## The Advent of Formal Diagrammatic Reasoning Systems

### BibTeX

@MISC{Dau_theadvent,

author = {Frithjof Dau},

title = {The Advent of Formal Diagrammatic Reasoning Systems},

year = {}

}

### OpenURL

### Abstract

In knowledge representation and reasoning systems, diagrams have many practical applications and are used in numerous settings. Indeed, it is widely accepted that diagrams are a valuable aid to intuition and help to convey ideas and information in a clear way. On the other side, logicians have viewed diagrams as informal tools, but which cannot be used in the manner of formal argumentation. Instead, logicians focused on symbolic representations of logics. Recently, this perception was overturned in the mid 1990s, first with seminal work by Shin on an extended version of Venn diagrams. Since then, certainly a growth in the research field of formal reasoning with diagrams can be witnessed. This paper discusses the evolution of formal diagrammatic logics, focusing on those systems which are based on Euler and Venn-Peirce diagrams, and Peirces existential graphs. Also discussed are some challenges faced in the area, some of which are specifically related to diagrams.

### Citations

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Citation Context ...er architecture, and diagrammatic logic systems have a two layer architecture. The general advantages of this two layer structure for the pragmatics of diagrams is already discussed to a large extent =-=[49, 66, 6, 68, 57]-=-. It is usually argued that the additional diagrammatic layer, often referred to as ‘secondary notation’, provides the essential means to improve the pragmatics of a representation system. As Oberland... |

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Citation Context ...clusions. Again, an important aspect of contextual logic is that its core notions, i.e. both judgments and conclusions, can be diagrammatically represented (for this purpose, Sowa’s conceptual graphs =-=[70]-=- are utilized). It is widely accepted that diagrams play an important role for representing information in accessible and intuitive ways. In mathematics, however, there does still exist a long-standin... |

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Citation Context ...tend this approach to more expressive DLs. 3 Historical Systems In this section, we shortly discuss the historical background of DRSs, namely Euler circles (a.k.a. Euler diagrams) [24], Venn diagrams =-=[77]-=-, Venn-Peirce diagrams [59] and Peirce’s existential graphs [33, 60, 31].An Euler diagram is a finite collection of closed curves drawn in the plane. For example, in Fig 3, d1 is an Euler diagram whi... |

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Citation Context ...s Spider diagrams combine features of Venn diagrams and the more user friendly Euler diagrams. They can be thought of as extending Venn-II diagrams. Various different systems exist today, for example =-=[35, 36, 38,40, 74]-=-. In Fig.8, two examples of so-called unary SDs are depicted. A B A B C Fig. 8. Two spider diagrams The left diagram contains two existential spiders. Each spider denotes a uniquely given object (i.e.... |

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Citation Context ...less, for untrained users, the symbolic notation of DLs can be hard to learn and comprehend. A main alternative to the symbolic notation is the development of a diagrammatic representation of DLs. In =-=[55]-=-, the introduction to the Description Logic Handbook, Nardi and Brachman write a “major alternative for increasing the usability of Description Logics as a modeling language” is to “implement interfac... |

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Citation Context ...son, developing heterogeneous or hybrid notations that incorporate both symbolic and diagrammatic parts is a promising approach. There has already some research been conducted in this direction, e.g. =-=[30, 76]-=-. In fact, when a formal language is developed with the goal of a high usability, the design of the language depends on the requirements of intended application areas. This might render finding the ri... |

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Citation Context ...the essential foundations for the acceptance of diagrams as formal tools. 4.2 Seminal Work on Existential Graphs The main treatises on existential graphs are probably the books of Zeman [80], Roberts =-=[65]-=- and Shin [68]). Each of this books focuses on different aspects of existential graphs. Probably the most prominent book on existential graphs is D. Robert’s ‘The Existential Graphs of Charles S. Peir... |

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Citation Context ...ily understandable diagrammatic parts of UML. Thus it is reasonable to develop a diagrammatic variant of OCL. A groundbreaking approach towards this aim has been undertaken by Kent, who introduced in =-=[41]-=- the class of constraint diagrams. An 1 http://www.uml.org 2 http://www.bpmn.orgexample for such a diagram is provided in Fig. 2. This diagrams expresses the same constraint as its symbolic counterpa... |

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Citation Context ... semantics, and diagrammatic reasoning facilities. 2.2 Knowledge Representation and the Semantic Web Is has long been argued that diagrams are particularly useful for knowledge representation systems =-=[56, 28, 48]-=-. In this section, we focus on knowledge representation within the Semantic Web, particularly on RDF(S) and OWL. The underlying layer for knowledge representation within the Semantic Web is the Resour... |

