the top of the starting material in the vessel. One or more flames of hydrocarbon fuel-oxygen mixture are directed to the particulate layer, while translating the vessel along a closed path, so as to heat the particulate to white-hot state and to mix the cold starting material with the particulate thus heated, for melting the starting material. A composition-control agent is added in the material thus melted for producing desired composition of molten metal, while maintaining the flames and the translation. A fluid curtain is formed around the flames to inhibit the flames from directly burning the inner surface of the vessel.
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paper_title: On the Completeness and Expressiveness of Spider Diagram Systems

paper_content:

Spider diagram systems provide a visual language that extends the popular and intuitive Venn diagrams and Euler circles. Designed to complement object-oriented modelling notations in the specification of large software systems they can be used to reason diagrammatically about sets, their cardinalities and their relationships with other sets. A set of reasoning rules for a spider diagram system is shown to be sound and complete. We discuss the extension of this result to diagrammatically richer notations and also consider their expressiveness. Finally, we show that for a rich enough system we can diagrammatically express the negation of any diagram.
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paper_title: Spider diagrams: A diagrammatic reasoning system

paper_content:

Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can be used in conjunction with object-oriented modelling notations such as the Unified Modelling Language. This paper summarises the main syntax and semantics of spider diagrams. It also introduces inference rules for reasoning with spider diagrams and a rule for combining spider diagrams. This system is shown to be sound but not complete. Disjunctive diagrams are considered as one way of enriching the system to allow combination of diagrams so that no semantic information is lost. The relationship of this system of spider diagrams to other similar systems, which are known to be sound and complete, is explored briefly.
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paper_title: Reasoning with spider diagrams

paper_content:

Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can usefully be used in conjunction with object-oriented modelling notations such as the Unified Modelling Language (UML). This paper summarises the main syntax and semantics of spider diagrams and introduces four inference rules for reasoning with spider diagrams and a rule governing the equivalence of the Venn and Euler forms of spider diagrams. This paper also details rules for combining two spider diagrams to produce a single diagram which retains as much of their combined semantic information as possible, and discusses disjunctive diagrams as one possible way of enriching the system in order to combine spider diagrams so that no semantic information is lost.
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paper_title: Reasoning with Extended Venn-Peirce Diagrammatic Systems

paper_content:

A method for making molten metal for casting directly from cold starting material. Cold starting material is placed in a movable vessel, and a layer of particulate material selected from the group consisting of coke and flux is spread on the top of the starting material in the vessel. One or more flames of hydrocarbon fuel-oxygen mixture are directed to the particulate layer, while translating the vessel along a closed path, so as to heat the particulate to white-hot state and to mix the cold starting material with the particulate thus heated, for melting the starting material. A composition-control agent is added in the material thus melted for producing desired composition of molten metal, while maintaining the flames and the translation. A fluid curtain is formed around the flames to inhibit the flames from directly burning the inner surface of the vessel.
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paper_title: On Diagram Tokens and Types

paper_content:

Rejecting the temptation to make up a list of necessary and sufficient conditions for diagrammatic and sentential systems, we present an important distinction which arises from sentential and diagrammatic features of systems. Importantly, the distinction we will explore in the paper lies at a meta-level. That is, we argue for a major difference in meta-theory between diagrammatic and sentential systems, by showing the necessity of a more fine-grained syntax for a diagrammatic system than for a sentential system. Unlike with sentential systems, a diagrammatic system requires two levels of syntax--token and type. Token-syntax is about particular diagrams instantiated on some physical medium, and type-syntax provides a formal definition with which a concrete representtation of a diagram must comply. While these two levels of syntax are closely related, the domains of type-syntax and token-syntax are distinct from each other. Euler diagrams are chosen as a case study to illustrate the following major points of the paper: (i) What kinds of diagrammatic features (as opposed to sentential features) require two different levels of syntax? (ii) What is the relation between these two levels of syntax? (iii) What is the advantage of having a two-tiered syntax?
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paper_title: Using DAG Transformations to Verify Euler/Venn Homogeneous and Euler/Venn FOL Heterogeneous Rules of Inference

paper_content:

