A diagrammatic reasoning system for the description logic
Introduction
Description logics (DLs) are a common family of knowledge representation languages tailored to express knowledge about concepts and concept hierarchies. They include sound and complete decision procedures for reasoning about such knowledge. One of the main applications of DLs is their use as the basis for an ontology language, especially popular for the Semantic Web. In particular, the Ontology Web Language (OWL)—a W3C recommendation for the knowledge language of the Semantic Web—is based on a specific and expressive DL termed .1
The basic building blocks of DLs are concepts (unary predicates), roles (binary relations) and sometimes individuals, which can be composed by language constructs such as intersection, union, value or number restrictions to build more complex well-formed formulas that themselves represent complex concepts and roles. For example, when man, female, male, rich, happy are predefined concepts and if haschild is a predefined role, thendescribes the concepts of men who have both male and female children, and where all the children are rich or happy. Let us call a concept defined in this way as happyman.
The formal notation of DLs has the flavour of a variable-free first order predicate logic (FOL). In fact, DLs correspond to (decidable) fragments of FOL, and like FOL, DLs have a well-defined, formal syntax, a semantics in the form of Tarski-style models, and a sound and complete calculi (based on Table aux-algorithms). It is often emphasised that DLs offer, in contrast to other knowledge representation languages, sound, complete and (empirically) tractable reasoning services. A comprehensive overview on DLs is given in the Description Logic Handbook [2].
The notation of DLs is in the style of the usual linear and symbolic2 notations of FOL. The fact that the notation of DLs is variable-free makes them easier to comprehend than the common FOL formulas which include free variables. Nonetheless, 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. It is well accepted that diagrams are in many cases easier to comprehend than symbolic notations (see for example [3], [4], [5], [6]), and in particular it has been argued that they are useful for knowledge representation systems [7], [8]. This has been acknowledged by the DL community and is a common view among the broader knowledge representation community [9]. In [10], the introduction to the Description Logic Handbook, Nardi and Brachman write that besides the possibility of “providing a syntax that resembles more closely natural language”, a “major alternative for increasing the usability of DL as a modeling language” is to “implement interfaces where the user can specify the representation structures through graphical operations.”
A first attempt at a diagrammatic representation for DL is can be found in [7], where Gaines elaborates a graph-based representation for the textual DL CLASSIC, part of the kl-one-framework. More recently, the focus has shifted from the development of proprietary diagrammatic representations 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, Brockmans et al. [11] provide a UML-based, diagrammatic representation for the OWL DL. In these approaches, the focus is on a graphical representation of DL, however, as emphasized in many works on DL (see for example [2]), reasoning is seen as a distinguishing feature of DL and such reasoning is not supported diagrammatically by that treatment. Correspondences between graphical representation of the DL and the DL reasoning system are therefore important but remain largely unelaborated to date. Similar arguments hold for other popular diagrammatic languages like UML and ORM3 the difference being that, that unlike DLs, these diagrammatic modeling languages provide no extensional, mathematical semantics, nor any automated reasoning facilities.
On the other hand, there are some candidate diagrammatic reasoning systems that have the expressiveness of fragments of FOL, or even full FOL. In this paper, we will evaluate two families of contemporary mathematical diagrammatic reasoning systems, which have the following two (historical) origins:
- (i)
The system of Euler circles and Venn-diagrams, the latter enriched by C.S. Peirce to Venn-Peirce-diagrams (see [12], [13]).
- (ii)
The system of Peirce's Existential Graphs (EG).4
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A well-defined syntax: usually, the syntax is defined at an abstract level (for example in terms of graph theory), such that the well-formed formulas have diagrammatic representations. Moreover, the syntax is—like the syntax for DL—variable-free.
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Extensional—Tarski-style semantics—and/or translations to formulas of FOL.
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Sound and complete calculi: usually the rules are defined on the abstract syntax but mostly they can be understood as manipulations of the represented diagrams.
As a specific DL is the corresponding logic behind the OWL, the results of this paper benefit not only DL, but the Semantic Web more generally. Developing a Semantic Web language as a mathematical diagrammatic reasoning system has already been carried out for a much simpler language: namely RDF. For RDF, mathematical elaborations based on graph theory, including a Tarski-style semantics and a sound and complete calculus based on “projections” (see [19], [20]) or via diagrammatic rules (see [21]) have been elaborated.