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Citation Context ...s Spider diagrams combine features of Venn diagrams and the more user friendly Euler diagrams. They can be thought of as extending Venn-II diagrams. Various different systems exist today, for example =-=[35, 36, 38,40, 74]-=-. In Fig.8, two examples of so-called unary SDs are depicted. A B A B C Fig. 8. Two spider diagrams The left diagram contains two existential spiders. Each spider denotes a uniquely given object (i.e.... |

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Citation Context ...is in progress to extend this approach to more expressive DLs. 3 Historical Systems In this section, we shortly discuss the historical background of DRSs, namely Euler circles (a.k.a. Euler diagrams) =-=[24]-=-, Venn diagrams [77], Venn-Peirce diagrams [59] and Peirce’s existential graphs [33, 60, 31].An Euler diagram is a finite collection of closed curves drawn in the plane. For example, in Fig 3, d1 is ... |

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Citation Context ...am to another. With regard to applications, SDs have been used to detect component failures in safety critical hardware [12] and (slight variations of them) to represent non-hierarchical file systems =-=[21]-=- and for viewing clusters which contain concepts from multiple ontologies [32]. As mentioned in section 2, one area that could benefit from the development of diagrammatic logics is software engineeri... |

28 |
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Citation Context ...difficult, as there is no natural order placed on the quantifiers. This difficulty is overcome by augmenting each diagram with a reading tree; the formalization of constraint diagrams can be found in =-=[25]-=-. An example can be seen in Fig. 10. This diagram expresses that Book, Title and Author are pairwise disjoint, Fiction and NonFiction form a partition of Book, and finally every book x has a unique na... |

28 |
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(Show Context)
Citation Context ...foundations for the acceptance of diagrams as formal tools. 4.2 Seminal Work on Existential Graphs The main treatises on existential graphs are probably the books of Zeman [80], Roberts [65] and Shin =-=[68]-=-). Each of this books focuses on different aspects of existential graphs. Probably the most prominent book on existential graphs is D. Robert’s ‘The Existential Graphs of Charles S. Peirce’. This book... |

27 | P.: Visual Modeling of OWL DL Ontologies using UML
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Citation Context ...to representations within the framework of UML (Unified Modeling Language). In 2003, the Object Management Group requested a metamodel for the purpose of defining ontologies. Following this proposal, =-=[7]-=- provides a UML-based, diagrammatic representation for OWL DL. In these approaches, the focus is on a graphical representation of DL, however, as often emphasized, reasoning is seen as a distinguishin... |

25 |
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Citation Context ... semantics, and diagrammatic reasoning facilities. 2.2 Knowledge Representation and the Semantic Web Is has long been argued that diagrams are particularly useful for knowledge representation systems =-=[56, 28, 48]-=-. In this section, we focus on knowledge representation within the Semantic Web, particularly on RDF(S) and OWL. The underlying layer for knowledge representation within the Semantic Web is the Resour... |

25 |
Reasoning and the Logic of Things
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- 1898
(Show Context)
Citation Context ...In this section, we shortly discuss the historical background of DRSs, namely Euler circles (a.k.a. Euler diagrams) [24], Venn diagrams [77], Venn-Peirce diagrams [59] and Peirce’s existential graphs =-=[33, 60, 31]-=-.An Euler diagram is a finite collection of closed curves drawn in the plane. For example, in Fig 3, d1 is an Euler diagram which expresses that nothing is both a car and a van. Venn diagrams differ ... |

24 |
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Citation Context ...OL or via direct model-theoretic approach) and reasoning facilities are provided. Examples are the works of Prediger and Dau, where the reasoning facilities come in form of graph transformation rules =-=[61, 62, 9, 52,15]-=-, works of Chein/Mugnier et al, where the entailment between CGs are described by meaning-preserving graph homomorphisms called projections [8, 52, 11], or works where so-called standard-models are co... |

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(Show Context)
Citation Context ...where the semantics for the graphs is based on triadic Formal Concept Analysis. Prediger provides a sound and complete set of rules for these graphs. A different approach has been taken by Simonet in =-=[69, 10]-=- by translating the graphs to FOL-formulas, where the contexts are modeled by assigning to each concept box an additional argument which models the nesting of the box. In this approach, the notion of ... |

23 |
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(Show Context)
Citation Context ...grams, the Visual Modelling Group has developed a diagrammatic theorem prover called Edith. 7 Edith automates the search for Euler diagram proofs using a variety of reasoning rule sets and heuristics =-=[71]-=-. For a given rule set, Edith finds a shortest proof. In Edith, users can create diagrams, apply rules to write a proof and ask Edith to seek a proof from one diagram to another. With regard to applic... |