Abstract In this paper we will present a graph-transformation based method for the verification of heterogeneous first order logic (FOL) and Euler/Venn proofs. It has been shown that a special collection of directed acyclic graphs (DAGs) can be used interchangeably with Euler/Venn diagrams in reasoning processes [4]. Thus, proofs which include Euler/Venn diagrams can be thought of as proofs with DAGs where steps involving only Euler/Venn diagrams can be treated as particular DAG transformations. In the work reported here, we will show how the characterization of these manipulations can be used to verify Euler/Venn proofs. Also, a method for verifying the use of heterogeneous Euler/Venn and FOL reasoning rules will be presented that is also based upon DAG transformations.
---
paper_title: Reasoning with Extended Venn-Peirce Diagrammatic Systems

paper_content:

A method for making molten metal for casting directly from cold starting material. Cold starting material is placed in a movable vessel, and a layer of particulate material selected from the group consisting of coke and flux is spread on the top of the starting material in the vessel. One or more flames of hydrocarbon fuel-oxygen mixture are directed to the particulate layer, while translating the vessel along a closed path, so as to heat the particulate to white-hot state and to mix the cold starting material with the particulate thus heated, for melting the starting material. A composition-control agent is added in the material thus melted for producing desired composition of molten metal, while maintaining the flames and the translation. A fluid curtain is formed around the flames to inhibit the flames from directly burning the inner surface of the vessel.
---
paper_title: Reasoning with Extended Venn-Peirce Diagrammatic Systems

paper_content:

A method for making molten metal for casting directly from cold starting material. Cold starting material is placed in a movable vessel, and a layer of particulate material selected from the group consisting of coke and flux is spread on the top of the starting material in the vessel. One or more flames of hydrocarbon fuel-oxygen mixture are directed to the particulate layer, while translating the vessel along a closed path, so as to heat the particulate to white-hot state and to mix the cold starting material with the particulate thus heated, for melting the starting material. A composition-control agent is added in the material thus melted for producing desired composition of molten metal, while maintaining the flames and the translation. A fluid curtain is formed around the flames to inhibit the flames from directly burning the inner surface of the vessel.
---
paper_title: Reasoning with Extended Venn-Peirce Diagrammatic Systems

paper_content:

A method for making molten metal for casting directly from cold starting material. Cold starting material is placed in a movable vessel, and a layer of particulate material selected from the group consisting of coke and flux is spread on the top of the starting material in the vessel. One or more flames of hydrocarbon fuel-oxygen mixture are directed to the particulate layer, while translating the vessel along a closed path, so as to heat the particulate to white-hot state and to mix the cold starting material with the particulate thus heated, for melting the starting material. A composition-control agent is added in the material thus melted for producing desired composition of molten metal, while maintaining the flames and the translation. A fluid curtain is formed around the flames to inhibit the flames from directly burning the inner surface of the vessel.
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paper_title: What can spider diagrams say

paper_content:

Spider diagrams are a visual notation for expressing logical statements. In this paper we identify a well known fragment of first order predicate logic, that we call \(\mathcal {ESD}\), equivalent in expressive power to the spider diagram language. The language \(\mathcal {ESD}\) is monadic and includes equality but has no constants or function symbols. To show this equivalence, in one direction, for each diagram we construct a sentence in \(\mathcal {ESD}\) that expresses the same information. For the more challenging converse we show there exists a finite set of models for a sentence S that can be used to classify all the models for S. Using these classifying models we show that there is a diagram expressing the same information as S.
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paper_title: The Expressiveness of Spider Diagrams Augmented with Constants

paper_content:

Spider diagrams are a visual language for expressing logical statements. Spiders represent the existence of elements and contours denote sets. Several sound and complete spider diagram systems have been developed and it is known that the spider diagram language is equivalent in expressive power to monadic first order logic with equality. However, these sound and complete spider diagram systems do not contain syntactic elements analogous to constants in first order predicate logic. We extend the spider diagram language to include constant spiders which represent specific individuals and give formal semantics for the extended diagram language. We then prove that this extended system is equivalent in expressive power to the language of spider diagrams without constants
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paper_title: Projections in Venn-Euler diagrams

paper_content:

Venn diagrams and Euler circles have long been used to express constraints on sets and their relationships with other sets. However, these notations can get very cluttered when we consider many closed curves or contours. In order to reduce this clutter, and to focus attention within the diagram appropriately, the notion of a projected contour, or projection, is introduced. Informally, a projected contour is a contour that describes a set of elements limited to a certain context. Through a series of examples, we develop a formal semantics of projections and discuss the issues involved in introducing these.
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paper_title: Towards a formalization of constraint diagrams

paper_content:

Geared to complement UML and to the specification of large software systems by non-mathematicians, constraint diagrams are a visual language that generalizes the popular and intuitive Venn diagrams and Euler circles, and adds facilities for quantifying over elements and navigating relations. The language design emphasizes scalability and expressiveness while retaining intuitiveness. Spider diagrams form a subset of the notation, leaving out universal quantification and the ability to navigate relations. Spider diagrams have been given a formal definition. This paper extends that definition to encompass the constraint diagram notation. The formalization of constraint diagrams is nontrivial: it exposes subtleties concerned with the implicit ordering of symbols in the visual language, which were not evident before a formal definition of the language was attempted. This has led to an improved design of the language.
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paper_title: Drawing graphs in Euler diagrams

paper_content:

We describe a method for drawing graph-enhanced Euler diagrams using a three stage method. The first stage is to lay out the underlying Euler diagram using a multicriteria optimizing system. The second stage is to find suitable locations for nodes in the zones of the Euler diagram using a force based method. The third stage is to minimize edge crossings and total edge length by swapping the location of nodes that are in the same zone with a multicriteria hill climbing method. We show a working version of the software that draws spider diagrams. Spider diagrams represent logical expressions by superimposing graphs upon an Euler diagram. This application requires an extra step in the drawing process because the embedded graphs only convey information about the connectedness of nodes and so a spanning tree must be chosen for each maximally connected component. Similar notations to Euler diagrams enhanced with graphs are common in many applications and our method is generalizable to drawing Hypergraphs represented in the subset standard, or to drawing Higraphs where edges are restricted to connecting with only atomic nodes.
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paper_title: Dynamic Euler Diagram Drawing

paper_content:

In this paper we describe a method to lay out a graph enhanced Euler diagram so that it looks similar to a previously drawn graph enhanced Euler diagram. This task is nontrivial when the underlying structures of the diagrams differ. In particular, if a structural change is made to an existing drawn diagram, our work enables the presentation of the new diagram with minor disruption to the user's mental map. As the new diagram can be generated from an abstract representation, its initial embedding may be very different from that of the original. We have developed comparison measures for Euler diagrams, integrated into a multicriteria optimizer, and applied a force model for associated graphs that attempts to move nodes towards their positions in the original layout. To further enhance the usability of the system, the transition between diagrams can be animated
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paper_title: A reading algorithm for constraint diagrams

paper_content:

Constraint diagrams are a visual notation designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantification and navigation of relations. Their design emphasizes scalability and expressiveness while retaining intuitiveness. The formalization of constraint diagrams is non-trivial: previous attempts have exposed subtleties concerned with the ordering of symbols in the visual language. Consequently, some constraint diagrams have more than one intuitive reading. We develop the concept of the dependence graph for a constraint diagram. From the dependence graph, we obtain a set of reading trees. A reading tree provides a partial ordering for some syntactic elements of the diagram. Given a reading tree for a constraint diagram, we present an algorithm that delivers a unique semantic reading.
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paper_title: Towards a default reading for constraint diagrams

paper_content:

Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They were designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantification and navigation of relations. Due to the lack of a linear ordering of symbols inherent in a diagrammatic language which expresses logical statements, some constraint diagrams have more than one intuitive meaning. We generalize, from an example based approach, to suggest a default reading for constraint diagrams. This reading is usually unique, but may require a small number of simple user choices.
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paper_title: Reasoning with Projected Contours

paper_content:

Projected contours enable Euler diagrams to scale better. They enable the representation of information using less syntax and can therefore increase visual clarity. Here informal reasoning rules are given that allow the transformation of spider diagrams with respect to projected contours.
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paper_title: Reasoning with spider diagrams

paper_content:

Spider diagrams combine and extend Venn diagrams and Euler circles to express constraints on sets and their relationships with other sets. These diagrams can usefully be used in conjunction with object-oriented modelling notations such as the Unified Modelling Language (UML). This paper summarises the main syntax and semantics of spider diagrams and introduces four inference rules for reasoning with spider diagrams and a rule governing the equivalence of the Venn and Euler forms of spider diagrams. This paper also details rules for combining two spider diagrams to produce a single diagram which retains as much of their combined semantic information as possible, and discusses disjunctive diagrams as one possible way of enriching the system in order to combine spider diagrams so that no semantic information is lost.
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paper_title: Computing Reading Trees for Constraint Diagrams

paper_content:

Constraint diagrams are a visual notation designed to complement the Unified Modeling Language in the development of software systems. They generalize Venn diagrams and Euler circles, and include facilities for quantification and navigation of relations. Their design emphasizes scalability and expressiveness while retaining intuitiveness. Due to subtleties concerned with the ordering of symbols in this visual language, the formalization of constraint diagrams is non-trivial; some constraint diagrams have more than one intuitive reading. A ‘reading’ algorithm, which associates a unique semantic interpretation to a constraint diagram, with respect to a reading tree, has been developed. A reading tree provides a partial ordering for syntactic elements of the diagram. Reading trees are obtainable from a partially directed graph, called the dependence graph of the diagram. In this paper we describe a ‘tree-construction’ algorithm, which utilizes graph transformations in order to produce all possible reading trees from a dependence graph. This work will aid the production of tools which will allow an advanced user to choose from a range of semantic interpretations of a diagram.
---
paper_title: Constraint diagrams: visualizing invariants in object-oriented models

paper_content:

A new visual notation is proposed for precisely expressing constraints on object-oriented models, as an alternative to mathematical logic notation used in methods such as Syntropy and Catalysis. The notation is potentially intuitive, expressive, integrates well with existing visual notations, and has a clear and unambiguous semantics. It is reminiscent of informal diagrams used by mathematicians for illustrating relations, and borrows much from Venn diagrams. It may be viewed as a generalization of instance diagrams.
---
paper_title: Generating Readable Proofs: A Heuristic Approach to Theorem Proving With Spider Diagrams

paper_content:

An important aim of diagrammatic reasoning is to make it easier for people to create and understand logical arguments. We have worked on spider diagrams, which visually express logical statements. Ideally, automatically generated proofs should be short and easy to understand. An existing proof generator for spider diagrams successfully writes proofs, but they can be long and unwieldy. In this paper, we present a new approach to proof writing in diagrammatic systems, which is guaranteed to find shortest proofs and can be extended to incorporate other readability criteria. We apply the A * algorithm and develop an admissible heuristic function to guide automatic proof construction. We demonstrate the effectiveness of the heuristic used. The work has been implemented as part of a spider diagram reasoning tool.
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paper_title: Layout metrics for Euler diagrams

paper_content:

We present an aesthetics based method for drawing Euler diagrams. Aesthetic layout metrics have been found to be useful in graph drawing algorithms, which use metrics motivated by aesthetic principles that aid user understanding of diagrams. We have taken a similar approach to Euler diagram drawing, and have defined a set of suitable metrics to be used within a hill climbing multicriteria optimiser to produce "good" drawings. There are added difficulties when drawing Euler diagrams as they are made up of contours whose structural properties of intersection and containment must be preserved under any layout improvements. We describe our Java implementation of a pair of hill climbing variants to find good drawings, a set of metrics that measure aesthetics for good diagram layout, and issues concerning the choice of weightings for a useful combination of the metrics.
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paper_title: Constraint diagrams: A step beyond UML

paper_content:

The Unified Modeling Language (UML) is a set of notations for modelling object-oriented systems. It has become the de facto standard. Most of its notations are diagrammatic. An exception to this is the Object Constraint Language (OCL) which is essentially a textual, stylised form of first order predicate logic. We describe a notation, constraint diagrams, which were introduced as a visual technique intended to be used in conjunction with the UML for object-oriented modelling. Constraint diagrams provide a diagrammatic notation for expressing constraints (e.g., invariants) that could only be expressed in UML using OCL.
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paper_title: Generating proofs with spider diagrams using heuristics

paper_content:

We apply the A¤ algorithm to guide a diagrammatic theorem proving tool. The algorithm requires a heuristic function, which provides a metric on the search space. In this paper we present a collection of metrics between two spider diagrams. We combine these metrics to give a heuristic function that provides a lower bound on the length of a shortest proof from one spider diagram to another, using a collection of sound reasoning rules. We compare the effectiveness of our approach with a breadth- first search for proofs.
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paper_title: Generating Euler Diagrams

paper_content:

This article describes an algorithm for the automated generation of any Euler diagram starting with an abstract description of the diagram. An automated generation mechanism for Euler diagrams forms the foundations of a generation algorithm for notations such as Harel's higraphs, constraint diagrams and some of the UML notation. An algorithm to generate diagrams is an essential component of a diagram tool for users to generate, edit and reason with diagrams.The work makes use of properties of the dual graph of an abstract diagram to identify which abstract diagrams are "drawable" within given wellformedness rules on concrete diagrams. A Java program has been written to implement the algorithm and sample output is included.
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paper_title: A constraint diagram reasoning system

paper_content:

The Unified Modeling Language (UML) is a collection of notations which are mainly diagrammatic. These notations are used by software engineers in the process of object oriented modelling. The only textual notation in the UML is the Object Constraint Language (OCL). The OCL is used to express logical constraints such as system invariants. Constraint diagrams are designed to provide a diagrammatic alternative to the OCL. Since constraint diagrams are visual they complement existing notations in the UML. Spider diagrams form the basis of constraint diagrams and sound and complete reasoning systems have been developed. Spider diagrams allow subset relations between sets and cardinality constraints on sets to be expressed. In addition to this, constraint diagrams allow universal quantification and relational navigation and hence are vastly more expressive. In this paper we present the first constraint diagram reasoning system. We give syntax and semantics for constraint diagrams we call CD1 diagrams. We identify syntactic criteria that allow us to determine whether a CD1 diagram is satisfiable. We give descriptions of a set of sound and complete reasoning rules for CD1 diagrams.
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Title: Euler Diagrams 2004 Preliminary Version: A Survey of Reasoning Systems Based on Euler Diagrams

Section 1: Introduction
Description 1: Introduce Euler diagrams, their semantics, limitations, and the purpose of the survey.

Section 2: Reasoning with Euler Diagrams
Description 2: Describe Hammer’s Euler diagram reasoning system and its reasoning rules.

Section 3: Venn Diagrams
Description 3: Outline the features of Venn diagrams and their differences from Euler diagrams.

Section 4: Venn-Peirce Diagrams
Description 4: Discuss modifications by Peirce to Venn diagrams to increase expressiveness.

Section 5: Venn-I Diagrams
Description 5: Explain Shin’s Venn-I diagram system and its reasoning rules.

Section 6: Venn-II Diagrams
Description 6: Describe Shin’s Venn-II diagrams, their rules, and expressive power.

Section 7: Euler/Venn Diagrams
Description 7: Outline Swoboda and Allwein's Euler/Venn diagram system and its features.

Section 8: Spider Diagrams
Description 8: Explain the development, features, and reasoning systems of spider diagrams.

Section 9: SD1 Diagrams
Description 9: Provide details on the SD1 spider diagram system, its syntax, and reasoning rules.

Section 10: SD2 Diagrams
Description 10: Discuss the SD2 system that extends SD1 with the ability to express upper bounds on cardinalities.

Section 11: ESD2 Diagrams
Description 11: Outline the ESD2 system that includes Euler-based diagrams and additional syntax.

Section 12: Further Spider Diagram Systems
Description 12: Explain additional spider diagram systems, including SD3, that further extend the capabilities of SD2.

Section 13: Constraint Diagrams
Description 13: Describe constraint diagrams, their expressiveness, and their application in software engineering.