Aus der Einleitung:
'Conceptual graphs (CGs) are an expressive formalism for representing knowledge about an application domain in a graphical way. Since CGs can express all of first-order predicate logic (FO), they can also be seen as a graphical notation for FO formulae. In knowledge representation, one is usually not only interested in representing knowledge, one also wants to reason about the represented knowledge. For CGs, one is, for example, interested in validity of a given graph, and in the question whether one graph subsumes another one. Because of the expressiveness of the CG formalism, these reasoning problems are undecidable for general CGs. In the literature [Sow84, Wer95, KS97] one can find complete calculi for validity of CGs, but implementations of these calculi have the same problems as
theorem provers for FO: they may not terminate for formulae that are not valid, and they are very ineficient. To overcome this problem, one can either employ
incomplete reasoners, or try to find decidable (or even tractable) fragments of the formalism. This paper investigates the second alternative.
The most prominent decidable fragment of CGs is the class of simple conceptual graphs (SGs), which corresponds to the conjunctive, positive, and existential fragment of FO (i.e., existentially quantified conjunctions of atoms). Even for this simple fragment, however, subsumption is still an NP-complete problem [CM92]. SGs that are trees provide for a tractable fragment of SGs, i.e., a class of simple conceptual graphs for which subsumption can be decided in polynomial time [MC93]. In this report, we will identify a tractable fragment of SGs that is larger than the class of trees.
Instead of trying to prove new decidability or tractability results for CGs from scratch, our idea was to transfer decidability results from description logics [DLNN97, DLNS96] to CGs. The goal was to obtain a \natural' sub-class of the class of all CGs in the sense that, on the one hand, this sub-class is defined directly by syntactic restrictions on the graphs, and not by conditions on the first-order formulae obtained by translating CGs into FO, and, on the other hand, is in some sense equivalent to a more or less expressive description logic.
Although description logics (DLs) and CGs are employed in very similar applications (e.g., for representing the semantics of natural language sentences), it turned out that these two formalisms are quite different for several reasons:
(1) conceptual graphs are interpreted as closed FO formulae, whereas DL concept descriptions are interpreted by formulae with one free variable; (2) DLs do not allow for relations of arity > 2 ; (3) SGs are interpreted by existential sentences, whereas almost all DLs considered in the literature allow for universal quantification; (4) because DLs use a variable-free syntax, certain identifications of variables expressed by cycles in SGs and by co-reference links in CGs cannot be expressed in DLs. As a consequence of these differences, we could not identify a natural fragment of CGs corresponding to an expressive DL whose decidability was already shown in the literature. We could, however, obtain a new tractability result for a DL corresponding to SGs that are trees. This correspondence result strictly extends the one in [CF98]. In addition, we have extended the tractability result from SGs that are trees to SGs that can be transformed into trees using a certain \cycle-cutting' operation. The report is structured as follows. We first introduce the description logic for which we will identify a subclass of equivalent SGs. In Section 3, we recall basic definitions and results on SGs. Thereafter, we introduce a syntactical variant of SGs which allows for directly encoding the support into the graphs (Section 4.1). In order to formalize the equivalence between DLs and SGs, we have to consider SGs with one distinguished node called root (Section 4.2). In Section 5, we finally identify a class of SGs corresponding to a DL that is a strict extension of the DL considered in [CF98].
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:78832 |
| Date | 20 May 2022 |
| Creators | Baader, Franz, Molitor, Ralf, Tobies, Stefan |
| Publisher | Aachen University of Technology |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:report, info:eu-repo/semantics/report, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | urn:nbn:de:bsz:14-qucosa2-785040, qucosa:78504 |
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