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(Show Context)
Citation Context ...nent failures in safety critical hardware [12] and (slight variations of them) to represent non-hierarchical file systems [21] and for viewing clusters which contain concepts from multiple ontologies =-=[32]-=-. As mentioned in section 2, one area that could benefit from the development of diagrammatic logics is software engineering. Constraint diagrams were designed with this application area in mind and e... |

22 | Visualization of Formal Specifications
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(Show Context)
Citation Context ...agrams were designed with this application area in mind and extend the spider diagram language. Constraint diagrams have been used in a variety of areas including formal object oriented specification =-=[37, 44]-=- and a visual semantic web editing environment [51, 81]. Prototype tools to support their use are available from [64]. 6.2 Conceptual Graphs John Sowa’s conceptual graphs (CGs) are based on Peirce’s e... |

21 |
The Logic System of Concept Graphs with Negations: And its Relationship to Prediacte Logic
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- 2003
(Show Context)
Citation Context ...can consider CGs with full negation, where whole subgraphs can be negated. Kerdiles considers in [43] full negation as well. A comprehensive approach to add full negation to CGs is provided by Dau in =-=[13, 14, 16]-=-. In [13], Dau argues that to express full negation to CGs, a new syntactical entity to express negation has to be added, and he suggests to use the negation ovals of Peirce’s existential graphs. Thes... |

20 |
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(Show Context)
Citation Context ...son, developing heterogeneous or hybrid notations that incorporate both symbolic and diagrammatic parts is a promising approach. There has already some research been conducted in this direction, e.g. =-=[30, 76]-=-. In fact, when a formal language is developed with the goal of a high usability, the design of the language depends on the requirements of intended application areas. This might render finding the ri... |

19 | Extensions of simple conceptual graphs: the complexity of rules and constraints
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(Show Context)
Citation Context ...omorphisms called projections [8, 52, 11], or works where so-called standard-models are considered [61, 62, 15]. One can even express “if-then”-statements with simple CGs, at is has been discussed in =-=[2, 52, 3]-=-. Simple CGs with contexts go beyond FOL. For these graphs, there exist different formalizations as well. We want to mention the work of Prediger [61, 63], where the semantics for the graphs is based ... |

19 |
RDF entailment as a graph homomorphism
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Citation Context ...ns.” For RDF, mathematical elaborations based on graph theory have been developed. They include Tarski-style semantics as well as sound and complete calculi, latter either based on “projections” (see =-=[4, 5]-=-) or on diagrammatic rules (see [19]). That is, RDF is developed as a fully-fledged diagrammatic logic. A first attempt at a diagrammatic representation for DL is can be found in [28], where Gaines el... |

17 | On Diagram Tokens and Types
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(Show Context)
Citation Context ...s which are represented in the diagrams, and the diagrams should be understood as (mere) representations of these structures. Two thorough discussions on the type-token-issue for DRSs can be found in =-=[34, 17]-=-. Both papers argue that it is essential to provide mathematical definitions for the type-level (called abstract syntax in [34]). For the token-level (called concrete syntax in [34]), the papers come ... |

17 | On the completeness and expressiveness of spider diagram systems
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(Show Context)
Citation Context ...s Spider diagrams combine features of Venn diagrams and the more user friendly Euler diagrams. They can be thought of as extending Venn-II diagrams. Various different systems exist today, for example =-=[35, 36, 38,40, 74]-=-. In Fig.8, two examples of so-called unary SDs are depicted. A B A B C Fig. 8. Two spider diagrams The left diagram contains two existential spiders. Each spider denotes a uniquely given object (i.e.... |

16 | On the Efficacy of Representation
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(Show Context)
Citation Context ...er architecture, and diagrammatic logic systems have a two layer architecture. The general advantages of this two layer structure for the pragmatics of diagrams is already discussed to a large extent =-=[49, 66, 6, 68, 57]-=-. It is usually argued that the additional diagrammatic layer, often referred to as ‘secondary notation’, provides the essential means to improve the pragmatics of a representation system. As Oberland... |

16 |
Restructuring mathematical logic: An approach based on peirce’s pragmatism
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(Show Context)
Citation Context ...and the representation and processing of knowledge. Lattice theory is reworked in order to integrate and to rationalize origins, connections to and interpretations in the real world. As Wille says in =-=[78]-=-: The aim is to reach a structured theory which unfolds the formal thoughts according to meaningful interpretations allowing a broad communication and critical discussion of the content. Wille, 1996 T... |