In the next three 2 Introduction into DL, 3 Spider and constraint diagrams, 4 Existential and conceptual graphs, the basic notions of DL, SD and CD, and EGs and CGs, are introduced. In the subsequent 5 SD or CD for DL, 6 Conceptual or EGs for DL, we investigate whether SD and CD and EGs and CGs, respectively, are suited as a starting point for developing DL as a diagrammatic reasoning system. It will turn out that there are prospects to SD and CD for this purpose. However, CGs and, as we will show, EG are a much better match. In the three sections that follow, is developed as a diagrammatic reasoning system based on Peirce's EGs. In Section 7, a formalization of by means of labeled trees is given. Section 8 provides the Peirce style representation of such trees. In Section 9, a calculus for -trees, based on Peirce's calculus for EGs, is given, and soundness and completeness proven. Finally, in Section 7, further research is discussed. As we use several abbreviations in this paper, these are presented in Appendix.
Section snippets
Introduction into DL
The vocabulary of any DL consists of concepts, which denote sets of individuals, and roles, which denote binary relationships between individuals. The starting points are atomic concepts and roles—concept and role names—from which more complex concepts and roles are built with constructs such as intersection, union, value or number restrictions, and so on. Moreover, we usually consider concept vocabularies where we have a universal concept . The tables of Fig. 1, Fig. 2 provide an overview of
Spider and constraint diagrams
SDs and CDs are a diagrammatic reasoning system based on Euler circles and Venn–Peirce diagrams. They are inspired by a fragment of UML which is used for software specifications as an alternative to UML's textual object constraint language (OCL). They are elaborated as abstract mathematical structures, including an extensional semantics and inference rules. In this section, SDs and CDs are briefly and informally introduced. Particularly, only the diagrams of SDs and CDs are provided, not the
Existential and conceptual graphs
In this section, we introduce Peirce's EGs, Sowa's CGs, particularly its most common fragment of simple conceptual graphs (SCGs) and the system of concept graphs with cuts (CGwCs) introduced by one of the authors. Similar to SDs and CDs, there exist mathematically precise elaborations for each of these systems. Again, like SDs and CDs, the diagrammatic benefits of EGs or CGs (used in teaching) has been investigated by Schäfe [16], [84] and also by Pollant [15]. Furthermore, a theorem prover for
SD or CD for DL
At a first glance, there are some striking similarities between DL and the system of SDs and CDs. Most importantly, both correspond to decidable fragments of FOL where only unary predicates and binary relations are used. Both systems have sound and complete calculi which are implemented on tableaux-based algorithms. It is therefore worth investigating whether SD and CD can be used as a diagrammatic representation for DL.
In our scrutiny of SDs and CDs, we first restrict ourselves to a discussion
Conceptual or EGs for DL
Both CGs and DLs have some common background: they are both developed and intended as knowledge representation systems to include reasoning facilities, they were developed at nearly the same time and they have an important common ancestor, namely the semantic networks of AI (and Minski's frame systems). In fact, in 1979, the well-known system kl-one was developed on the basis of informal use of semantic networks in AI (see [60], [61]). kl-one was itself a diagrammatic formalism and in 1987 a
A tree-based formalization of the DL
In this section, the syntax and semantics for the DL is provided. First of all, as mentioned earlier in Section 2, the DL has conjunction, negation and value restriction as constructors. As the RGs of Peirce have conjunction, negation and existential quantification as constituents, we can consider made up from conjunction, negation and existential restriction (instead of value restriction).
Let denote the class of RGs that will be used to formalize . The graphs of will
A diagrammatic representation for
In Section 4.1, we briefly discussed that C.S. Peirce distinguished between graphs and graph replicas. As it is more generally argued in [42], [43], any formalization of logic by means of diagrams has to distinguish these two levels, usually called types or abstract syntax (which correspond to Peirce's graphs), and tokens or concrete syntax (corresponding to (Peirce's replicas). The -trees, as they have been defined in the last section, are obviously the abstract syntax. In this section we
The calculus for
C.S. Peirce provided a set of five rules for the system of EGs termed erasure, insertion, iteration, deiteration, double cut. It is possible to elaborate Peirce's graphical calculus, including a proof of its soundness and completeness, in a mathematical precise manner, and to extend the calculus for the system of RGs, but this goes beyond the scope of this paper.
is a fragment of the full system of RGs. As one would expect, the rules for RGs are also sound rules for . However it is not
Conclusion and further research
This paper has discussed different diagrammatic reasoning systems—spider and constraint diagrams on the one hand, existential and conceptual graphs on the other—as candidates for a diagrammatic version of DL. It has been argued that spider and constraint diagrams are not as well suited to a diagrammatic version of DLs as conceptual graphs and existential graphs. More specifically, a version of conceptual graphs including negation and free variables, and relation graphs—existential graphs with
Acknowledgements
We like to thank the reviewers for their helpful remarks and also Andrew Fish for his valuable comments on using CDs for .
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