15 |
Precise visual modelling
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Citation Context ...agrams were designed with this application area in mind and extend the spider diagram language. Constraint diagrams have been used in a variety of areas including formal object oriented specification =-=[37, 44]-=- and a visual semantic web editing environment [51, 81]. Prototype tools to support their use are available from [64]. 6.2 Conceptual Graphs John Sowa’s conceptual graphs (CGs) are based on Peirce’s e... |

12 | Knowledge Representation and Reasonings Based on Graph Homomorphisms
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(Show Context)
Citation Context ...OL or via direct model-theoretic approach) and reasoning facilities are provided. Examples are the works of Prediger and Dau, where the reasoning facilities come in form of graph transformation rules =-=[61, 62, 9, 52,15]-=-, works of Chein/Mugnier et al, where the entailment between CGs are described by meaning-preserving graph homomorphisms called projections [8, 52, 11], or works where so-called standard-models are co... |

12 | Grice for graphics: pragmatic implicature in network diagrams
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(Show Context)
Citation Context ...er architecture, and diagrammatic logic systems have a two layer architecture. The general advantages of this two layer structure for the pragmatics of diagrams is already discussed to a large extent =-=[49, 66, 6, 68, 57]-=-. It is usually argued that the additional diagrammatic layer, often referred to as ‘secondary notation’, provides the essential means to improve the pragmatics of a representation system. As Oberland... |

11 | Conceptual graphs are also graphs
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- 1995
(Show Context)
Citation Context ...OL or via direct model-theoretic approach) and reasoning facilities are provided. Examples are the works of Prediger and Dau, where the reasoning facilities come in form of graph transformation rules =-=[61, 62, 9, 52,15]-=-, works of Chein/Mugnier et al, where the entailment between CGs are described by meaning-preserving graph homomorphisms called projections [8, 52, 11], or works where so-called standard-models are co... |

11 |
Failure mode modular de-composition using spider diagrams
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(Show Context)
Citation Context ...grams, apply rules to write a proof and ask Edith to seek a proof from one diagram to another. With regard to applications, SDs have been used to detect component failures in safety critical hardware =-=[12]-=- and (slight variations of them) to represent non-hierarchical file systems [21] and for viewing clusters which contain concepts from multiple ontologies [32]. As mentioned in section 2, one area that... |

11 | Simple concept graphs: A logic approach
- Prediger
- 1998
(Show Context)
Citation Context |

10 |
Types and tokens for logic with diagrams: A mathematical approach
- Dau
(Show Context)
Citation Context ...s which are represented in the diagrams, and the diagrams should be understood as (mere) representations of these structures. Two thorough discussions on the type-token-issue for DRSs can be found in =-=[34, 17]-=-. Both papers argue that it is essential to provide mathematical definitions for the type-level (called abstract syntax in [34]). For the token-level (called concrete syntax in [34]), the papers come ... |

10 | Reasoning with projected contours
- JOHN
(Show Context)
Citation Context |

10 |
Towards a Visual Editing Environment for the Languages of the Semantic Web
- Lovdahl
- 2002
(Show Context)
Citation Context ... and extend the spider diagram language. Constraint diagrams have been used in a variety of areas including formal object oriented specification [37, 44] and a visual semantic web editing environment =-=[51, 81]-=-. Prototype tools to support their use are available from [64]. 6.2 Conceptual Graphs John Sowa’s conceptual graphs (CGs) are based on Peirce’s existential graphs. Sowa writes that they are an extensi... |

10 | Reasoning with Constraint Diagrams
- Stapleton
- 2004
(Show Context)
Citation Context ...ll constraint diagrams is developed but a sound and complete calculus is elusive. The first ever constraint reasoning system (i.e., including a sound and complete calculus) was developed by Stapleton =-=[73]-=- but compared to Kent’s approach it has several limitations. An example for a constraint diagram was already given in Fig. 2. In the formal approach to constraint diagrams, we have both existential an... |

9 |
Homomorphismes d’hypergraphes pour la subsomption en RDF/RDFS. In 10e conférence sur langages et modèles à objets
- BAGET
(Show Context)
Citation Context ...ns.” For RDF, mathematical elaborations based on graph theory have been developed. They include Tarski-style semantics as well as sound and complete calculi, latter either based on “projections” (see =-=[4, 5]-=-) or on diagrammatic rules (see [19]). That is, RDF is developed as a fully-fledged diagrammatic logic. A first attempt at a diagrammatic representation for DL is can be found in [28], where Gaines el... |

9 |
di Luzio. Patching up a logic of Venn diagrams
- Scotto
- 2000
(Show Context)
Citation Context ...hat Shin defined the syntax of Venn-II at the concrete (drawn) diagram level. This resulted in a lack of mathematical preciseness in her work, leading to unclear definitions and errors in proofs (see =-=[23]-=-). Of course, this lack of rigor should not detract from the importance of Shin’s work because she laid the essential foundations for the acceptance of diagrams as formal tools. 4.2 Seminal Work on Ex... |

8 | On querying simple conceptual graphs with negation, Data Knowledge Engineering 60
- Mugnier, Leclère
- 2007
(Show Context)
Citation Context ...le subgraphs, can be negated. A first step has been taken bay Kerdiles in [43]. Kerdiles ex-tends the projections of simple CGs to simple CGs with atomic negation. This approach has been extended in =-=[50, 54]-=-, where for this framework, different logical approaches –classical logic, intuitionistic logic, logic with a closed-world semantics– are elaborated and compared. Finally, Klinger provides in [45–47] ... |

8 |
Collected Papers, volume 4
- Peirce
- 1933
(Show Context)
Citation Context ...expressive DLs. 3 Historical Systems In this section, we shortly discuss the historical background of DRSs, namely Euler circles (a.k.a. Euler diagrams) [24], Venn diagrams [77], Venn-Peirce diagrams =-=[59]-=- and Peirce’s existential graphs [33, 60, 31].An Euler diagram is a finite collection of closed curves drawn in the plane. For example, in Fig 3, d1 is an Euler diagram which expresses that nothing i... |

7 |
Two FOL Semantics for Simple and Nested Conceptual Graphs
- Simonet
- 1998
(Show Context)
Citation Context ...where the semantics for the graphs is based on triadic Formal Concept Analysis. Prediger provides a sound and complete set of rules for these graphs. A different approach has been taken by Simonet in =-=[69, 10]-=- by translating the graphs to FOL-formulas, where the contexts are modeled by assigning to each concept box an additional argument which models the nesting of the box. In this approach, the notion of ... |

6 |
Concept types and coreference in simple conceptual graphs
- Chein, Mugnier
(Show Context)
Citation Context ...ome in form of graph transformation rules [61, 62, 9, 52,15], works of Chein/Mugnier et al, where the entailment between CGs are described by meaning-preserving graph homomorphisms called projections =-=[8, 52, 11]-=-, or works where so-called standard-models are considered [61, 62, 15]. One can even express “if-then”-statements with simple CGs, at is has been discussed in [2, 52, 3]. Simple CGs with contexts go b... |

6 |
Negations in Simple Concept Graphs
- Dau
- 2000
(Show Context)
Citation Context ...can consider CGs with full negation, where whole subgraphs can be negated. Kerdiles considers in [43] full negation as well. A comprehensive approach to add full negation to CGs is provided by Dau in =-=[13, 14, 16]-=-. In [13], Dau argues that to express full negation to CGs, a new syntactical entity to express negation has to be added, and he suggests to use the negation ovals of Peirce’s existential graphs. Thes... |

6 | Fixing shin’s reading algorithm for peirce’s existential graphs
- Dau
- 2006
(Show Context)
Citation Context ...r on, theorems and proofs) cannot be considered to be mathematical. This leads to a mistake in her reading algorithm, and –even worse– some of her newly implemented transformation rules are not sound =-=[18]-=-. Finally, Shin does not provide an extensional semantics for Peirce’s graphs: her reading algorithms are translations to symbolic logic, thus translations from one formal system to another. 5 Methodo... |

6 |
Visual languages for konwledge representation
- Kremer
- 1998
(Show Context)
Citation Context ... semantics, and diagrammatic reasoning facilities. 2.2 Knowledge Representation and the Semantic Web Is has long been argued that diagrams are particularly useful for knowledge representation systems =-=[56, 28, 48]-=-. In this section, we focus on knowledge representation within the Semantic Web, particularly on RDF(S) and OWL. The underlying layer for knowledge representation within the Semantic Web is the Resour... |

5 |
Constraint Diagrams: Visualizing Assertions in Object-Oriented Models
- Kent
- 1997
(Show Context)
Citation Context ...ed objects. Constraint diagrams are essentially an extension of SDs with universal spiders (quantifiers) and arrows which represent binary relations. A full constraint notation was introduced by Kent =-=[42]-=- in an informal manner. Since then, several papers attempt to elaborate a full mathematical treatment of Kent’s vision, including syntax, semantics, and a sound and complete calculus for constraint di... |